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An open access journal in theoretical computer science |
Volume 4
2025
1. Finite-valued Streaming String Transducers
A transducer is finite-valued if for some bound k, it maps any given input to at most k outputs. For classical, one-way transducers, it is known since the 80s that finite valuedness entails decidability of the equivalence problem. This decidability result is in contrast to the general case, which makes finite-valued transducers very attractive. For classical transducers, it is also known that finite valuedness is decidable and that any k-valued finite transducer can be decomposed as a union of k single-valued finite transducers. In this paper, we extend the above results to copyless streaming string transducers (SSTs), answering questions raised by Alur and Deshmukh in 2011. SSTs strictly extend the expressiveness of one-way transducers via additional variables that store partial outputs. We prove that any k-valued SST can be effectively decomposed as a union of k (single-valued) deterministic SSTs. As a corollary, we obtain equivalence of SSTs and two-way transducers in the finite-valued case (those two models are incomparable in general). Another corollary is an elementary upper bound for checking equivalence of finite-valued SSTs. The latter problem was already known to be decidable, but the proof complexity was unknown (it relied on Ehrenfeucht's conjecture). Finally, our main result is that finite valuedness of SSTs is decidable. The complexity is PSpace, and even PTime when the number of variables is fixed.
2. An Alternate Proof of Near-Optimal Light Spanners
In 2016, a breakthrough result of Chechik and Wulff-Nilsen [SODA '16] established that every $n$-node graph $G$ has a $(1+\varepsilon)(2k-1)$-spanner of lightness $O_{\varepsilon}(n^{1/k})$, and recent followup work by Le and Solomon [STOC '23] generalized the proof strategy and improved the dependence on $\varepsilon$. We give a new proof of this result, with the improved $\varepsilon$-dependence. Our proof is a direct analysis of the often-studied greedy spanner, and can be viewed as an extension of the folklore Moore bounds used to analyze spanner sparsity.
3. Approximate Counting for Spin Systems in Sub-Quadratic Time
We present two randomised approximate counting algorithms with $\widetilde{O}(n^{2-c}/\varepsilon^2)$ running time for some constant $c>0$ and accuracy $\varepsilon$: (1) for the hard-core model with fugacity $\lambda$ on graphs with maximum degree $\Delta$ when $\lambda=O(\Delta^{-1.5-c_1})$ where $c_1=c/(2-2c)$; (2) for spin systems with strong spatial mixing (SSM) on planar graphs with quadratic growth, such as $\mathbb{Z}^2$. For the hard-core model, Weitz's algorithm (STOC, 2006) achieves sub-quadratic running time when correlation decays faster than the neighbourhood growth, namely when $\lambda = o(\Delta^{-2})$. Our first algorithm does not require this property and extends the range where sub-quadratic algorithms exist. Our second algorithm appears to be the first to achieve sub-quadratic running time up to the SSM threshold, albeit on a restricted family of graphs. It also extends to (not necessarily planar) graphs with polynomial growth, such as $\mathbb{Z}^d$, but with a running time of the form $\widetilde{O}\left(n^2\varepsilon^{-2}/2^{c(\log n)^{1/d}}\right)$ where $d$ is the exponent of the polynomial growth and $c>0$ is some constant.
4. On Approximability of Steiner Tree in $\ell_p$-metrics
In the Continuous Steiner Tree problem (CST), we are given as input a set of points (called terminals) in a metric space and ask for the minimum-cost tree connecting them. Additional points (called Steiner points) from the metric space can be introduced as nodes in the solution. In the Discrete Steiner Tree problem (DST), we are given in addition to the terminals, a set of facilities, and any solution tree connecting the terminals can only contain the Steiner points from this set of facilities. Trevisan [SICOMP'00] showed that CST and DST are APX-hard when the input lies in the $\ell_1$-metric (and Hamming metric). Chleb\'ik and Chleb\'iková [TCS'08] showed that DST is NP-hard to approximate to factor of $96/95\approx 1.01$ in the graph metric (and consequently $\ell_\infty$-metric). Prior to this work, it was unclear if CST and DST are APX-hard in essentially every other popular metric. In this work, we prove that DST is APX-hard in every $\ell_p$-metric. We also prove that CST is APX-hard in the $\ell_{\infty}$-metric. Finally, we relate CST and DST, showing a general reduction from CST to DST in $\ell_p$-metrics.
5. Computing Threshold Budgets in Discrete-Bidding Games
In a two-player zero-sum graph game, the players move a token throughout a graph to produce an infinite play, which determines the winner of the game. Bidding games are graph games in which in each turn, an auction (bidding) determines which player moves the token: the players have budgets, and in each turn, both players simultaneously submit bids that do not exceed their available budgets, the higher bidder moves the token, and pays the bid to the lower bidder (called Richman bidding). We focus on discrete-bidding games, in which, motivated by practical applications, the granularity of the players' bids is restricted, e.g., bids must be given in cents. A central quantity in bidding games is threshold budgets: a necessary and sufficient initial budget for winning the game. Previously, thresholds were shown to exist in parity games, but their structure was only understood for reachability games. Moreover, the previously-known algorithms have a worst-case exponential running time for both reachability and parity objectives, and output strategies that use exponential memory. We describe two algorithms for finding threshold budgets in parity discrete-bidding games. The first is a fixed-point algorithm. It reveals, for the first time, the structure of threshold budgets in parity discrete-bidding games. Based on this structure, we develop a second algorithm that shows that the problem of finding threshold budgets is in NP and coNP for both reachability and parity objectives. […]