Perfect Matching in Random Graphs is as Hard as Tseitin

We study the complexity of proving that a sparse random regular graph on an odd number of vertices does not have a perfect matching, and related problems involving each vertex being matched some pre-specified number of times. We show that this requires proofs of degree $\Omega(n / \log n)$ in the Polynomial Calculus (over fields of characteristic $\ne 2$) and Sum-of-Squares proof systems, and exponential size in the bounded-depth Frege proof system. This resolves a question by Razborov asking whether the Lov\'asz-Schrijver proof system requires $n^\delta$ rounds to refute these formulas for some $\delta>0$. The results are obtained by a worst-case to average-case reduction of these formulas relying on a topological embedding theorem which may be of independent interest.


Introduction
Proof complexity is the study of certi cates of unsatis ability, initiated by Cook and Reckhow [20] as a program to separate NP from coNP.The main goal of this program is to prove size lower bounds on proofs of unsatis ability of logical formulas.This is a daunting job -indeed we are far from proving general size lower bounds on certi cates of unsatis ability.As an intermediate step we study proof systems with restricted deductive power and prove size lower bounds for such restricted certi cates of unsatis ability.The most studied such proof system is resolution [10] which is fairly well understood by now, see e.g., the proof complexity book by Krajíček [42].
But resolution is by far not the only proof system.A closely related and quite general proof system is the bounded depth Frege proof system [20] which manipulates propositional formulas of bounded depth.While we have some results for the bounded depth Frege proof system, in this introduction we instead focus on two other systems as these were the primary motivation behind our work.These are the two proof systems Polynomial Calculus (PC) [19,2] and Sum-of-Squares (SoS) [65,53,45].These proof systems do not rely on propositional logic, like resolution or Frege, but rather on algebraic reasoning and are examples of so-called (semi-)algebraic proof systems (see e.g.[33]).
Both PC and SoS provide refutations of (satis ability of) a set of polynomial equations Q = {   () = 0 |  ∈ [] } over  variables  1 , . . .,   .In the case of PC, these polynomials can be over any eld F ( nite or in nite), and in the case of SoS, these polynomials are over R. A key complexity measure of a PC F or SoS refutation of Q is its degree, de ned as the maximum degree of any polynomial appearing in the refutation.The degree of refuting Q in PC F or SoS, which we denote by Deg(Q PC F ⊥) and Deg(Q SoS ⊥) respectively, is the minimum degree of any PC F or SoS refutation of Q.For Boolean systems of equations, meaning that Q contains the equations  2   −   = 0 for all  ∈ [], strong enough degree lower bounds imply size lower bounds in both PC F [19,36] and SoS [5], where the size of a refutation is the total number of monomials appearing in it.For nite F the proof system PC F is incomparable to SoS [58,32,33] whereas SoS can simulate PC R by the recent result of Berkholz [9].
There is by now a large number of lower bound results for both PC [58, 36, and SoS [32, 63, , with SoS in particular having received considerable attention in recent years due to its close connection to the Sum-of-Squares hierarchy of semide nite programming, a powerful "meta-algorithm" for combinatorial optimization problems [7].
In this paper we study the power (or lack thereof) of these proof systems when it comes to refuting the perfect matching formula PM() de ned over sparse random graphs  = ( , ) on an odd number of vertices.This formula can be viewed as a system of linear equations over R on a set of Boolean variables: for each edge  ∈  there is a variable   ∈ {0, 1} (indicating whether the edge is used in the matching) and for each vertex  ∈  there is an equation     = 1.
Apart from being a natural well-studied problem on its own, the perfect matching formula is interesting because of its close relation to two other widely studied families of formulas, namely the pigeonhole principle (PHP), and Tseitin formulas.
PHP asserts that  pigeons cannot t in  <  holes (where each hole can t at most one pigeon).This can be viewed as a bipartite matching problem on the complete bipartite graph with  +  vertices, where each vertex on the large side (with  vertices) must be matched at least once, and each vertex on the small side (with  vertices) can be matched at most once.
There are many variants of PHP (see e.g. the survey [59]), and the one closest to the perfect matching formula is the so-called "onto functional PHP", in which each vertex on both sides must be matched exactly once (rather than at least/at most once).Equivalently, this formula is simply the perfect matching formula on a complete bipartite graph with  +  vertices.While most variants of PHP are hard for PC [58,50], the onto functional PHP variant is in fact easy to refute in PC over any eld [60].In SoS, all variants of PHP are easy to refute [33].
The Tseitin formula over a graph  claims that there is a subgraph of  such that each vertex has odd degree.As the sum of the degrees of a graph is even, this formula is not satis able if  has an odd number of vertices.In contrast to the PHP, the Tseitin formula is (almost) always hard: for PC F over elds F of characteristic distinct from 2 [15,3] and SoS [32] these formulas require linear degree if  is a good vertex expander.We cannot hope to prove degree lower bounds over elds of characteristic 2 as the constraints become linear and we can thus refute the Tseitin formula using Gaussian elimination.As the perfect matching formula PM() implies the Tseitin formula, PC over elds of characteristic 2 can also easily refute PM() for  with an odd number of vertices.
In summary, the perfect matching formula lies somewhere in between PHP and Tseitin, of which the former is easy to refute in SoS (and easy to refute in PC in the onto functional variant), and the latter is hard to refute in SoS (as well as in PC with characteristic ≠ 2).Hence it is natural to wonder whether SoS or PC requires large degree to refute the perfect matching formula over non-bipartite graphs.
The case of perfect matching in the complete graph on an odd number of vertices (sometimes called the "MOD 2 principle") is well-understood in both PC [15] and SoS [32,56], requiring degree Ω() in both proof systems unless the underlying eld of PC is of characteristic 2. For sparse graphs, less is known.Buss et al. [15] obtained worst-case lower bounds in PC showing that there exist bounded degree graphs on  vertices requiring Ω() degree refutations.This is obtained by a reduction from Tseitin formulas and while the work of Buss et al. predates the current interest in the SoS system, it is not hard to see that the same reduction yields a similar Ω() degree lower bound for SoS (details provided in Appendix A).
However, for random graphs  little is known about the hardness of the perfect matching formula and, e.g., Razborov [57] asked whether it is true that the Lov ász-Schrijver hierarchy [47] (which is weaker than SoS) requires   rounds to refute the perfect matching principle on a random sparse regular graph with high probability.

Our results
We show that indeed the perfect matching principle requires large size on random -regular graphs (for some constant ) in the Sum-of-Squares, Polynomial Calculus, and bounded-depth Frege proof systems.Our results apply more generally to Tseitin-like formulas de ned by linear equations over the reals induced by some graph, so let us now de ne these.
For a graph  = ( , ) and integer vector  ∈ Z  , consider the system of linear equations over the reals having a variable   for each  ∈ , and the equation     =   for each  ∈  .Let Card(, ) denote this system of linear equations along with the Boolean constraints   ∈ {0, 1} (viewed as a quadratic equation  2  −   = 0) for each edge -in Section 2.2 the encoding is discussed in more detail.Note that Card(, ì 1) corresponds to the perfect matching problem in  and in general Card(, ) can be viewed as asserting that  has a "matching" where each vertex is matched exactly   times.Note that whenever ∈   is odd, Card(, ) is unsatis able (since the equations imply    = 2    which is even) 1 .
We focus on the special case of Card(, ) where  is -regular and  = ì  = (, , . . ., ) is the all- vector for some  ∈ [].If in this scenario both  and  are odd (implying  is even) then as observed above Card(, ì ) is unsatis able.On the other hand if  is odd and  is even then Card(, ì ) is always satis able (because such  admits a 2-factorization).The remaining case when  is even may be either satis able or unsatis able, but for a random -regular  with  ≥ 3, Card(, ) will be satis able with high probability (because such  can be partitioned into perfect matchings with high probability).
If we let F  denote a Frege system restricted to depth- formulas (see Section 2.1), then our main theorem is as follows.

T H E O R E M 1 .1.
There is a constant  0 such that for all constants  ≥  0 and  ∈ [], the following holds asymptotically almost surely over a random -regular graph  on  vertices.
The interesting case of the above theorem is when both  and  are odd so that Card(, ) is unsatis able; in the other cases Card(, ì ) is satis able with high probability and the lower bounds are vacuous.
By known size-degree tradeo s for Polynomial Calculus [36,19] and Sum-of-Squares [5] the degree lower bounds in Theorem 1.1 imply near-optimal size lower bounds of exp Ω(/log 2 ) .
Apart from the perfect matching formula, another special case of Card(, ì ) is the so-called even coloring formula, introduced by Markström partially resolves this open problem, establishing that it is hard on random graphs (rather than on all spectral expanders).See Section 6 for some further remarks on what parts of our proof use the randomness assumption.
We will give a more detailed overview of how the results are obtained in Section 1.3 below, but for now let us mention that we obtain them using embedding techniques, as introduced to proof complexity by Pitassi et al. [54] (see discussion of related work in Section 1.2).In particular for, say, the SoS lower bound, our starting point is the Ω() worst-case degree lower As pointed out to us by Aleksa Stanković, decidability of Card(, ) is in polynomial time: starting with the all 0 assignment, iteratively build up an assignment that may match some vertices fewer times than required.If there is a satisfying assignment, then there is always an augmenting path along which the current assignment can be improved, i.e., more edges set to 1, by a similar argument as for matchings [8].Such a path can be found in polynomial time by an adaptation of the blossom algorithm [23].
bound in sparse graphs, and we then prove that these hard instances can be embedded in a random -regular graph in such a way that the hardness of refuting the formula is preserved.
To achieve this, one of the components we need is a new graph embedding theorem which may be of independent interest.Very loosely speaking, we show that any bounded-degree graph with (/log ) edges can be embedded as a topological minor in any bounded-degree -expander on  vertices and su ciently many edges.In addition, for our application to perfect matching (and more generally the Card(, ì ) formulas), we need to be able to control the parities of the path lengths used in the topological embedding, and we show that as long as every large linear-sized subgraph contains an odd cycle of length Ω(1/), this is indeed possible.
Somewhat informally, we prove the following. 2 •  (), then  contains  as a topological minor.Furthermore, if all large vertex induced subgraphs of  contain an odd cycle of length Ω(1/), then one can choose the parities of the length of all the edge embeddings in the minor.
This generalizes various classical results of a similar avor (e.g.[39,44,18,43]).See the next subsection for a discussion comparing these (and other) existing embedding results to ours.
As a further illustration of the applicability of this theorem we partially resolve a question of Filmus et al. [24].They prove that with high probability for random -regular graphs , where  ≥ 4, PC requires space Ω( √ ) to refute the Tseitin formula, and conjecture that PC in fact requires space Ω().On the other hand, Galesi et al. [28] considered it plausible that the Ω( √ ) bound is optimal.We (almost) resolve this question by proving Ω(/log ) space lower bounds for the Tseitin formula de ned on vertex expanders, but only of large enough (constant) average degree.
T H E O R E M 1 .3. For all  > 0 there is a  0 such that the following holds.Let  be a bounded degree -expander on  vertices of average degree at least  0 .Then over any eld F it holds that PC F requires space Ω(/log ) to refute the Tseitin formula de ned on .
Let us mention that the constant hidden in the lower bound Ω(/log ) depends on the maximum degree of .Unlike Theorem 1.1, vertex expansion is su cient and we require no randomness.This lower bound is obtained by embedding a worst-case instance, due to Filmus et al., into a vertex expander.We provide more details in Section 6.1.3 ), where NAE 3 is the not-all-equal predicate on three bits.In contrast to their work we show (almost) linear degree lower bounds for the stronger Sum-of-Squares hierarchy, but only for a very wide predicate of some large (but constant) arity.

Embedding Theorems
There is a rich literature on embeddings of graphs as minors or topological minors into expander graphs.We focus here on the ones most closely related to Theorem 1.2.

3
A distribution  over {0, 1}  is said to be pairwise uniform if for all 1 ≤  <  ≤ , the marginal distribution of  restricted to coordinates  and  is uniform.
The classical result of Kleinberg and Rubinfeld [39] shows that a regular expander  on  vertices contains every graph  with (/polylog()) vertices and edges as a minor.Krivelevich and Nenadov [44] simpli ed and strengthened this by improving the bound on the size of  to (/log ).These results di er from ours in two key ways: (i) we want topological minors, and (ii) we want to be able to control the parities of the path lengths in the embedding.We now discuss these two aspects separately.
Results on topological minors, while somewhat less common, also exist.A result similar to ours is the result of Broder et al. length and the embedding even works in an adversarial setting.Namely, the adversary is allowed to x the embedding of the vertices, as long as no neighborhood in  contains too many vertex embeddings.
The embedding theorem of Dragani ć et al. can be used to implement our proof strategy.
The results are una ected by this change except in the setting of Theorem 1.3.There, instead of considering vertex expanders, we need to consider regular spectral expanders with the bene t that the required average degree  0 is considerably decreased.
Extended Formulations There has been a fair amount of work studying the extension complexity of the perfect matching polytope [67, 62], but these lower bounds do not have any direct implications for the PC and SoS degree of the perfect matching formula.Let us elaborate.
Suppose we have a convex polytope P consisting of many facets.A natural question is whether there is simpler polytope Q in a higher dimensional space so that P is the "shadow" of Q, or a bit more formally that there is a linear projection  such that (Q) = P.Such a Q is then called a linear extension of P and the extension complexity of a polytope P is the minimum number of facets of any linear extension of P.
Rothvoss [62] proved that the perfect matching polytope of a complete -node graph has extension compexity exp(Ω()) for  even.This result is incomparable to our lower bounds: as the graphs we consider do not contain a perfect matching, their perfect matching polytope is empty and thus has extension complexity 0. Rather than linear programs, i.e., polytopes, we consider semide nite programs which are more expressive.The extension complexity in the semide nite setting has also been studied before [46,12] but these results are incomparable for the same reason just mentioned.While these results are incomparable, it is worth mentioning that there is a connection between Sherali-Adams (a proof system weaker than SoS) and extended formulations [17,40].

Overview of Proof Techniques
As previously mentioned, our high level approach is to rst obtain worst-case perfect matching lower bounds and to then embed these into the Card(, ì ) formula for  a random regular graph.The worst-case lower bounds are obtained by a gadget reduction from Tseitin to perfect matching, due to Buss et al. [15].Using known lower bounds for the Tseitin formula in the corresponding proof systems [15,32,34] we then obtain the desired worst-case lower bounds for the perfect matching formula.
A naïve attempt to obtain average-case lower bounds from a sparse worst-case instance  on  vertices is to topologically embed the worst-case instance into a random regular graph  on ( log ) vertices using Theorem 1.2.One would then like to argue that PM() is hard.
Suppose each path   in the embedding of  in  corresponding to some edge {, } ∈ () is of odd length.Then it is straightforward to verify that the perfect matching formula de ned over the embedding is at least as hard to refute as the worst-case instance PM(): map each variable   , for  ∈   , alternatingly to   or x such that the rst and last edges of   are mapped to   (using that   is of odd length).This simple projection maps the perfect matching formula de ned over the embedding of  to PM() and thus shows that the hardness of PM() should be inherited.
But having such a worst-case instance as a topological minor is not su cient to conclude that PM() is hard.For instance  may contain an isolated vertex and it is then trivial to refute PM().On the other hand if we could guarantee that there is a perfect matching  in the subgraph of  induced by the vertices not used in the embedding of , we can conclude that PM() is hard: hit the formula with the restriction corresponding to the matching  and by the argument from the previous paragraph we are basically left with the worst-case formula.
Thus if we can ensure that  is a topological minor of  with the two additional properties that (i) every path used in the embedding of  has odd length, and (ii) there exists a perfect matching in the subgraph of  induced by the vertices not used in the embedding of , then we obtain average-case lower bounds for the perfect matching formula PM() ≡ Card(, ì 1).
The lower bounds for Card(, ì ) for  > 1 can then be obtained by a reduction to the  = 1 case: after xing the value of the edges in /2 cycle covers of  to 1, a restriction of Card(, ì ) is obtained which behaves like Card( , ì 1) for a somewhat sparser random regular graph  .
Let us elaborate a bit further on the properties required from the topological minor of  in .As mentioned previously, our embedding theorem can ensure that all paths are of odd length.To ensure the second property, we in fact do not embed  directly into  but rather into a suitably chosen vertex induced subgraph [ ] with the crucial property that for any set of vertices  ⊆  of odd cardinality the induced subgraph [ \ ] has a perfect matching.As the embedding of  will consist of an odd number of vertices we then obtain property (ii) above.
Since we now want to apply Theorem 1.2 not to  but to [ ], we have to ensure that  [𝑇 ] satis es all the conditions of that theorem.We prove what we refer to as the Partition Lemma, which asserts that an induced subgraph [ ] exists that satis es both the perfect matching property described above, as well as all conditions of Theorem 1.2.The proof of the Partition Lemma relies primarily on the Lov ász Local Lemma and spectral bounds to obtain the desired properties.
For the proof of our embedding theorem (Theorem 1.2), we extend an argument due to Krivelevich and Nenadov [44] (see also [43]) for ordinary minors (rather than topological minors).In order to obtain a minor embedding of  in , the idea there is to embed the vertices one by one from  in  while maintaining an "unused" subgraph  of  which is a slightly worse expander than  is.During this process it may happen that some vertex embedding cannot be connected to a neighbor.If this happens, the embedding of that vertex is removed and it needs to be embedded again.
In order to obtain topological embeddings, we need to adapt this procedure.Since we now want vertex-disjoint paths connecting the embedded vertices, we would ideally like to embed each vertex of  as a large star, and then embed the edges of  as paths connecting di erent leaves of these stars.In order to make this work out, rather than embedding the vertices as actual stars, we embed them as "star-like" subgraphs of  (more precisely de ned in De nition 5.3) that consist of a central vertex connected to many large vertex-disjoint connected subgraphs of  and show (Lemma 5.4) that we can always embed the vertices of  as such "star-like" subraphs of .
With this in place, obtaining control of the parities of the path lengths used in the embedding (under the assumption on odd cycles in Theorem 1.2) is relatively straightforward: almost by de nition, when embedding an edge of  into a path of , we can route it via an odd cycle and can then choose which of the two halves of the odd cycles to use, obtaining two possible embeddings with di erent path length parity, and can choose the one with the appropriate parity.

Organization
We give some preliminaries in Section 2, formally de ning the used proof systems and encodings used, and recalling some general background results.In Section 3 we provide most of the proof of Theorem 1.1 while deferring the proofs of two key results, the aforementioned Partition Lemma and our embedding theorem.The proof of the Partition Lemma is given in Section 4, and the proof of the embedding theorem can be found in Section 5.
In Appendix A we recall the reduction of Buss et al. [15] from Tseitin to perfect matching and show that it yields lower bounds not only for Polynomial Calculus but also for Sum-of-Squares and bounded depth Frege.

Preliminaries
Natural logarithms (base e) are denoted by ln, whereas base 2 logarithms are denoted by log.For integers  ≥ 1 we introduce the shorthand [] = {1, 2, . . ., } and sometimes identify singletons {} with the element .For a set  we denote the power set of  by 2  and a transversal  of a family of sets B = { 1 ,  2 , . . .  } is a set such that there is a bijective function  :  → B satisfying that  ∈  () for all elements  ∈ .

Proof Systems
Let P = { 1 = 0, . . .,   = 0} be a system of polynomial equations over the set of variables  = { 1 , . . .,   , x1 , . . ., x }.Each   is called an axiom, and throughout the paper we always assume P includes all axioms  2  −   and x2  − x , ensuring that the variables are boolean, as well as the axioms 1 −   − x , making sure that the "bar" variables are in fact the negation of the "non-bar" variables.

Sum-of-Squares (SoS
) is a static semi-algebraic proof system.An SoS proof of  ≥ 0 from P is a sequence of polynomials  = ( 1 , . . .,   ;  1 , . . .,   ) such that The degree of a proof  is An SoS refutation of P is an SoS proof of −1 ≥ 0 from P, and the SoS degree to refute P is the minimum degree of any SoS refutation of P: if we let  range over all SoS refutations of P, we can write Deg(P SoS ⊥) = min  Deg().

D E F I N I T I O N 2 .1 (Pseudoexpectation).
A degree  pseudo-expectation for P is a linear operator Ẽ on the space of real polynomials of degree at most , such that () It is easy to check that if there is a degree  pseudo-expectation for P, then there is no SoS refutation of P of degree at most : if Ẽ is applied to both sides of (1), where  = −1, then the right side is equal to −1 while the left is greater than or equal to 0.
The size of an SoS refutation , Size(), is the sum of the number of monomials in each polynomial in  and the size of refuting P is the minimum size over all refutations Size(P SoS ⊥) = min  Size().
Polynomial Calculus is a dynamic proof system operating on polynomial equations over a eld F. Let P be over F. Polynomial Calculus over F (PC F ) consists of the derivation rules , where ,  ∈ F[] and ,  ∈ F, and multiplication  = 0   = 0 , where  ∈ F[] and  ∈ .
A PC refutation of P is a sequence of polynomials  =  1 , . . .,  such that  = 1 and each polynomial   is either in P or can be derived by one of the derivation rules from earlier polynomials.The degree of a refutation is the maximum degree appearing in the sequence ) and the PC F degree of refuting P is the minimum degree required of any refutation Deg(P PC F ⊥) = min  Deg().Similarly, the size of a refutation  is the sum of the number of monomials in each line of  and the PC F size of refuting P is the minimum size required of any refutation Size(P PC F ⊥) = min  Size().
Frege System Let us describe a Frege system due to Shoen eld, as presented in [66].As Frege systems over the basis ∨, ∧ and ¬ can polynomially simulate each other [20], the details of the system are not essential and hold for any Frege system over the mentioned basis.
Schoen eld's Frege system works over the basis ∨ and ¬.We treat the conjunction  ∧  as an abbreviation for the formula ¬(¬ ∨ ¬) and let 0, 1 denote "false" and "true" respectively.
The Frege system F that we consider consists of the following rules: An F-refutation of an unsatis able formula The size of a formula is the number of connectives in the formula and the size of a refutation , denoted by Size(), is the sum of the sizes of all formulas in the refutation.The depth of  is the maximum depth of any formula  ∈ .We denote by F  the proof system F restricted to formulas of depth at most .

Propositional Formulas
As we are only interested in constant degree graphs all our axioms are of constant size.Hence the precise encoding of the axioms is not signi cant as we can change the encoding in constant size/degree.() for all  ≥ 2 it holds that Size(F Suppose we have a refutation  of F in one of the mentioned proof systems.We want to show that if we hit the proof with the restriction  such that F  ≡ F then we obtain a proof  =   of F .First we need to ensure that we can derive all the axioms of F .These may be encoded in a di erent manner, but as these proof systems are implicationally complete, and each axiom only depends on a constant number of variables, this can be done in constant degree (constant size).This shows that the SoS degree of the resulting refutation is at most a constant factor larger.
For Polynomial Calculus and Frege the statement is readily veri ed by an inductive argument over the proof.
For concreteness let us also de ne the encoding of the formulas that we are interested in.

Perfect Matching and Card(𝑮, ì
) The Perfect Matching formula PM() encodes the claim that the graph  contains a perfect matching.For every edge  ∈ () introduce a boolean variable   ∈ {0, 1} and add for every vertex  ∈  () an axiom claiming that precisely one incident edge is set to true.As a polynomial over R, we encode this claim as which is satis ed under an assignment  if  PM  () = 0.Over other elds we encode this as a sum over indicator polynomials (see example for Tseitin below).For the Frege proof system we encode the vertex axiom as the propositional formula The formula Card(, ì ) is encoded in a similar fashion: in the polynomial encoding replace the 1 with   , whereas in the propositional encoding we let the latter ∧ range over edge-tuples of size   + 1.

Tseitin Formula
The Tseitin formula () claims that the edges of the graph  can be labeled by 0, 1 such that the number of 1-labeled edges incident to any vertex is odd.For every edge  ∈ () introduce a boolean variable   ∈ {0, 1}, denote the set of variables corresponding to edges incident to  by   = {   |  ∈ } and let   ⊆ {0, 1}   contain all assignments to the variables   that set an odd number of variables to 1.We encode the claim that an odd number of edges incident to  ∈  () are set to 1 as the polynomial where ȳ is the indicator polynomial that is 1 i the variables in   are set according to .As before, we also add the boolean axioms to ensure that the variables take values in {0, 1}.
For the Frege system we encode the claim that an odd number of edges incident to  ∈  () is set to 1 as the propositional formula where the indicator is now encoded as the formula

Graph Theory
This paper only considers simple, undirected graphs: all graphs have no self-loops nor multiple edges.For a graph  = ( , ) the neighborhood of a vertex is the length of the shortest path from  to  and the distance between two sets ,  ⊂  is the minimum distance between any pair of vertices  ∈  and  ∈ .Let diam() denote the diameter of , that is, the maximum distance between any two vertices in .For a vertex set  ⊆  , and an integer  ∈ N, let    () ⊆  () be the ball around  of radius  in :    () contains all vertices  ∈  that are at distance at most  from .
A graph  on  vertices is an -expander (has vertex expansion ) if for all sets  ⊆  () We denote the uniform distribution over -regular graphs on  vertices by G(, ) and tacitly assume throughout this paper that  is even.A graph  contains  as a topological minor if there is an injective map  :  () →  () and for every {, } ∈ () there is a path   ⊆  from () to () that is pairwise vertexdisjoint from all other paths except in the endpoints.The paths   are the edge embeddings of the minor.
Let us record the well-known fact that vertex expanders have small diameter.

L E M M A 2 . 3 ([43]
).Let  be an -expander on  vertices.Then the diameter of  is upper bounded by 2(log −1) As this constant will show up in a few places, let  ø  = 2 log(1+) + 3 and hence diam() ≤  ø  • log , if  is an -expander.
The following lemma states that even if a small set of vertices is removed from a vertex expander, large sets still have many vertices at small distance./2 log  between  and .

Probabilistic Bounds
We use the following version of the multiplicative Cherno bound.
T H E O R E M 2 .6 (Cherno ).Suppose  1 , . . .,   are independent random variables taking values in {0, 1}.Let  denote their sum and let  = E[].Then, for every 0 ≤  ≤ 1 we have We also need a similar bound for Poisson random variables.

Lower Bounds on Average
In this section we establish our main result Theorem 1.1 giving average-case lower bounds in PC, SoS and bounded depth Frege for the Card(, ì ) formulas.

Lower Bounds for Perfect Matching
Recall that we aim to prove that any sparse graph  (in particular a graph where PM() is hard to refute) can be topologically embedded into a random graph such that all paths in the embedding have odd length.In order to do this, we need to assume that the graph is far from bipartite (since otherwise  would need to be bipartite as well, and PM() is easy for bipartite graphs).Furthermore our embedding theorem relies on all large induced subgraphs of  having su ciently large maximum degree.The two following de nitions capture that both properties hold for all large induced subgraphs of ./2 log .But a priori these may all be shorter than .

D E F I N I T I O N
The de nition asks for short odd cycles of length at least , at the cost of a slightly worse upper bound on the cycle length.
Both properties are clearly monotone in : if the properties hold for some  0 > 0, then they also hold for all  ≥  0 .With these de nitions at hand we can state our embedding theorem.
T H E O R E M 3 .3 (Embedding Theorem).For  > 0 there are ,  0 > 0 such that the following holds.Let  be an -expander on  >  0 vertices, let  ≥ 6, and let  be a graph on at most / log  vertices and edges.If  is (1−4/, 550Δ()/ 2 )-max-degree-robust, then  contains  as a topological minor.Furthermore, if  is also (1 − 2/, , 1 + 2/)-odd-cycle-robust, for  =  3(1+) , then one can choose the parities of the lengths of all the edge embeddings in the minor.
Let us highlight that  may depend on the graph .We have made no attempt to optimize the constants.The proof of the embedding theorem can be found in Section 5.
As mentioned before we need to ensure that once we obtain an embedding of the worstcase graph  in , that there is a matching in the graph  with the embedding of  removed.
To ensure this we will in fact not embed  directly in  but rather in a subgraph of : rst we identify a set of vertices  ⊆  () such that no matter what set  ⊆  of odd cardinality is removed from , the graph  \ still contains a perfect matching.We then proceed to show that the graph [ ] satis es all the properties required in order to embed  into it.The following lemma captures these properties 4 .

L E M M A 3 . 4 (Partition Lemma).
There is a  0 such that for all  >  0 there is an  0 such that the following holds.Let  >  0 be odd and  ∼ G(, ).we can now easily state and prove our lower bounds for the perfect matching formula (i.e., the special case  = 1 of Theorem 1.1).

T H E O R E M 3 . 5.
There is a  0 and an  > 0 such that for all  >  0 the following holds.For  and  ≤  log  both odd, let  ∼ G(, ) and  be any graph on  vertices of degree Δ() ≤ 5.
Then, asymptotically almost surely, PM() is an a ne restriction of PM().
Using the graphs from Appendix A (i.e., the graphs from Theorems A.

Lower Bounds for Card(𝑮, ì 𝒕)
In the following we prove the average-case lower bounds on the Card(, ì ) formulas for  ∼ G(, ).We consider the special case when  and  ≤  are odd and thus  is even.Without loss of generality, assume that  ≤ /2: otherwise " ip" the roles of 0 and 1.
The idea is to split the edge set of the graph  into /2 2-regular graphs  1 , . . .,  /2 and one  0 -regular graph  0 , where  0 =  − 2 /2 .Then we want to set all variables that correspond to an edge in any of the 2-regular graphs  1 , . . .,  /2 to 1 so that we are left with the perfect matching formula PM( 0 ), on which we will embed the worst-case instance of Appendix A.
In order to be able to apply Theorem 3.5 to PM( 0 ), we need to argue that  0 is a random  0 -regular graph.Also, we need to show that it is in fact possible to decompose a random -regular graph into /2 2-regular graphs plus a  0 -regular graph.For this, we use the notion of contiguity.Intuitively, two sequences of probability measures are contiguous, if all properties that hold with high probability in one also hold with high probability in the other measure.

D E F I N I T I O N 3 . 6. Let (𝑃 𝑛 ) ∞
1 and (  ) ∞ 1 be two sequences of probability measures, such that for each ,   and   both are de ned on the same measurable space (Ω  , F  ).
In other words, if we can show that e.g.SoS requires linear degree for a formula over  ∼ G(,  0 ) ⊕ ∈ /2 G(, 2) with high probability, then this also holds for the same formula over graphs  ∼ G(, ).Implementing our idea in the former probability distribution is straightforward and we have the following theorem.

T H E O R E M 3 . 8.
There is a  0 and an  > 0 such that for all  ≥  0 the following holds.Let ,  ≤  log  and  ∈ [] all be odd, let  ∼ G(, ) and  be a graph on  vertices of degree Δ() ≤ 5.Then, asymptotically almost surely, PM() is an a ne restriction of Card(, ì ).
Analogously to how Theorem 3.5 implied the  = 1 case of Theorem 1.1, this theorem implies the general case of Theorem 1.1.

P R O O F O F T H E O R E M 3 . 8 .
As  is odd  must be even.Note that we may assume that  ≤ /2: if  > /2, let us ip the role of 1 and 0 in the formula to obtain Card(, By Theorem 3.7, if we show the statement for  , then it also holds for  ∼ G(, ).
Set all variables in  1 , . . . /2 to 1.When Card(, ì ) is hit with this restriction we are left with the formula PM( 0 ).As  0 is distributed according to G(,  0 ), we may apply Theorem 3.5 to conclude that PM() is an a ne restriction of Card(, ì ).

Proof of the Partition Lemma
In this section we prove Lemma 3.4, restated here for convenience.

L E M M A 3 . 4 (Partition Lemma).
There is a  0 such that for all  >  0 there is an  0 such that the following holds.Let  >  0 be odd and  ∼ G(, ).() the graph  \  has a perfect matching for any  ⊆  of odd cardinality.
Deferring the proof of this lemma to Section 4.2, let us rst show that (, )-degree-balanced cuts always exist in regular graphs of large enough degree.
Note that the event   depends only on   for  within distance 2 of  in , and there are at most  2 many such 's.We want to apply the Lov ász local lemma (Lemma 2.8) to the events {  |  ∈  ()} and   =  for some parameter .The local lemma conditions then require Pr [  ] ≤  (1 − )  2 and this right hand side is maximized at  = 1  2 +1 where, using the bound For large enough  = Ω( The statement follows.
All that remains is to prove Lemma 4.2.In the following section we recall some results from spectral graph theory needed for the proof of Lemma 4.2 which is then given in Section 4.2.

Spectral Bounds
Let us establish some notation and recall some results from spectral graph theory.
We denote the adjacency matrix of a graph  by   and by   its Laplacian   =   −   (where   is the diagonal matrix containing the degrees of the vertices of ).For a matrix (𝐴) the eigenvalues of  in non-decreasing order.
The edge expansion of a graph  on  vertices is It is well-known that if the second smallest eigenvalue of the Laplacian is large, then the graph is a good expander.Note that the following theorem does not require that  is regular.Another well-known result from spectral graph theory is that the smallest eigenvalue of the adjacency matrix puts a limit on the maximum size of an independent set.
Let  be a -regular graph on  vertices.For any set  ⊆  ( For the sake of contradiction suppose that there is an  ⊆  () such that [] is bipartite and || > − Let us recall the mixing lemma; it states that between linearly sized sets of vertices there are about as many edges as expected in a random regular graph.
L E M M A 4 .9 (Expander Mixing Lemma [35]).Let  be a -regular graph on  vertices.Then for all ,  ⊆  (): We also rely on the following theorem that relates the spectrum of the Laplacian and the existence of a perfect matching.() the two paths  0 ,  1 from  0 to  1 on  are longer than .
Note that such a subpath  exists as the two paths connecting  to  on  are both longer than : if no such path  exists, then each potential  can be replaced by a part of , thereby obtaining a walk from  to  on  of length at most | |; a contradiction.
But note that such a  gives rise to a shorter odd cycle: either  0 ∪  or  1 ∪  is an odd cycle, of length less than .This is in contradiction to the initial assumption that  is a shortest odd cycle.The statement follows.

Proof of Lemma 4.2
Recall that by Theorem 4.6, with high probability all but the largest eigenvalue of the adjacency matrix of  are bounded in magnitude by 2 √  − 1 + (1).In the following we assume that  is large enough such that the (1) term is small.Let us argue each property separately.
() Let  ⊆  () be any set of size .Apply the mixing lemma (Lemma 4.9) to the graph As  is a -regular graph, we conclude that the average degree in [] is at least ).By the observation that if the average degree is at least , then there is a vertex of degree at least  , the statement follows for  large enough.() Recall that a sum of independent Poisson variables  1 , . . .,   with means  1 , . . .,   is again a Poisson variable with mean ∈[]   .Hence the number of cycles in  of length at most is, according to Theorem 4.14, a Poisson random variable  with mean where we used that  −1 + ( − 1) ≤  .Theorem 2.7 then tells us that for any  > 0, independent of , it holds that where the strict inequality holds for  large enough.Hence we may assume that  <  log .Let  ⊆  () be a set of vertices that contains one vertex from each cycle of length at most .By assumption || <  log  and the shortest cycle in  \  is of length at least .We also know that all but the largest eigenvalue of the adjacency matrix of  are bounded in magnitude by 2

√
− 1 + (1).() Let  ⊆  of odd cardinality be as in the statement, and denote by  the number of vertices in  \ .By Theorem 4.10, it is su cient to establish the bound   ( \ ) ≤ 2 2 ( \ ) on the eigenvalues of the Laplacian of  \ .Applying Proposition 4.13 to  \ , we can bound these eigenvalues in terms of the eigenvalues of the adjacency matrix of , obtaining As  1 (  ) and  −1 (  ) are both bounded in absolute value by 2 Since  > 1/2 +  and we assumed that  ≥ ( − 1/2 − ) −2 , we have that   ( \ ) ≤ 2 1 ( \ ) as desired.

Embedding Theorem
In this section we prove our embedding theorem (Theorem 3.3).Before starting with the proof, let us establish some notation and recall some facts from graph theory.

Further Graph Theory Preliminaries
In a graph  = ( , ) on  vertices a vertex set  ⊆  is a balanced separator in  if there is ∪  of the vertex set of  such that ||, || ≤ 2/3, and  has no edges between  and .
Large vertex expansion implies that balanced separators are large: the next lemma makes this well-known connection precise.
L E M M A 5 .1.Let  be an -expander on  vertices, and let  be a balanced separator in .Then || ≥  We also require the following lemma on vertex-disjoint paths in expanders.

Proof of Theorem 3.3
We now proceed with the proof of Theorem 3.3, restated here for convenience.
T H E O R E M 3 .3 (Embedding Theorem).For  > 0 there are ,  0 > 0 such that the following holds.Let  be an -expander on  >  0 vertices, let  ≥ 6, and let  be a graph on at most / log  vertices and edges.If  is (1−4/, 550Δ()/ 2 )-max-degree-robust, then  contains  as a topological minor.Furthermore, if  is also (1 − 2/, , 1 + 2/)-odd-cycle-robust, for  =  3(1+) , then one can choose the parities of the lengths of all the edge embeddings in the minor.
When embedding a high degree vertex  ∈  () into , we want to nd a vertex  ∈  () of high degree such that many neighbors are connected to large, disjoint sets of vertices.These large sets are very useful as they guarantee that there are many vertices to which we can connect a vertex embedding.The following de nition makes this intuition precise.

D E F I N I T I O N 5 . 3 (Cross
).An (, )-cross in a graph  = ( , ) is a tuple (, U), where  ∈  is a vertex and U ⊆ 2  consists of  pairwise disjoint vertex sets  ⊆  \ {}, each of size | | = , such that  () ∩  ≠ ∅ and the graph [] is connected.We refer to  as the center of the cross and to U as the branches of the cross.
The following lemma shows that crosses always exist in expanders with su ciently large maximum degree.
L E M M A 5 .4. For all  > 0 and  =  3(1+) the following holds.Let  be a -expander on  vertices that is (1 − 2/, (1 + 1/))-max-degree-robust, for some  ≥ 3 and  > 0 such that  ≤  3  (1+) .Then  contains an (, )-cross, for all  that satisfy The proof is an adaptation of a proof by Krivelevich and Nenadov [44] and is deferred to Section 5.3.We also have the following lemma which is what allows us to choose the path length parities in the "furthermore" part of Theorem 3.3.It states that if there is an odd cycle in the graph, then there is an odd and even path between any vertex  and a large enough set  of vertices.Note that this does not necessarily hold if  is too small: the vertex  may have degree 1 and  may be the single neighbor of .Similarly a lower bound on the length of the odd cycle is needed.
L E M M A 5 .5. For all  > 0 the following holds.Let  be a -expander on  vertices that contains an odd cycle of length ≥ 1 + 2/.Then, for all  ∈  () and  ⊆  (), of size || ≥ ( ø  log  + 1) (1 + 2/), there is a vertex  ∈  such that  and  are connected by both an odd and an even path, each of length at most (15 ø /2 /) log  + .
We defer the proof of Lemma 5.5 to Section 5.4.
We now prove Theorem 3.3 with the assumption of odd-cycle-robustness. Furthermore, the proof makes all paths of odd length, though it is immediate that one can choose the parities.
To get the theorem without the assumption of odd-cycle-robustness, one just has to replace the application of Lemma 5.5 by any shortest path (which, by Lemma 2.3 is short).
The main idea is due to Krivelevich and Nenadov [44] (see also [43]).In contrast to their work we cannot directly embed the vertices into the graph but rather take a detour by embedding appropriately sized crosses for each vertex and then connect branches of crosses that correspond to embeddings of adjacent vertices.The reason for this di erence is that the present theorem deals with topological minors rather than plain graph minors (the di erence is that in topological minors vertices are connected by vertex disjoint paths while in graph minors subgraphs are connected).
In order for this to work we need to make some further changes to the embedding process used.In their work, three sets of vertices are maintained throughout the process: one set  of "discarded" vertices, one set  of vertices used in the embedding, and the remaining set  of vertices.A key invariant which is maintained is that the set of discarded vertices expand poorly into the set of remaining vertices, which together with expansion implies that not too many vertices can be discarded.In our case, some of the discarded vertices may in fact have good expansion into , but we can maintain the property that there are not too many such vertices.This process is illustrated in Figure 1 and can be found as pseudo code in Algorithm 4.
If all branches of a vertex embedding   have either too few neighbors in  or are already adjacent to an edge embedding (i.e., have already been used to embed some other edge), then we want to remove the embedding of .This has to be done in a careful manner in order not to break the invariants.First, move all branches that are not used to connect  to a neighbor to .
Note that each such branch  satis es | (, )| < | |.Next, move the remaining branches along with the adjacent edge embeddings to  .Last, the center of   is moved to  and  is removed from .Note that at most 2(deg  () − 1) many vertices are moved to  : at most deg  () − 1 many branches of size  and as many edge embeddings, each again of size at most .

Crosses in Expanders
Let us now turn to the proof of Lemma 5.4, restated here for convenience.
The proof follows a similar algorithm as the proof of Theorem 3. Note that such a transversal  exists by Hall's marriage theorem, using that  ≥ 1/ and that every subset of B is -expanding into .Each path   connects  () to  (  , ) and we thus have that, as required, each branch intersects  () and that the branches are connected, where we use that the sets   ∈ B are by de nition connected.We also need to verify that the branches are pairwise vertex-disjoint.
To this end recall that the sets   ∈ B are pairwise disjoint and, furthermore, each such set is disjoint from .As the pairwise vertex-disjoint paths   live in [], these paths do not intersect ∪ ∈[]   and we may thus conclude that the branches are pairwise vertex-disjoint.Finally, we also need to check that each branch is of size : each set   is of size  and thus each branch is of size at least .Shrinking the branches to the appropriate size recovers the statement.

Odd and Even Paths
In this section we prove Lemma 5.5.
L E M M A 5 .5. For all  > 0 the following holds.Let  be a -expander on  vertices that contains an odd cycle of length ≥ 1 + 2/.Then, for all  ∈  () and  ⊆  (), of size || ≥ ( ø  log  + 1) (1 + 2/), there is a vertex  ∈  such that  and  are connected by both an odd and an even path, each of length at most (15 ø /2 /) log  + .
The lemma is a corollary of a more general statement about short paths in -expanders.Fix an arbitrary  ★ ∈  +1 \  +1 .By the induction hypothesis the claim holds for  ★ ∈   \   , so we may assume 5 that either  ★ ∈   , or  ★ ∈    [    ,    +1 ].If  ★ ∈   (excluding its endpoint    +1 ) then we simply extend the path in P ending in   with the subpath of   from  ★ to   , increasing the total length of P by at most |  |.On the other hand if  ★ ∈    [    ,    +1 ] then we reroute the path from   in P to   via   and then use the now unused part of    to connect  ★ to , again increasing the total length of P by at most |  |.There is an illustration of the two cases in Figure 3.
In either case, we can connect  and  ★ to  via vertex-disjoint paths of length at most  [ +1

P R O O F S K E T C H .
Consider the worst-case instance  from Filmus et al. [24] on /log  vertices, for some small enough  > 0. Apply Theorem 3.3 to  and .This gives a topological embedding of  in , with no control of the parities of the length of the paths.Consider a restriction  that sets the variables outside the embedding of  such that no axiom is falsi ed (see, e.g., [54]).By appropriately substituting the variables on each path of the topological embedding we obtain that the worst-case instance () is an a ne restriction of ().As an a ne restriction only reduces the amount of space needed to verify a proof, we see that () requires PC space Ω(/log ).

Paths in Expanders
The arguments used in the proof of Theorem 3.3 can be adapted to make partial progress on a question by Friedman and Krivelevich [26].They asked, given a positive integer , whether it is possible to guarantee the existence of a cycle whose length is divisible by  in every -expander.
We can show that for all primes  satisfying 1/poly()  √︁ /log , this indeed holds.In fact, for all  ∈ Z  , we can show that there is a cycle of length  mod .
The idea is to embed a cycle   2 of length  2 into  such that between any two vertices there are two paths whose length di erence is non-zero modulo .If we can ensure this, as all 0 ≠  ∈ Z  are generators, we can choose one path between all embedded vertices such that the length of the cycle is  mod  for any  ∈ Z  .
In order to obtain paths of di erent length modulo , let us embed a cycle   (of length 1/poly()) for each edge  = {, }.We then want to connect the vertex embeddings   ,   to   such that the two resulting paths are of di erent length modulo .Note that once a vertex is connected to the cycle, there are only about 2/ vertices in   such that both paths are of equal length modulo .As  is rather large and thus there are few such "bad" vertices, when an edge embedding has to be moved to the sets ,  , we can ensure that the set  remains relatively small compared to .

Open Problems
The main concrete problem left open is to reduce the degree of the hard graphs: the embedding approach taken in the worst-case to average-case reduction results in very large degree ; while Theorem 1.1 does not give an explicit estimate on  0 one can trace through the proofs and get an estimate of around 15 000.The main bottleneck that prevents us from reducing this is the Partition Lemma and in particular the dependence of  0 on  and  in Lemma 4.3.If this part could be signi cantly improved or circumvented we believe that the degree of the graph could be signi cantly reduced, although it would still be relatively large (at least a few hundreds).It would be interesting to see what happens for very small degrees, e.g., 4-regular graph (recall that since  is odd,  must be even) -is PM() hard with high probability even for these graphs?
We denote by deg() = | ()| the degree of a vertex  ∈  , by Δ() the maximum degree, () the minimum degree and by  () the average degree of .The edges between two vertex sets ,  ⊆  are denoted by (, ) = { {, } ∈  |  ∈ ,  ∈  }.For a set  ⊆  , we denote by [] = (, (, )) the induced subgraph of  in .For a set  ⊆  we also use  \  as a shorthand for the induced subgraph [ \  ].For a path  in  we denote by | | the number of edges and by  ( ) ⊆  () the set of vertices of .For two vertices vertices ,  ∈  ( ), we let [, ] denote the subpath of  between (and including) the vertices  and .The distance between two vertices ,  ∈ has a perfect matching.The partition lemma is proved in Section 4. The constants in Lemma 3.4 are rather arbitrarily chosen and their precise values are not signi cant -the interested reader can nd the precise dependencies between them in the proof.With Theorem 3.3 and Lemma 3.4 at hand,

Furthermore
, while || ≤ / we have that | | < ||/2 ≤ /2 .Note that this also holds the rst time || becomes larger than /.This shows, in particular, that the invariant || ≥ (1 − 2/) is maintained throughout the execution of the algorithm.For the sake of contradiction, suppose that the algorithm terminates because of || ≥ /.Note that | ()| ≤ | | + |∪ ∈B | + | (, )| < (|| + /).We do a case distinction, depending on the size of .In both cases we derive contradiction and thus show that the algorithm only terminates after having embedded all of  into .

Apply Lemma 5 .
2 to [] and the vertex sets  () and  to conclude that there are pairwise vertex-disjoint paths {   |  ∈ [] } each connecting  () to a set  (  , ), for some   ∈ B. We let the (, )-cross have center  and branches {  (  ) ∪   |  ∈ [] }.Let us verify that this is indeed a valid (, )-cross.
[61]embedding techniques in connection with algebraic systems like PC or SoS.As far as we are aware the rst such work is that ofPitassi et al. [54], who apply embedding techniques to obtain Tseitin lower bounds for Frege Systems, and their use is most similar to ours.They rely on a result ofKleinberg and Rubinfeld [39]that guarantees that any small enough graph is a minor of an expander (note that we require topological minors).This is in contrast to the other results that rely on the fundamental result that a graph of large enough treewidth contains the grid graph as a minor[61].
There are a few other papers that employ embedding techniques in proof complexity [54, 31, 27, 37], though none of these use For a -ary predicate  : {0, 1}  → {0, 1}, an instance of the CSP() problem consists of a set of constraints over  Boolean variables  1 , . . .,   , each constraint being an application of  on a list of  variables.The Card(, ì ) formulas we study can be viewed as instances of CSP() where each variable appears in exactly two constraints and  : {0, 1}  → {0, 1} is the constraint that exactly  of the  inputs are 1.
[41]1]oblems have been extensively studied throughout the years, and fairly general conditions under which CSP() is hard for PC and SoS are known[3,41].To be more accurate, these results are for the more general CSP( ± ) problem in which each constraint is an application of  on  literals rather than variables.In particular, Alekhnovich and Razborov[3]showed that if  is, say, 8-immune 2 over the underlying eld F, then any PC F refutation of a random CSP( ± ) instance with a linear number of constraints requires degree Ω().For SoS, Kothari et al.[41]showed that, if there exists a pairwise uniform distribution 3  over {0, 1}  supported on satisfying assignments of , then with high probability a random CSP( ± ) instance on  = Δ constraints needs degree Ω(/Δ 2 ) to be refuted by the SoS proof system.The predicates we study are linear equations over R and are neither immune nor do they support a pairwise uniform distribution.As such,

[13] that
with high probability the random graph G(, ) on  vertices and  = Ω( log ) edges contains any graph  with Δ() = (/) and at most =  1 ∧ . . .∧   is a sequence of formulas  1 ,  2 , . . .,  such that  = 0 and every formula   is either one of  1 , . . .,   or inferred from formulas   1 , . . .,    earlier in the sequence by a rule in F. As F is sound and complete a formula  has a refutation if and only if it is unsatis able.
We extend this notation to formulas in the obvious way, i.e., F  = {  1  ,  2  , . . .,    }.Two formulas F and F are equivalent, denoted by F ≡ F if the formulas are elementwise equivalent, disregarding functions that are constant True.We say that a formula F is an a ne restriction of F if there is a map  : { 1 , . . .,   } → {0, 1,  1 , . . .,   , x1 , . . ., x } such that F ≡ F  .The following lemma states that a formula F is at least as hard as any of its a ne restrictions.
Note that in the latter de nition, assuming that [] is an -expander, the diameter of[] is at most  ø log() ≤  ø /2 log() which means that, unless [] is bipartite, it certainly has short odd cycles of length at most 1 + 2 ø 3 .1.A graph  on  vertices is (, )-max-degree-robust if for all  ⊆  () of size | | ≥  it holds that the maximum degree of the induced subgraph [] is Δ( []) ≥ .D E F I N I T I O N 3 .2. A graph  on  vertices is (, , )-odd-cycle-robust if for all  ⊆  () of size | | ≥  and such that [] is an -expander it holds that the induced subgraph [] contains an odd cycle  of length ≤ || ≤ 3 ø /2 log .
1, A.3 and A.4) as our choice of  and combining Theorem 3.5 with Lemma 2.2 nishes the proof of Theorem 1.1 for the perfect matching formula.Let  ∼ G(, ) as in the statement.Apply Lemma 3.4 to  to obtain a set  with the mentioned properties.In order to apply Theorem 3.3 to [ ] and the graph  to obtain a topological minor   ⊆ [ ], where all edge embeddings in   are of odd length, we need to check that (for our choice  =1/3,  = 6) [ ] is (1/3, 550 • 5 • 9)-max-degree-robust, () [ ] is (2/3, 1/12, 1 + 2 • 12)-odd-cycle-robust, and()  is a graph on at most /6 log  vertices and edges, for some  > 0.From the guarantees of Lemma 3.4 we see that () is satis ed, that for  large () holds and also that () holds as odd-cycle-robustness is monotone in the rst argument.Lastly, () holds if we let  = /6.thetopologicalminor   of  in  at hand, we proceed to construct a restriction  to argue that PM() is an a ne restriction of PM().As all edge embeddings in   are of odd length and the number of vertices in  is odd, we see that | (  )| is odd.HenceLemma 3.4guarantees that there exists a perfect matching  in the graph  =  \  (  ).The restriction  sets all variables outside of   to 0 or 1 depending on whether the edge  ∈ .still need to specify how  maps the variables in   .For every edge embedding   of   , choose an arbitrary edge   ∈   and map the edge variables   , for  ∈   , alternatingly along   to either    or x  such that the rst and last edge of   are mapped to P R O O F O F T H EO R E M 3 .5.()[ ] is a 1/3-expander,()  (where we use that |   | is odd).By inspection we see that PM() is an a ne restriction of PM() as claimed.
The two sequences are contiguous if for every sequence of sets (  ) ∞ 1 , where   ∈ F  , it holds that lim For two random graphs G  and H  on the same set of  vertices, we denote by G  ⊕ H  the union of two independent samples conditioned on the result being simple.If G  = G(, ) and H  = G(,  ) are uniform distributions over random regular graphs we can think of this as a proccess where we rst sample  ∼ G  and then repeatedly sample  ∼ H  until the union of  and  is simple.For all constants  ≥ 3,  ≥ 1 and  1 , . . .,   ≥ 1 satisfying  =  =1   it holds that [38] ( ) = 0 ⇔ lim →∞   (  ) = 0 .We denote contiguity of two sequences by   ≈   .T H E O R E M 3 .7 (Corollary 9.44,[38]).
Apply Corollary 4.8 to conclude that no subset  ⊆  () of size at least | | ≥ 5  √  induces a bipartite subgraph, in other words any such [] contains an odd cycle.Let  ⊆  () be of size at least 6  √  .For  large, it holds that [ \ ] is of size at least 5  √  and thus contains an odd cycle of length at least .Let  denote such a cycle.What remains is to show that there is a cycle in [] that is simultaneously of length at least as well as bounded in length by 3 ø /2 log .Towards contradiction suppose that || ≥ 3 ø /2 log  and that  is a shortest odd cycle in [ \ ].Arbitrarily split the cycle  into four paths  1 ,  1 ,  2 and  2 , such that  1 and  2 separate  1 from  2 on , both   s are of size at least 1 4  ø /2 log , and both   s are of size at least 9 8  ø /2 log .We may assume that  ≤  9  ø /2 so that for  large we can apply Corollary 2.5 to [], ,  1 and  2 to conclude that in [ \ ] there is a path  connecting  1 to  2 of length at most  ø /2 log .This contradicts Lemma 4.15 as both paths of  connecting  1 to  2 As there are no cycles of length at most in [ \ ] we see that this cycle is also of length at least , as required.
We conclude that there is an odd cycle of length at most 3 ø /2 log  in [ \ ].
of the vertices of .sets , , and  play the same roles as in the informal description above, and  is an additional set of discarded vertices which may have large expansion into .When the algorithm terminates, every vertex  ∈  () (edge  ∈ (), respectively) has a vertex embedding   ∈ B (an edge embedding   ∈ B) giving a topological minor of  in .Initially, all sets except  =  () are empty.Let  =  3(1+) be the constant from Lemma 5.1 for the lower bound on the size of a balanced separator in an -expander.At several points in the algorithm we want to ensure that [] is a -expander.This is achieved by removing any subset  ⊆  of size | | ≤ ||/2 with small neighborhood | (,  \ )| < | | from  and adding it to  (i.e., letting  ←  \  and  ←  ∪ ).Clearly once there are no sets  ⊆  left as above, [] is a -expander.The algorithm maintains the set  ⊆  () to keep track of the vertices already embedded.Let () =  (1 + 4/) − 1 < 25/ and  = 18 ø /2 / log .In what follows we assume  is su ciently small as a function of .

Figure 1 .
The vertex embedding   is connected to    by the path   which connects the two branches   and    .The dotted branches have an edge embedding adjacent and can thus not be used to connect    to   . 3(1+) .Since || ≥ (1 − 2/) and Δ() ≤ | ()| ≤   log  the rst bound clearly holds for  large enough, and provided  is su ciently small as a function of  the second bound also holds.Thus we can indeed apply Lemma 5.4 on [] with the desired choice of  and .embedding , we need to connect the embedding   to the embeddings of the neighbors   () ∩  = {  1 , . . .,   } that are already embedded.Suppose, for now, that the vertex embeddings have branches   ∈   and    ∈    that are -expanding into  (i.e.| (  , )|, | (   , )| ≥ ), and such that neither of the two branches are already used to connect , resp.  , to a neighbor.By the assumption on odd-cycle-robustness, we see that [] is non-bipartite and contains an odd cycle  of length 1 + 2/ ≤ || ≤ 3 ø /2 log  .As each branch is rather large, of size , we can apply Lemma 5.5 to [],  (  , ) and  (   , ) to conclude that in [] there is an odd path   connecting   to    of length 18 ø /2 / log  ≤ .Remove   from , add it to B as the edge embedding  {,   } and restore -expansion in [].
where  = The high-level idea of the proof is as follows.First, using the embedding argument of Krivelevich and Nenadov, we nd some number  >  pairwise disjoint sets  1 , ...,   of vertices of  and a nal set  disjoint from all   s such that (i) each   is a connected subgraph of  on  vertices, (ii) the   s have many neighbors in , and (iii) [] is expanding.Having these subsets, we can then choose a representative   ∈  (  , ) of each   , take a vertex  ∈  of high degree (which exists by the max-degree-robustness of ), and apply Lemma 5.2 to nd vertex-disjoint paths connecting  () to the   s.This establishes the existence of a cross with  as the center and the   s together with the respective paths as branches.See Figure2for an illustration.Let us proceed with the details.In case there is some ambiguity in the verbal description there is also a pseudo code description in Appendix B of what follows.Fix , set  = (1 + 1/) and choose  ∈ N maximal such that  ≤  • .Note that  ≥ 1/ and if the statement holds for this maximal , then it also holds for smaller values of , as one can always shrink the branches to the appropriate size.Let us describe an algorithm to identify the sets B = {   ⊆  () |  ∈ [ ] }.  of the vertices of .Initially, all sets except  =  () are empty.After running the procedure, the set B contains  pairwise vertex-disjoint sets such that for each   ∈ B it holds that |  | =  and the induced subgraph [  ] is a single connected component.Further, for all subfamilies F ⊆ B it holds that ∈F  (, ) ≥ |F |. the execution of the algorithm the following invariants are maintained ()  never increases in size and || ≥ (1 − 2/),() [] is a -expander (by restoring expansion whenever needed), The algorithm terminates if B contains  vertex sets as described, or if the size of  reaches || ≥ /.The latter case can only occur if there is a small balanced separator in .But  is a -expander, so we know from Lemma 5.1 that there are no small balanced separators and hence when the algorithm terminates, B must contain  sets as described above.Like in the main algorithm used in the proof of Theorem 3.3, we want to ensure that [] is a -expander throughout the algorithm, which is achieved by removing any subset  ⊆  of size | | ≤ ||/2 with small neighborhood | (,  \ )| < | | from  =  \  and adding it to  =  ∪ .Repeat the following while there are less than  sets in B. Choose a set of vertices  ⊆  As || ≥ / this is a contradiction.Case 2: || ≥ /2.Note that the rst time || ≥ /, it also holds that || ≤ (1 + 1/)/2 as the sets added to  are of size at most ||/2 ≤ ( − ||)/2.Hence we get (using  ≥ 3) Note that  () is a balanced separator, separating  from  () \ .But this is a contradiction, since Lemma 5.1 states that any balanced separator of  has size at It remains to obtain an (, )-cross from the sets   and the remaining part .Choose a vertex  ∈  of degree at least deg [] () ≥  .Such a vertex  exists, as || ≥ (1 − 2/) is large ( rst invariant) and the statement assumes that there is a vertex of degree  in every induced subgraph of size at least (1 − 2/).Let  be a transversal of the family {  (, ) |  ∈ B }.
The lemma states that if sets ,  , where || | |/, are connected by | | many short vertexdisjoint paths, then for any large set  there is again a set of short vertex-disjoint paths that does not only connect every vertex of  to  but also a vertex from  to .order to state the lemma, let us introduce some notation.For a graph  and vertex sets ,  ⊆  (), denote by   disj ( , ) the minimum total length of connecting all vertices of  to    |  ∈  } ranges over all sets of pairwise vertex-disjoint paths such that   connects  to  (note the paths {  } are not allowed to intersect even in ).If no such set of paths exists, the value of the minimum is taken to be ∞.If the graph  is clear from context, we omit the superscript.Figure 3.Given the set   , the first figure depicts the process of obtaining the set  +1 .The following figures indicate how to route the paths, as in the proof of Claim 5.7, depending on where  ★ is located.Suppose the statement is true for some  ∈ {0, . . .− 1} and let us prove that is then true for  + 1 as well.By the inductive hypothesis,  [  ∪ (P)]   disj ( , ) + |  |, and this bound can be achieved by a set of paths P which follow P outside   \   .