45th Annual IEEE Symposium on Foundations of Computer Science

For general antiferromagnetic 2-spin systems, including the hardcore model on weighted independent sets and the antiferromagnetic Ising model, there is an for the partition function on graphs of maximum degree when the infinite regular tree lies in the uniqueness region by Li, Lu, and Yin [Correlation Decay up to Uniqueness in Spin Systems, preprint, https://arxiv.org/abs/1111.7064, 2021]. Moreover, in the tree nonuniqueness region, Sly in [Computational transition at the uniqueness threshold, in Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science, 2010, pp. 287–296] showed that there is no to estimate the partition function unless . The algorithmic results follow from the correlation decay approach due to Weitz [Counting independent sets up to the tree threshold, in Proceedings of the 38th Annual ACM Symposium on Theory of Computing, 2006, pp. 140–149] or the polynomial interpolation approach developed by Barvinok [Combinatorics and Complexity of Partition Functions, Springer, 2016]. However, the running time is only polynomial for constant . For the hardcore model, recent work of Anari, Liu, and Oveis Gharan [Spectral independence in high-dimensional expanders and applications to the hardcore model, in Proceedings of the 61st Annual IEEE Symposium on Foundations of Computer Science, 2020, pp. 1319–1330] establishes rapid mixing of the simple single-site Markov chain, known as the Glauber dynamics, in the tree uniqueness region. Our work simplifies their analysis of the Glauber dynamics by considering the total pairwise influence of a fixed vertex on other vertices, as opposed to the total influence of other vertices on , thereby extending their work to all 2-spin models and improving the mixing time. More important, our proof ties together the three disparate algorithmic approaches: we show that contraction of the so-called tree recursions with a suitable potential function, which is the primary technique for establishing efficiency of Weitz’s correlation decay approach and Barvinok’s polynomial interpolation approach, also establishes rapid mixing of the Glauber dynamics. We emphasize that this connection holds for all 2-spin models (both antiferromagnetic and ferromagnetic), and existing proofs for the correlation decay and polynomial interpolation approaches immediately imply rapid mixing of the Glauber dynamics. Our proof utilizes the fact that the graph partition function is a divisor of the partition function for Weitz’s self-avoiding walk tree. This fact leads to new tools for the analysis of the influence of vertices and may be of independent interest for the study of complex zeros.

We examine how to induce selfish heterogeneous users in a multicommodity network to reach an equilibrium that minimizes the social cost. In the absence of centralized coordination, we use the classical method of imposing appropriate taxes (tolls) on the edges of the network. We significantly generalize previous work (Yang and Huang in Transp. Res. Part B 38:1–15, [2004]; Karakostas and Kolliopoulos in Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, pp. 268–276, [2004]; Fleischer et al. in Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, pp. 277–285, [2004]) by allowing user demands to be elastic. In this setting the demand of a user is not fixed a priori but it is a function of the routing cost experienced, a most natural assumption in traffic and data networks.

The distributed $(\Delta + 1)$-coloring problem is one of the most fundamental and well-studied problems in distributed algorithms. Starting with the work of Cole and Vishkin in 1986, a long line of gradually improving algorithms has been published. The state-of-the-art running time, prior to our work, is $O(\Delta \log \Delta + \log^* n)$, due to Kuhn and Wattenhofer [Proceedings of the $25$th Annual ACM Symposium on Principles of Distributed Computing, Denver, CO, 2006, pp. 7--15]. Linial [Proceedings of the $28$th Annual IEEE Symposium on Foundation of Computer Science, Los Angeles, CA, 1987, pp. 331--335] proved a lower bound of $\frac{1}{2} \log^* n$ for the problem, and Szegedy and Vishwanathan [Proceedings of the 25th Annual ACM Symposium on Theory of Computing, San Diego, CA, 1993, pp. 201--207] provided a heuristic argument that shows that algorithms from a wide family of locally iterative algorithms are unlikely to achieve a running time smaller than $\Theta(\Delta \log \Delta)$. We present a deterministic $(\Delta + 1)$-coloring distributed algorithm with running time $O(\Delta) + \frac{1}{2} \log^* n$. We also present a trade-off between the running time and the number of colors, and devise an $O(\lambda\cdot\Delta)$-coloring algorithm, with running time $O(\Delta / \lambda + \log^* n)$, for any parameter $\lambda > 1$. Our algorithm breaks the heuristic barrier of Szegedy and Vishwanathan and achieves running time which is linear in the maximum degree $\Delta$. On the other hand, the conjecture of Szegedy and Vishwanathan may still be true, as our algorithm does not belong to the family of locally iterative algorithms. On the way to this result we study a generalization of the notion of graph coloring, which is called defective coloring [L. Cowen, R. Cowen, and D. Woodall, J. Graph Theory, 10 (1986), pp. 187--195]. In an $m$-defective $p$-coloring the vertices are colored with $p$ colors so that each vertex has up to $m$ neighbors with the same color. We show that an $m$-defective $p$-coloring with reasonably small $m$ and $p$ can be computed very efficiently in the distributed setting. We also develop a technique to employ multiple defective colorings of various subgraphs of the original graph $G$ for computing a $(\Delta+1)$-coloring of $G$. We believe that these techniques are of independent interest.

We consider two of the most fundamental theorems in cryptography. The first, due to Håstad et al. [SIAM J. Comput., 28 (1999), pp. 1364–1396] is that pseudorandom generators can be constructed from any one-way function. The second, due to Yao [Proceedings of the $23$rd Annual Symposium on Foundations of Computer Science (FOCS), 1982, pp. 80–91], states that the existence of weak one-way functions implies the existence of full-fledged one-way functions. These powerful plausibility results shape our understanding of hardness and randomness in cryptography, but unfortunately their proofs are not as tight (i.e., security preserving) as one may desire. This work revisits a technique that we call the randomized iterate, introduced by Goldreich, Krawczyk, and Luby [SIAM J. Comput., 22 (1993), pp. 1163–1175]. This technique was used by Goldreich, Krawczyk, and Luby [SIAM J. Comput., 22 (1993), pp. 1163–1175] to give a construction of pseudorandom generators from regular one-way functions. We simplify and strengthen this technique in order to obtain a similar construction, where the seed length of the resulting generators is as short as $\Theta(n \log n)$ (rather than $\Theta(n^3)$ achieved by Goldreich, Krawczyk, and Luby [SIAM J. Comput., 22 (1993), pp. 1163–1175]). Our technique has the potential of implying seed length $\Theta(n)$, and the only bottleneck for such a result are the parameters of current generators against bounded-space computations. We give a construction with similar parameters for security amplification of regular one-way functions. This improves upon the construction of Goldreich et al. [Proceedings of the $31$st Annual Symposium on Foundations of Computer Science, (FOCS), 1990, pp. 318–326] in that the construction does not need to “know" the regularity parameter of the functions (in terms of security, the two reductions are incomparable). In addition, we use the randomized iterate to show a construction of a pseudorandom generator based on an exponentially hard one-way function that has a seed length of only $\Theta(n^2)$. This improves a recent result of Holenstein [Proceedings of the Theory of Cryptography, Third Theory of Cryptography Conference (TCC), 2006] that shows a construction with seed length $\Theta(n^5)$ based on such one-way functions. Finally, we show that the randomized iterate may even be useful in the general context of Håstad et al. [SIAM J. Comput., 28 (1999), pp. 1364–1396]. In particular, we use the randomized iterate to replace the basic building block of the Håstad et al. [SIAM J. Comput., 28 (1999), pp. 1364–1396] construction. Interestingly, this modification improves efficiency by an $\Theta(n^2)$ factor and reduces the seed length to $\Theta(n^7)$ (which also implies improvement in the security of the construction).

48th Annual IEEE Symposium on Foundations of Computer Science - Title

Copyright and Reprint Permissions: Abstracting is permitted with credit to the source. Libraries may photocopy beyond the limits of US copyright law, for private use of patrons, those articles in this volume that carry a code at the bottom of the first page, provided that the per-copy fee indicated in the code is paid through the Copyright Clearance Center. The papers in this book comprise the proceedings of the meeting mentioned on the cover and title page. They reflect the authors' opinions and, in the interests of timely dissemination, are published as presented and without change. Their inclusion in this publication does not necessarily constitute endorsement by the editors or the Institute of Electrical and Electronics Engineers, Inc.

46th Annual IEEE Symposium on Foundations of Computer Science - Table of Contents

46th Annual IEEE Symposium on Foundations of Computer Science - Title Page

Special FOCS issue—twenty-second annual IEEE symposium on foundations of computer science

The 2-way quantum finite automaton introduced by Kondacs and Watrous [Proceedings of the 38th Annual Symposium on Foundations of Computer Science, 1997, IEEE Computer Society, pp. 66--75] can accept nonregular languages with bounded error in polynomial time. If we restrict the head of the automaton to moving classically and to moving only in one direction, the acceptance power of this 1-way quantum finite automaton is reduced to a proper subset of the regular languages. In this paper we study two different models of 1-way quantum finite automata. The first model, termed measure-once quantum finite automata, was introduced by Moore and Crutchfield [ Theoret. Comput. Sci., 237 (2000), pp. 275--306], and the second model, termed measure-many quantum finite automata, was introduced by Kondacs and Watrous [Proceedings of the38th Annual Symposium on Foundations of Computer Science, 1997, IEEE Computer Society, pp. 66--75]. We characterize the measure-once model when it is restricted to accepting with bounded error and show that, without that restriction, it can solve the word problem over the free group. We also show that it can be simulated by a probabilistic finite automaton and describe an algorithm that determines if two measure-once automata are equivalent. We prove several closure properties of the classes of languages accepted by measure-many automata, including inverse homomorphisms, and provide a new necessary condition for a language to be accepted by the measure-many model with bounded error. Finally, we show that piecewise testable sets can be accepted with bounded error by a measure-many quantum finite automaton, introducing new construction techniques for quantum automata in the process.

The natural relaxation for the group Steiner tree problem, as well as for its generalization, the directed Steiner tree problem, is a flow‐based linear programming relaxation. We prove new lower bounds on the integrality ratio of this relaxation. For the group Steiner tree problem, we show that the integrality ratio is $\Omega(\log^2 k)$, where k denotes the number of groups; this holds even for input graphs that are hierarchically well‐separated trees, introduced by Bartal [in Proceedings of the 37th Annual IEEE Symposium on Foundations of Computer Science, 1996, pp. 184–193], in which case this lower bound is tight. This also applies for the directed Steiner tree problem. In terms of the number n of vertices, our results for the directed Steiner problem imply an $\Omega(\frac{\log^2 n}{(\log \log n)^2})$ integrality ratio. For both problems, these are the first lower bounds on the integrality ratio that are superlogarithmic in the input size. This exhibits, for the first time, a relaxation of a natural optimization problem whose integrality ratio is known to be superlogarithmic but subpolynomial. Our results and techniques have been used by Halperin and Krauthgamer [in Proceedings of the 35th Annual ACM Symposium on Theory of Computing, 2003, pp. 585–594] to show comparable inapproximability results, assuming that NP has no quasi‐polynomial Las Vegas algorithms. We also show algorithmically that the integrality ratio for the group Steiner tree problem is much better for certain families of instances, which helps pinpoint the types of instances (parametrized by optimal solutions to their flow‐based relaxations) that appear to be most difficult to approximate.

The min-rank of a graph was introduced by Haemers (Algebr. Methods Graph Theory 25:267–272, 1978) to bound the Shannon capacity of a graph. This parameter of a graph has recently gained much more attention from the research community after the work of Bar-Yossef et al. (in Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 197–206, 2006). In their paper, it was shown that the min-rank of a graph \(\mathcal {G}\) characterizes the optimal scalar linear solution of an instance of the Index Coding with Side Information (ICSI) problem described by the graph \(\mathcal {G}\).It was shown by Peeters (Combinatorica 16(3):417–431, 1996) that computing the min-rank of a general graph is an NP-hard problem. There are very few known families of graphs whose min-ranks can be found in polynomial time. In this work, we introduce a new family of graphs with efficiently computed min-ranks. Specifically, we establish a polynomial time dynamic programming algorithm to compute the min-ranks of graphs having simple tree structures. Intuitively, such graphs are obtained by gluing together, in a tree-like structure, any set of graphs for which the min-ranks can be determined in polynomial time. A polynomial time algorithm to recognize such graphs is also proposed.

Alphabet-independent linear-time construction of compressed suffix arrays using [formula omitted]-bit working space

2009 50th Annual IEEE Symposium on Foundations of Computer Science

The "direct-product code" of a function $f$ gives its values on all $k$-tuples $(f(x_1),\dots, f(x_k))$. This basic construct underlies "hardness amplification" in cryptography, circuit complexity, and probabilistically checkable proofs (PCPs). Goldreich and Safra [SIAM J. Comput., 29 (2000), pp. 1132--1154] pioneered its local testing and its PCP application. A recent result by Dinur and Goldenberg [Proceedings of the Forty-Ninth Annual IEEE Symposium on Foundations of Computer Science, 2008, pp. 613--622] enabled for the first time testing proximity to this important code in the "list-decoding" regime. In particular, they give a $2$-query test which works for polynomially small success probability $1/k^{\alpha}$ and show that no such test works below success probability $1/k$. Our main result is a $3$-query test which works for exponentially small success probability $\exp({-k^{\alpha}})$. Our techniques (based on recent simplified decoding algorithms for the same code [R. Impagliazzo et al., Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, 2008, pp. 579--588]) also allow us to considerably simplify the analysis of the 2-query test of [Proceedings of the Forty-Ninth Annual IEEE Symposium on Foundations of Computer Science, 2008, pp. 613--622]. We then show how to derandomize their test, achieving a code of polynomial rate, independent of $k$, and success probability $1/k^{\alpha}$. Finally, we show the applicability of the new tests to PCPs. Starting with a 2-query PCP with a projection property over an alphabet $\Sigma$ and with soundness error $1-\delta$, Rao [Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, 2008, pp. 1--10] (building on Raz's ($k$-fold) parallel repetition theorem [R. Raz, SIAM J. Comput., 27 (1998), pp. 763--803] and Holenstein's proof [T. Holenstein, Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing, 2007, pp. 411--419] obtains a new 2-query PCP over the alphabet $\Sigma^k$ with soundness error $\exp(-\delta^2 k)$. Our techniques yield a 2-query PCP with soundness error $\exp(-\delta \sqrt{k})$. Our PCP construction turns out to be essentially the same as the miss-match proof system defined and analyzed by Feige and Kilian [SIAM J. Comput., 30 (2000), pp. 324--346] but with simpler analysis and exponentially better soundness error.