New Direct-Product Testers and 2-Query PCPs

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.

Article Free Access Share on Probabilistically checkable proofs with zero knowledge Authors: Joe Kilian NEC Research Institute NEC Research InstituteView Profile , Erez Petrank DIMACS Center, P.O.Box 1179, Piscataway, NJ DIMACS Center, P.O.Box 1179, Piscataway, NJView Profile , Gábor Tardos Mathematical Institute of the Hungarian Academy of Sciences, Pf. 127, Budapest, H-1364 Hungary and Institute for Advanced Study, Princeton, NJ Mathematical Institute of the Hungarian Academy of Sciences, Pf. 127, Budapest, H-1364 Hungary and Institute for Advanced Study, Princeton, NJView Profile Authors Info & Claims STOC '97: Proceedings of the twenty-ninth annual ACM symposium on Theory of computingMay 1997 Pages 496–505https://doi.org/10.1145/258533.258643Online:04 May 1997Publication History 24citation435DownloadsMetricsTotal Citations24Total Downloads435Last 12 Months34Last 6 weeks3 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

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.

We give a simple explicit hitting set generator for read-once branching programs of width $w$ and length $r$ with known variable order and acceptance probability at least $\epsilon$. When $r = w$, our generator has seed length $O(\log^2 r + \log(1/\epsilon))$. When $r = \text{polylog } w$, our generator has optimal seed length $O(\log w + \log(1/\epsilon))$. For intermediate values of $r$, our generator's seed length smoothly interpolates between these two extremes. Our generator's seed length improves on recent work by Braverman, Cohen, and Garg [SIAM J. Comput., (2020), doi:10.1137/18M1197734]. In addition, our generator and its analysis are dramatically simpler than the work by Braverman et al. When $\epsilon$ is small, our generator's seed length improves on all the classic generators for space-bounded computation [N. Nisan, Combinatorica, 12 (1992), pp. 449--461; R. Impagliazzo, N. Nisan, and A. Wigderson, in Proceedings of the 26th Annual ACM Symposium on Theory of Computing, ACM, 1994, pp. 356--364; N. Nisan and D. Zuckerman, J. Comput. System Sci., 52 (1996), pp. 43--52]. However, all of these other works construct more general objects than we do. As a corollary of our construction, we show that every ${RL}$ algorithm that uses $r$ random bits can be simulated by an ${NL}$ algorithm that uses only $O(r/\log^c n)$ nondeterministic bits, where $c$ is an arbitrarily large constant. Finally, we show that any ${RL}$ algorithm with small success probability $\epsilon$ can be simulated deterministically in space $O(\log^{3/2} n + \log n \log \log(1/\epsilon))$. This space bound improves on work by Saks and Zhou [J. Comput. System Sci., 58 (1999), pp. 376--403], who gave an algorithm for the more general “two-sided” problem that runs in space $O(\log^{3/2} n + \sqrt{\log n} \log(1/\epsilon))$.

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.

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.

We consider a model of game-theoretic network design initially studied by Anshelevich et al. (Proceedings of the 45th Annual Symposium on Foundations of Computer Science (FOCS), pp. 295–304, 2004), where selfish players select paths in a network to minimize their cost, which is prescribed by Shapley cost shares. If all players are identical, the cost share incurred by a player for an edge in its path is the fixed cost of the edge divided by the number of players using it. In this special case, Anshelevich et al. (Proceedings of the 45th Annual Symposium on Foundations of Computer Science (FOCS), pp. 295–304, 2004) proved that pure-strategy Nash equilibria always exist and that the price of stability—the ratio between the cost of the best Nash equilibrium and that of an optimal solution—is Θ(log k), where k is the number of players. Little was known about the existence of equilibria or the price of stability in the general weighted version of the game. Here, each player i has a weight w i ≥1, and its cost share of an edge in its path equals w i times the edge cost, divided by the total weight of the players using the edge. This paper presents the first general results on weighted Shapley network design games. First, we give a simple example with no pure-strategy Nash equilibrium. This motivates considering the price of stability with respect to α-approximate Nash equilibria—outcomes from which no player can decrease its cost by more than an α multiplicative factor. Our first positive result is that O(log w max )-approximate Nash equilibria exist in all weighted Shapley network design games, where w max is the maximum player weight. More generally, we establish the following trade-off between the two objectives of good stability and low cost: for every α=Ω(log w max ), the price of stability with respect to O(α)-approximate Nash equilibria is O((log W)/α), where W is the sum of the players’ weights. In particular, there is always an O(log W)-approximate Nash equilibrium with cost within a constant factor of optimal. Finally, we show that this trade-off curve is nearly optimal: we construct a family of networks without o(log w max / log log w max )-approximate Nash equilibria, and show that for all α=Ω(log w max /log log w max ), achieving a price of stability of O(log W/α) requires relaxing equilibrium constraints by an Ω(α) factor.

Sums of squares based approximation algorithms for MAX-SAT

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).

We introduce and study the notion of probabilistically checkable proofs (PCP) for real number algorithms. Our starting point is the computational model of Blum, Shub, and Smale (BSS) and the real analogue NPR of NP in that model. We define in a straightforward manner verifiers as well as complexity classes PCPR(r(n),q(n)) for the BSS model. Our main result is, to the best of our knowledge, the first PCP theorem for NPR. It states that each problem in NPR has transparent long proofs, i.e.,NPR \subseteq PCPR(poly,1), where poly denotes the class of univariate polynomial functions. The techniques used extend ideas from [12] for self-testing and self-correcting certain functions over so-called rational domains to more general domains over the real numbers. The latter arise from the particular NPR-complete problem for which we construct a verifier of the required form.

Random languages for nonuniform complexity classes

We propose a new method for designing quantum search algorithms for finding a "marked" element in the state space of a classical Markov chain. The algorithm is based on a quantum walk \'a la Szegedy (2004) that is defined in terms of the Markov chain. The main new idea is to apply quantum phase estimation to the quantum walk in order to implement an approximate reflection operator. This operator is then used in an amplitude amplification scheme. As a result we considerably expand the scope of the previous approaches of Ambainis (2004) and Szegedy (2004). Our algorithm combines the benefits of these approaches in terms of being able to find marked elements, incurring the smaller cost of the two, and being applicable to a larger class of Markov chains. In addition, it is conceptually simple and avoids some technical difficulties in the previous analyses of several algorithms based on quantum walk.

formula omitted]-Regular languages are testable with a constant number of queries

Hardness amplification is the fundamental task of converting a $\delta$-hard function $f:\{0,1\}^n\to\{0,1\}$ into a $(1/2-\epsilon)$-hard function $\mathit{Amp}(f)$, where $f$ is $\gamma$-hard if small circuits fail to compute $f$ on at least a $\gamma$ fraction of the inputs. In this paper we study the complexity of black-box proofs of hardness amplification. A class of circuits $\mathcal{D}$ proves a hardness amplification result if for any function $h$ that agrees with $\mathit{Amp}(f)$ on a $1/2+\epsilon$ fraction of the inputs there exists an oracle circuit $D\in\mathcal{D}$ such that $D^h$ agrees with $f$ on a $1-\delta$ fraction of the inputs. We focus on the case where every $D\in\mathcal{D}$ makes nonadaptive queries to $h$. This setting captures most hardness amplification techniques. We prove two main results: (1) The circuits in $\mathcal{D}$ “can be used” to compute the majority function on $1/\epsilon$ bits. In particular, when $\epsilon\leq1/\log^{\omega(1)}n$, $\mathcal{D}$ cannot consist of oracle circuits that have unbounded fan-in, size $\mathrm{poly}(n)$, and depth $O(1)$. (2) The circuits in $\mathcal{D}$ must make $\Omega\left(\log(1/\delta)/\epsilon^2\right)$ oracle queries. Both our bounds on the depth and on the number of queries are tight up to constant factors. Our results explain why hardness amplification techniques have failed to transform known lower bounds against constant-depth circuit classes into strong average-case lower bounds. Our results reveal a contrast between Yao's XOR lemma ($\mathit{Amp}(f):=f(x_1)\oplus\cdots\oplus f(x_t)\in\{0,1\}$) and the direct-product lemma ($\mathit{Amp}(f):=f(x_1)\circ\cdots\circ f(x_t)\in\{0,1\}^t$; here $\mathit{Amp}(f)$ is non-Boolean). Our results (1) and (2) apply to Yao's XOR lemma, whereas known proofs of the direct-product lemma violate both (1) and (2). One of our contributions is a new technique for handling “nonuniform” reductions, i.e., the case when $\mathcal{D}$ contains many circuits.