Proceedings of the forty-sixth annual ACM symposium on Theory of computing最新文献

{"title":"Breaking the minsky-papert barrier for constant-depth circuits","authors":"Alexander A. Sherstov","doi":"10.1145/2591796.2591871","DOIUrl":"https://doi.org/10.1145/2591796.2591871","url":null,"abstract":"The threshold degree of a Boolean function f is the minimum degree of a real polynomial p that represents f in sign: f(x) ≡ sgn p(x). In a seminal 1969 monograph, Minsky and Papert constructed a polynomial-size constant-depth {∧, ∨)-circuit in n variables with threshold degree Ω(n1/3). This bound underlies some of today's strongest results on constant-depth circuits. It has been an open problem (O'Donnell and Servedio, STOC 2003) to improve Minsky and Papert's bound to nΩ(1)+1/3. We give a detailed solution to this problem. For any fixed k ≥ 1, we construct an {∧, ∨)-formula of size n and depth k with threshold degree Ω(n k-1/2k-1). This lower bound nearly matches a known O(√n) bound for arbitrary formulas, and is exactly tight for regular formulas. Our result proves a conjecture due to O'Donnell and Servedio (STOC 2003) and a different conjecture due to Bun and Thaler (2013). Applications to communication complexity and computational learning are given.","PeriodicalId":123501,"journal":{"name":"Proceedings of the forty-sixth annual ACM symposium on Theory of computing","volume":"113 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122684431","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Computing with a full memory: catalytic space","authors":"H. Buhrman, R. Cleve, M. Koucký, B. Loff, F. Speelman","doi":"10.1145/2591796.2591874","DOIUrl":"https://doi.org/10.1145/2591796.2591874","url":null,"abstract":"We define the notion of a catalytic-space computation. This is a computation that has a small amount of clean space available and is equipped with additional auxiliary space, with the caveat that the additional space is initially in an arbitrary, possibly incompressible, state and must be returned to this state when the computation is finished. We show that the extra space can be used in a nontrivial way, to compute uniform TC1-circuits with just a logarithmic amount of clean space. The extra space thus works analogously to a catalyst in a chemical reaction. TC1-circuits can compute for example the determinant of a matrix, which is not known to be computable in logspace. In order to obtain our results we study an algebraic model of computation, a variant of straight-line programs. We employ register machines with input registers x1,..., xn and work registers r1,..., rm. The instructions available are of the form ri ← ri±u×v, with u, v registers (distinct from ri) or constants. We wish to compute a function f(x1,..., xn) through a sequence of such instructions. The working registers have some arbitrary initial value ri = τi, and they may be altered throughout the computation, but by the end all registers must be returned to their initial value τi, except for, say, r1 which must hold τ1 + f(x1,..., xn). We show that all of Valiant's class VP, and more, can be computed in this model. This significantly extends the framework and techniques of Ben-Or and Cleve [6]. Upper bounding the power of catalytic computation we show that catalytic logspace is contained in ZPP. We further construct an oracle world where catalytic logpace is equal to PSPACE, and show that under the exponential time hypothesis (ETH), SAT can not be computed in catalytic sub-linear space.","PeriodicalId":123501,"journal":{"name":"Proceedings of the forty-sixth annual ACM symposium on Theory of computing","volume":"111 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115878102","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Embedding and canonizing graphs of bounded genus in logspace","authors":"Michael Elberfeld, K. Kawarabayashi","doi":"10.1145/2591796.2591865","DOIUrl":"https://doi.org/10.1145/2591796.2591865","url":null,"abstract":"Graph embeddings of bounded Euler genus (that means, embeddings with bounded orientable or nonorientable genus) help to design time-efficient algorithms for many graph problems. Since linear-time algorithms are known to compute embeddings of any bounded Euler genus, one can always assume to work with embedded graphs and, thus, obtain fast algorithms for many problems on any class of graphs of bounded Euler genus. Problems on graphs of bounded Euler genus are also important from the perspective of finding space-efficient algorithms, mostly focusing on problems related to finding paths and matchings in graphs. So far, known space-bounded approaches needed the severe assumption that an embedding is given as part of the input since no space-efficient embedding procedure for nonplanar graphs was known. The present work sidesteps this assumption and shows that embeddings of any bounded Euler genus can be computed in deterministic logarithmic space (logspace); allowing to generalize results on the space complexity of path and matching problems from embedded graphs to graphs of bounded Euler genus. The techniques developed for the embedding procedure also allow to compute canonical representations and, thus, solve the isomorphism problem for graphs of bounded Euler genus in logspace.","PeriodicalId":123501,"journal":{"name":"Proceedings of the forty-sixth annual ACM symposium on Theory of computing","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124191942","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"How to delegate computations: the power of no-signaling proofs","authors":"Y. Kalai, R. Raz, R. Rothblum","doi":"10.1145/2591796.2591809","DOIUrl":"https://doi.org/10.1145/2591796.2591809","url":null,"abstract":"We construct a 1-round delegation scheme (i.e., argument system) for every language computable in time t = t(n), where the running time of the prover is poly(t) and the running time of the verifier is n · polylog(t). In particular, for every language in P we obtain a delegation scheme with almost linear time verification. Our construction relies on the existence of a computational sub-exponentially secure private information retrieval (PIR) scheme. The proof exploits a curious connection between the problem of computation delegation and the model of multi-prover interactive proofs that are sound against no-signaling (cheating) strategies, a model that was studied in the context of multi-prover interactive proofs with provers that share quantum entanglement, and is motivated by the physical principle that information cannot travel faster than light. For any language computable in time t = t(n), we construct a multi-prover interactive proof (MIP) that is sound against no-signaling strategies, where the running time of the provers is poly(t), the number of provers is polylog(t), and the running time of the verifier is n · polylog(t). In particular, this shows that the class of languages that have polynomial-time MIPs that are sound against no-signaling strategies, is exactly EXP. Previously, this class was only known to contain PSPACE. To convert our MIP into a 1-round delegation scheme, we use the method suggested by Aiello et al (ICALP, 2000). This method relies on the existence of a sub-exponentially secure PIR scheme, and was proved secure by Kalai et al (STOC, 2013) assuming the underlying MIP is secure against no-signaling provers.","PeriodicalId":123501,"journal":{"name":"Proceedings of the forty-sixth annual ACM symposium on Theory of computing","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114485778","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Dichotomies in equilibrium computation, and complementary pivot algorithms for a new class of non-separable utility functions","authors":"J. Garg, R. Mehta, V. Vazirani","doi":"10.1145/2591796.2591863","DOIUrl":"https://doi.org/10.1145/2591796.2591863","url":null,"abstract":"After more than a decade of work in TCS on the computability of market equilibria, complementary pivot algorithms have emerged as the best hope of obtaining practical algorithms. So far they have been used for markets under separable, piecewise-linear concave (SPLC) utility functions [23] and SPLC production sets [25]. Can his approach extend to non-separable utility functions and production sets? A major impediment is rationality, i.e., if all parameters are set to rational numbers, there should be a rational equilibrium. Recently, [35] introduced classes of non-separable utility functions and production sets, called Leontief-free, which are applicable when goods are substitutes. For markets with these utility functions and production sets, and satisfying mild sufficiency conditions, we obtain the following results: • Proof of rationality. • Complementary pivot algorithms based on a suitable adaptation of Lemke's classic algorithm. • A strongly polynomial bound on the running time of our algorithms if the number of goods is a constant, despite the fact that the set of solutions is disconnected. • Experimental verification, which confirms that our algorithms are practical. • Proof of PPAD-completeness. Next we give a proof of membership in FIXP for markets under piecewise-linear concave (PLC) utility functions and PLC production sets by capturing equilibria as fixed points of a continuous function via a nonlinear complementarity problem (NCP) formulation. Finally we provide, for the first time, dichotomies for equilibrium computation problems, both Nash and market; in particular, the results stated above play a central role in arriving at the dichotomies for exchange markets and for markets with production. We note that in the past, dichotomies have played a key role in bringing clarity to the complexity of decision and counting problems.","PeriodicalId":123501,"journal":{"name":"Proceedings of the forty-sixth annual ACM symposium on Theory of computing","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123651740","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Improved approximation algorithms for degree-bounded network design problems with node connectivity requirements","authors":"Alina Ene, A. Vakilian","doi":"10.1145/2591796.2591837","DOIUrl":"https://doi.org/10.1145/2591796.2591837","url":null,"abstract":"We consider degree bounded network design problems with element and vertex connectivity requirements. In the degree bounded Survivable Network Design (SNDP) problem, the input is an undirected graph G = (V, E) with weights w(e) on the edges and degree bounds b(v) on the vertices, and connectivity requirements r(uv) for each pair uv of vertices. The goal is to select a minimum-weight subgraph H of G that meets the connectivity requirements and it satisfies the degree bounds on the vertices: for each pair uv of vertices, H has r(uv) disjoint paths between u and v; additionally, each vertex v is incident to at most b(v) edges in H. We give the first (O(1), O(1) · b(v)) bicriteria approximation algorithms for the degree-bounded SNDP problem with element connectivity requirements and for several degree-bounded SNDP problems with vertex connectivity requirements. Our algorithms construct a subgraph H whose weight is at most O(1) times the optimal such that each vertex v is incident to at most O(1) · b(v) edges in H. We can also extend our approach to network design problems in directed graphs with out-degree constraints to obtain (O(1), O(1) · b+(v)) bicriteria approximation.","PeriodicalId":123501,"journal":{"name":"Proceedings of the forty-sixth annual ACM symposium on Theory of computing","volume":"91 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127163825","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Super-polynomial lower bounds for depth-4 homogeneous arithmetic formulas","authors":"N. Kayal, N. Limaye, Chandan Saha, S. Srinivasan","doi":"10.1145/2591796.2591823","DOIUrl":"https://doi.org/10.1145/2591796.2591823","url":null,"abstract":"We show that any depth-4 homogeneous arithmetic formula computing the Iterated Matrix Multiplication polynomial IMMn,d -- the (1, 1)-th entry of the product of d generic n × n matrices -- has size nΩ(log n), if d = Ω (log2 n). More-over, any depth-4 homogeneous formula computing the determinant polynomial Detn -- the determinant of a generic n × n matrix -- has size nΩ(log n).","PeriodicalId":123501,"journal":{"name":"Proceedings of the forty-sixth annual ACM symposium on Theory of computing","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121510328","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Toward better formula lower bounds: an information complexity approach to the KRW composition conjecture","authors":"Dmitry Gavinsky, Or Meir, Omri Weinstein, A. Wigderson","doi":"10.1145/2591796.2591856","DOIUrl":"https://doi.org/10.1145/2591796.2591856","url":null,"abstract":"One of the major open problems in complexity theory is proving super-logarithmic lower bounds on the depth of circuits (i.e., P ⊈ NC1). This problem is interesting for two reasons: first, it is tightly related to understanding the power of parallel computation and of small-space computation; second, it is one of the first milestones toward proving super-polynomial circuit lower bounds. Karchmer, Raz, and Wigderson [21] suggested to approach this problem by proving the following conjecture: given two boolean functions f and g, the depth complexity of the composed function g o f is roughly the sum of the depth complexities of f and g. They showed that the validity of this conjecture would imply that P ⊈ NC1. As a starting point for studying the composition of functions, they introduced a relation called \"the universal relation\", and suggested to study the composition of universal relations. This suggestion proved fruitful, and an analogue of the KRW conjecture for the universal relation was proved by Edmonds et. al. [12]. An alternative proof was given later by Håstad and Wigderson [18]. However, studying the composition of functions seems more difficult, and the KRW conjecture is still wide open. In this work, we make a natural step in this direction, which lies between what is known and the original conjecture: we show that an analogue of the conjecture holds for the composition of a function with a universal relation. We also suggest a candidate for the next step and provide initial results toward it. Our main technical contribution is developing an approach based on the notion of information complexity for analyzing KW relations -- communication problems that are closely related to questions on circuit depth and formula complexity. Recently, information complexity has proved to be a powerful tool, and underlined some major progress on several long-standing open problems in communication complexity. In this work, we develop general tools for analyzing the information complexity of KW relations, which may be of independent interest.","PeriodicalId":123501,"journal":{"name":"Proceedings of the forty-sixth annual ACM symposium on Theory of computing","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124079435","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Solving SDD linear systems in nearly mlog1/2n time","authors":"Michael B. Cohen, Rasmus Kyng, G. Miller, J. Pachocki, Richard Peng, Anup B. Rao, S. Xu","doi":"10.1145/2591796.2591833","DOIUrl":"https://doi.org/10.1145/2591796.2591833","url":null,"abstract":"We show an algorithm for solving symmetric diagonally dominant (SDD) linear systems with m non-zero entries to a relative error of ε in O(m log1/2 n logc n log(1/ε)) time. Our approach follows the recursive preconditioning framework, which aims to reduce graphs to trees using iterative methods. We improve two key components of this framework: random sampling and tree embeddings. Both of these components are used in a variety of other algorithms, and our approach also extends to the dual problem of computing electrical flows. We show that preconditioners constructed by random sampling can perform well without meeting the standard requirements of iterative methods. In the graph setting, this leads to ultra-sparsifiers that have optimal behavior in expectation. The improved running time makes previous low stretch embedding algorithms the running time bottleneck in this framework. In our analysis, we relax the requirement of these embeddings to snowflake spaces. We then obtain a two-pass approach algorithm for constructing optimal embeddings in snowflake spaces that runs in O(m log log n) time. This algorithm is also readily parallelizable.","PeriodicalId":123501,"journal":{"name":"Proceedings of the forty-sixth annual ACM symposium on Theory of computing","volume":"119 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126617120","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An excluded half-integral grid theorem for digraphs and the directed disjoint paths problem","authors":"K. Kawarabayashi, Yusuke Kobayashi, S. Kreutzer","doi":"10.1145/2591796.2591876","DOIUrl":"https://doi.org/10.1145/2591796.2591876","url":null,"abstract":"The excluded grid theorem, originally proved by Robertson and Seymour in Graph Minors V, is one of the most central results in the study of graph minors. It has found numerous applications in algorithmic graph structure theory, for instance as the basis for bidimensionality theory on graph classes excluding a fixed minor. In 1997, Reed [25] and later Johnson, Robertson, Seymour and Thomas [17] conjectured an analogous theorem for directed graphs, i.e. the existence of a function f: N → N such that every digraph of directed tree-width at least f(k) contains a directed grid of order k. In this paper, we make significant progress toward this conjecture. Namely, we prove that every digraph of directed tree-width at least f(k) contains a \"half-integral\" directed grid of order k. This structural result allows us to contribute to the disjoint paths problem. We show that the following can be done in polynomial time: Suppose that we are given a digraph G and k terminal pairs (s1, t1), (s2, t2),..., (sk, tk), where k is a fixed constant. In polynomial time, either • we can find k paths P1,..., Pk such that Pi is from si to ti for i = 1,..., k and every vertex in G is in at most four of the paths, or • we can conclude that G does not contain disjoint paths P1,..., Pk such that Pi is from si to ti for i = 1,..., k. To the best of our knowledge, this is the first positive result for the general directed disjoint paths problem. Note that the directed disjoint paths problem is NP-hard even for k = 2. Therefore, polynomial-time algorithms for semiintegral disjoint paths is the best one can hope for.","PeriodicalId":123501,"journal":{"name":"Proceedings of the forty-sixth annual ACM symposium on Theory of computing","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130607667","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}