25 Search Results for "Saks, Michael E."


Document
NP-Hardness of Circuit Minimization for Multi-Output Functions

Authors: Rahul Ilango, Bruno Loff, and Igor C. Oliveira

Published in: LIPIcs, Volume 169, 35th Computational Complexity Conference (CCC 2020)


Abstract
Can we design efficient algorithms for finding fast algorithms? This question is captured by various circuit minimization problems, and algorithms for the corresponding tasks have significant practical applications. Following the work of Cook and Levin in the early 1970s, a central question is whether minimizing the circuit size of an explicitly given function is NP-complete. While this is known to hold in restricted models such as DNFs, making progress with respect to more expressive classes of circuits has been elusive. In this work, we establish the first NP-hardness result for circuit minimization of total functions in the setting of general (unrestricted) Boolean circuits. More precisely, we show that computing the minimum circuit size of a given multi-output Boolean function f : {0,1}^n → {0,1}^m is NP-hard under many-one polynomial-time randomized reductions. Our argument builds on a simpler NP-hardness proof for the circuit minimization problem for (single-output) Boolean functions under an extended set of generators. Complementing these results, we investigate the computational hardness of minimizing communication. We establish that several variants of this problem are NP-hard under deterministic reductions. In particular, unless 𝖯 = 𝖭𝖯, no polynomial-time computable function can approximate the deterministic two-party communication complexity of a partial Boolean function up to a polynomial. This has consequences for the class of structural results that one might hope to show about the communication complexity of partial functions.

Cite as

Rahul Ilango, Bruno Loff, and Igor C. Oliveira. NP-Hardness of Circuit Minimization for Multi-Output Functions. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 22:1-22:36, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


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@InProceedings{ilango_et_al:LIPIcs.CCC.2020.22,
  author =	{Ilango, Rahul and Loff, Bruno and Oliveira, Igor C.},
  title =	{{NP-Hardness of Circuit Minimization for Multi-Output Functions}},
  booktitle =	{35th Computational Complexity Conference (CCC 2020)},
  pages =	{22:1--22:36},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-156-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{169},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.22},
  URN =		{urn:nbn:de:0030-drops-125744},
  doi =		{10.4230/LIPIcs.CCC.2020.22},
  annote =	{Keywords: MCSP, circuit minimization, communication complexity, Boolean circuit}
}
Document
Counting Induced Subgraphs: An Algebraic Approach to #W[1]-hardness

Authors: Julian Dörfler, Marc Roth, Johannes Schmitt, and Philip Wellnitz

Published in: LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)


Abstract
We study the problem #IndSub(Phi) of counting all induced subgraphs of size k in a graph G that satisfy the property Phi. This problem was introduced by Jerrum and Meeks and shown to be #W[1]-hard when parameterized by k for some families of properties Phi including, among others, connectivity [JCSS 15] and even- or oddness of the number of edges [Combinatorica 17]. Very recently [IPEC 18], two of the authors introduced a novel technique for the complexity analysis of #IndSub(Phi), inspired by the "topological approach to evasiveness" of Kahn, Saks and Sturtevant [FOCS 83] and the framework of graph motif parameters due to Curticapean, Dell and Marx [STOC 17], allowing them to prove hardness of a wide range of properties Phi. In this work, we refine this technique for graph properties that are non-trivial on edge-transitive graphs with a prime power number of edges. In particular, we fully classify the case of monotone bipartite graph properties: It is shown that, given any graph property Phi that is closed under the removal of vertices and edges, and that is non-trivial for bipartite graphs, the problem #IndSub(Phi) is #W[1]-hard and cannot be solved in time f(k)* n^{o(k)} for any computable function f, unless the Exponential Time Hypothesis fails. This holds true even if the input graph is restricted to be bipartite and counting is done modulo a fixed prime. A similar result is shown for properties that are closed under the removal of edges only.

Cite as

Julian Dörfler, Marc Roth, Johannes Schmitt, and Philip Wellnitz. Counting Induced Subgraphs: An Algebraic Approach to #W[1]-hardness. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 26:1-26:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{dorfler_et_al:LIPIcs.MFCS.2019.26,
  author =	{D\"{o}rfler, Julian and Roth, Marc and Schmitt, Johannes and Wellnitz, Philip},
  title =	{{Counting Induced Subgraphs: An Algebraic Approach to #W\lbrack1\rbrack-hardness}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{26:1--26:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-117-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{138},
  editor =	{Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.26},
  URN =		{urn:nbn:de:0030-drops-109703},
  doi =		{10.4230/LIPIcs.MFCS.2019.26},
  annote =	{Keywords: counting complexity, edge-transitive graphs, graph homomorphisms, induced subgraphs, parameterized complexity}
}
Document
Approximating the Orthogonality Dimension of Graphs and Hypergraphs

Authors: Ishay Haviv

Published in: LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)


Abstract
A t-dimensional orthogonal representation of a hypergraph is an assignment of nonzero vectors in R^t to its vertices, such that every hyperedge contains two vertices whose vectors are orthogonal. The orthogonality dimension of a hypergraph H, denoted by overline{xi}(H), is the smallest integer t for which there exists a t-dimensional orthogonal representation of H. In this paper we study computational aspects of the orthogonality dimension of graphs and hypergraphs. We prove that for every k >= 4, it is NP-hard (resp. quasi-NP-hard) to distinguish n-vertex k-uniform hypergraphs H with overline{xi}(H) <= 2 from those satisfying overline{xi}(H) >= Omega(log^delta n) for some constant delta>0 (resp. overline{xi}(H) >= Omega(log^{1-o(1)} n)). For graphs, we relate the NP-hardness of approximating the orthogonality dimension to a variant of a long-standing conjecture of Stahl. We also consider the algorithmic problem in which given a graph G with overline{xi}(G) <= 3 the goal is to find an orthogonal representation of G of as low dimension as possible, and provide a polynomial time approximation algorithm based on semidefinite programming.

Cite as

Ishay Haviv. Approximating the Orthogonality Dimension of Graphs and Hypergraphs. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 39:1-39:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{haviv:LIPIcs.MFCS.2019.39,
  author =	{Haviv, Ishay},
  title =	{{Approximating the Orthogonality Dimension of Graphs and Hypergraphs}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{39:1--39:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-117-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{138},
  editor =	{Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.39},
  URN =		{urn:nbn:de:0030-drops-109836},
  doi =		{10.4230/LIPIcs.MFCS.2019.39},
  annote =	{Keywords: orthogonal representations of hypergraphs, orthogonality dimension, hardness of approximation, Kneser and Schrijver graphs, semidefinite programming}
}
Document
On the Symmetries of and Equivalence Test for Design Polynomials

Authors: Nikhil Gupta and Chandan Saha

Published in: LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)


Abstract
In a Nisan-Wigderson design polynomial (in short, a design polynomial), every pair of monomials share a few common variables. A useful example of such a polynomial, introduced in [Neeraj Kayal et al., 2014], is the following: NW_{d,k}({x}) = sum_{h in F_d[z], deg(h) <= k}{ prod_{i=0}^{d-1}{x_{i, h(i)}}}, where d is a prime, F_d is the finite field with d elements, and k << d. The degree of the gcd of every pair of monomials in NW_{d,k} is at most k. For concreteness, we fix k = ceil[sqrt{d}]. The family of polynomials NW := {NW_{d,k} : d is a prime} and close variants of it have been used as hard explicit polynomial families in several recent arithmetic circuit lower bound proofs. But, unlike the permanent, very little is known about the various structural and algorithmic/complexity aspects of NW beyond the fact that NW in VNP. Is NW_{d,k} characterized by its symmetries? Is it circuit-testable, i.e., given a circuit C can we check efficiently if C computes NW_{d,k}? What is the complexity of equivalence test for NW, i.e., given black-box access to a f in F[{x}], can we check efficiently if there exists an invertible linear transformation A such that f = NW_{d,k}(A * {x})? Characterization of polynomials by their symmetries plays a central role in the geometric complexity theory program. Here, we answer the first two questions and partially answer the third. We show that NW_{d,k} is characterized by its group of symmetries over C, but not over R. We also show that NW_{d,k} is characterized by circuit identities which implies that NW_{d,k} is circuit-testable in randomized polynomial time. As another application of this characterization, we obtain the "flip theorem" for NW. We give an efficient equivalence test for NW in the case where the transformation A is a block-diagonal permutation-scaling matrix. The design of this algorithm is facilitated by an almost complete understanding of the group of symmetries of NW_{d,k}: We show that if A is in the group of symmetries of NW_{d,k} then A = D * P, where D and P are diagonal and permutation matrices respectively. This is proved by completely characterizing the Lie algebra of NW_{d,k}, and using an interplay between the Hessian of NW_{d,k} and the evaluation dimension.

Cite as

Nikhil Gupta and Chandan Saha. On the Symmetries of and Equivalence Test for Design Polynomials. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 53:1-53:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{gupta_et_al:LIPIcs.MFCS.2019.53,
  author =	{Gupta, Nikhil and Saha, Chandan},
  title =	{{On the Symmetries of and Equivalence Test for Design Polynomials}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{53:1--53:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-117-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{138},
  editor =	{Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.53},
  URN =		{urn:nbn:de:0030-drops-109979},
  doi =		{10.4230/LIPIcs.MFCS.2019.53},
  annote =	{Keywords: Nisan-Wigderson design polynomial, characterization by symmetries, Lie algebra, group of symmetries, circuit testability, flip theorem, equivalence test}
}
Document
Distributed Minimum Vertex Coloring and Maximum Independent Set in Chordal Graphs

Authors: Christian Konrad and Viktor Zamaraev

Published in: LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)


Abstract
We give deterministic distributed (1+epsilon)-approximation algorithms for Minimum Vertex Coloring and Maximum Independent Set on chordal graphs in the LOCAL model. Our coloring algorithm runs in O( (1 / epsilon) log n) rounds, and our independent set algorithm has a runtime of O( (1/epsilon) log(1/epsilon)log^* n) rounds. For coloring, existing lower bounds imply that the dependencies on 1/epsilon and log n are best possible. For independent set, we prove that Omega(1/epsilon) rounds are necessary. Both our algorithms make use of the tree decomposition of the input chordal graph. They iteratively peel off interval subgraphs, which are identified via the tree decomposition of the input graph, thereby partitioning the vertex set into O(log n) layers. For coloring, each interval graph is colored independently, which results in various coloring conflicts between the layers. These conflicts are then resolved in a separate phase, using the particular structure of our partitioning. For independent set, only the first O(log (1/epsilon)) layers are required as they already contain a large enough independent set. We develop a (1+epsilon)-approximation maximum independent set algorithm for interval graphs, which we then apply to those layers. This work raises the question as to how useful tree decompositions are for distributed computing.

Cite as

Christian Konrad and Viktor Zamaraev. Distributed Minimum Vertex Coloring and Maximum Independent Set in Chordal Graphs. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 21:1-21:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{konrad_et_al:LIPIcs.MFCS.2019.21,
  author =	{Konrad, Christian and Zamaraev, Viktor},
  title =	{{Distributed Minimum Vertex Coloring and Maximum Independent Set in Chordal Graphs}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{21:1--21:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-117-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{138},
  editor =	{Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.21},
  URN =		{urn:nbn:de:0030-drops-109651},
  doi =		{10.4230/LIPIcs.MFCS.2019.21},
  annote =	{Keywords: local model, approximation algorithms, minimum vertex coloring, maximum independent set, chordal graphs}
}
Document
Better Bounds for Online Line Chasing

Authors: Marcin Bienkowski, Jarosław Byrka, Marek Chrobak, Christian Coester, Łukasz Jeż, and Elias Koutsoupias

Published in: LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)


Abstract
We study online competitive algorithms for the line chasing problem in Euclidean spaces R^d, where the input consists of an initial point P_0 and a sequence of lines X_1, X_2, ..., X_m, revealed one at a time. At each step t, when the line X_t is revealed, the algorithm must determine a point P_t in X_t. An online algorithm is called c-competitive if for any input sequence the path P_0, P_1 , ..., P_m it computes has length at most c times the optimum path. The line chasing problem is a variant of a more general convex body chasing problem, where the sets X_t are arbitrary convex sets. To date, the best competitive ratio for the line chasing problem was 28.1, even in the plane. We improve this bound by providing a simple 3-competitive algorithm for any dimension d. We complement this bound by a matching lower bound for algorithms that are memoryless in the sense of our algorithm, and a lower bound of 1.5358 for arbitrary algorithms. The latter bound also improves upon the previous lower bound of sqrt{2}~=1.412 for convex body chasing in 2 dimensions.

Cite as

Marcin Bienkowski, Jarosław Byrka, Marek Chrobak, Christian Coester, Łukasz Jeż, and Elias Koutsoupias. Better Bounds for Online Line Chasing. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 8:1-8:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{bienkowski_et_al:LIPIcs.MFCS.2019.8,
  author =	{Bienkowski, Marcin and Byrka, Jaros{\l}aw and Chrobak, Marek and Coester, Christian and Je\.{z}, {\L}ukasz and Koutsoupias, Elias},
  title =	{{Better Bounds for Online Line Chasing}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{8:1--8:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-117-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{138},
  editor =	{Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.8},
  URN =		{urn:nbn:de:0030-drops-109521},
  doi =		{10.4230/LIPIcs.MFCS.2019.8},
  annote =	{Keywords: convex body chasing, line chasing, competitive analysis}
}
Document
A Sound Foundation for the Topological Approach to Task Solvability

Authors: Jérémy Ledent and Samuel Mimram

Published in: LIPIcs, Volume 140, 30th International Conference on Concurrency Theory (CONCUR 2019)


Abstract
The area of fault-tolerant distributed computability is concerned with the solvability of decision tasks in various computational models where the processes might crash. A very successful approach to prove impossibility results in this context is that of combinatorial topology, started by Herlihy and Shavit’s paper in 1999. They proved that, for wait-free protocols where the processes communicate through read/write registers, a task is solvable if and only if there exists some map between simplicial complexes satisfying some properties. This approach was then extended to many different contexts, where the processes have access to various synchronization and communication primitives. Usually, in those cases, the existence of a simplicial map from the protocol complex to the output complex is taken as the definition of what it means to solve a task. In particular, no proof is provided of the fact that this abstract topological definition agrees with a more concrete operational definition of task solvability. In this paper, we bridge this gap by proving a version of Herlihy and Shavit’s theorem that applies to any kind of object. First, we start with a very general way of specifying concurrent objects, and we define what it means to implement an object B in a computational model where the processes are allowed to communicate through shared objects A_1, ..., A_k. Then, we derive the notion of a decision task as a special case of concurrent object. Finally, we prove an analogue of Herlihy and Shavit’s theorem in this context. In particular, our version of the theorem subsumes all the uses of the combinatorial topology approach that we are aware of.

Cite as

Jérémy Ledent and Samuel Mimram. A Sound Foundation for the Topological Approach to Task Solvability. In 30th International Conference on Concurrency Theory (CONCUR 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 140, pp. 34:1-34:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{ledent_et_al:LIPIcs.CONCUR.2019.34,
  author =	{Ledent, J\'{e}r\'{e}my and Mimram, Samuel},
  title =	{{A Sound Foundation for the Topological Approach to Task Solvability}},
  booktitle =	{30th International Conference on Concurrency Theory (CONCUR 2019)},
  pages =	{34:1--34:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-121-4},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{140},
  editor =	{Fokkink, Wan and van Glabbeek, Rob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2019.34},
  URN =		{urn:nbn:de:0030-drops-109365},
  doi =		{10.4230/LIPIcs.CONCUR.2019.34},
  annote =	{Keywords: Fault-tolerant protocols, Asynchronous computability, Combinatorial topology, Protocol complex, Distributed task}
}
Document
Stronger Connections Between Circuit Analysis and Circuit Lower Bounds, via PCPs of Proximity

Authors: Lijie Chen and R. Ryan Williams

Published in: LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)


Abstract
We considerably sharpen the known connections between circuit-analysis algorithms and circuit lower bounds, show intriguing equivalences between the analysis of weak circuits and (apparently difficult) circuits, and provide strong new lower bounds for approximately computing Boolean functions with depth-two neural networks and related models. - We develop approaches to proving THR o THR lower bounds (a notorious open problem), by connecting algorithmic analysis of THR o THR to the provably weaker circuit classes THR o MAJ and MAJ o MAJ, where exponential lower bounds have long been known. More precisely, we show equivalences between algorithmic analysis of THR o THR and these weaker classes. The epsilon-error CAPP problem asks to approximate the acceptance probability of a given circuit to within additive error epsilon; it is the "canonical" derandomization problem. We show: - There is a non-trivial (2^n/n^{omega(1)} time) 1/poly(n)-error CAPP algorithm for poly(n)-size THR o THR circuits if and only if there is such an algorithm for poly(n)-size MAJ o MAJ. - There is a delta > 0 and a non-trivial SAT (delta-error CAPP) algorithm for poly(n)-size THR o THR circuits if and only if there is such an algorithm for poly(n)-size THR o MAJ. Similar results hold for depth-d linear threshold circuits and depth-d MAJORITY circuits. These equivalences are proved via new simulations of THR circuits by circuits with MAJ gates. - We strengthen the connection between non-trivial derandomization (non-trivial CAPP algorithms) for a circuit class C, and circuit lower bounds against C. Previously, [Ben-Sasson and Viola, ICALP 2014] (following [Williams, STOC 2010]) showed that for any polynomial-size class C closed under projections, non-trivial (2^{n}/n^{omega(1)} time) CAPP for OR_{poly(n)} o AND_{3} o C yields NEXP does not have C circuits. We apply Probabilistic Checkable Proofs of Proximity in a new way to show it would suffice to have a non-trivial CAPP algorithm for either XOR_2 o C, AND_2 o C or OR_2 o C. - A direct corollary of the first two bullets is that NEXP does not have THR o THR circuits would follow from either: - a non-trivial delta-error CAPP (or SAT) algorithm for poly(n)-size THR o MAJ circuits, or - a non-trivial 1/poly(n)-error CAPP algorithm for poly(n)-size MAJ o MAJ circuits. - Applying the above machinery, we extend lower bounds for depth-two neural networks and related models [R. Williams, CCC 2018] to weak approximate computations of Boolean functions. For example, for arbitrarily small epsilon > 0, we prove there are Boolean functions f computable in nondeterministic n^{log n} time such that (for infinitely many n) every polynomial-size depth-two neural network N on n inputs (with sign or ReLU activation) must satisfy max_{x in {0,1}^n}|N(x)-f(x)|>1/2-epsilon. That is, short linear combinations of ReLU gates fail miserably at computing f to within close precision. Similar results are proved for linear combinations of ACC o THR circuits, and linear combinations of low-degree F_p polynomials. These results constitute further progress towards THR o THR lower bounds.

Cite as

Lijie Chen and R. Ryan Williams. Stronger Connections Between Circuit Analysis and Circuit Lower Bounds, via PCPs of Proximity. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 19:1-19:43, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{chen_et_al:LIPIcs.CCC.2019.19,
  author =	{Chen, Lijie and Williams, R. Ryan},
  title =	{{Stronger Connections Between Circuit Analysis and Circuit Lower Bounds, via PCPs of Proximity}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{19:1--19:43},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-116-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{137},
  editor =	{Shpilka, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.19},
  URN =		{urn:nbn:de:0030-drops-108419},
  doi =		{10.4230/LIPIcs.CCC.2019.19},
  annote =	{Keywords: PCP of Proximity, Circuit Lower Bounds, Derandomization, Threshold Circuits, ReLU}
}
Document
Hardness Magnification near State-Of-The-Art Lower Bounds

Authors: Igor Carboni Oliveira, Ján Pich, and Rahul Santhanam

Published in: LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)


Abstract
This work continues the development of hardness magnification. The latter proposes a new strategy for showing strong complexity lower bounds by reducing them to a refined analysis of weaker models, where combinatorial techniques might be successful. We consider gap versions of the meta-computational problems MKtP and MCSP, where one needs to distinguish instances (strings or truth-tables) of complexity <= s_1(N) from instances of complexity >= s_2(N), and N = 2^n denotes the input length. In MCSP, complexity is measured by circuit size, while in MKtP one considers Levin’s notion of time-bounded Kolmogorov complexity. (In our results, the parameters s_1(N) and s_2(N) are asymptotically quite close, and the problems almost coincide with their standard formulations without a gap.) We establish that for Gap-MKtP[s_1,s_2] and Gap-MCSP[s_1,s_2], a marginal improvement over the state-of-the-art in unconditional lower bounds in a variety of computational models would imply explicit super-polynomial lower bounds. Theorem. There exists a universal constant c >= 1 for which the following hold. If there exists epsilon > 0 such that for every small enough beta > 0 (1) Gap-MCSP[2^{beta n}/c n, 2^{beta n}] !in Circuit[N^{1 + epsilon}], then NP !subseteq Circuit[poly]. (2) Gap-MKtP[2^{beta n}, 2^{beta n} + cn] !in TC^0[N^{1 + epsilon}], then EXP !subseteq TC^0[poly]. (3) Gap-MKtP[2^{beta n}, 2^{beta n} + cn] !in B_2-Formula[N^{2 + epsilon}], then EXP !subseteq Formula[poly]. (4) Gap-MKtP[2^{beta n}, 2^{beta n} + cn] !in U_2-Formula[N^{3 + epsilon}], then EXP !subseteq Formula[poly]. (5) Gap-MKtP[2^{beta n}, 2^{beta n} + cn] !in BP[N^{2 + epsilon}], then EXP !subseteq BP[poly]. (6) Gap-MKtP[2^{beta n}, 2^{beta n} + cn] !in (AC^0[6])[N^{1 + epsilon}], then EXP !subseteq AC^0[6]. These results are complemented by lower bounds for Gap-MCSP and Gap-MKtP against different models. For instance, the lower bound assumed in (1) holds for U_2-formulas of near-quadratic size, and lower bounds similar to (3)-(5) hold for various regimes of parameters. We also identify a natural computational model under which the hardness magnification threshold for Gap-MKtP lies below existing lower bounds: U_2-formulas that can compute parity functions at the leaves (instead of just literals). As a consequence, if one managed to adapt the existing lower bound techniques against such formulas to work with Gap-MKtP, then EXP !subseteq NC^1 would follow via hardness magnification.

Cite as

Igor Carboni Oliveira, Ján Pich, and Rahul Santhanam. Hardness Magnification near State-Of-The-Art Lower Bounds. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 27:1-27:29, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{oliveira_et_al:LIPIcs.CCC.2019.27,
  author =	{Oliveira, Igor Carboni and Pich, J\'{a}n and Santhanam, Rahul},
  title =	{{Hardness Magnification near State-Of-The-Art Lower Bounds}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{27:1--27:29},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-116-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{137},
  editor =	{Shpilka, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.27},
  URN =		{urn:nbn:de:0030-drops-108494},
  doi =		{10.4230/LIPIcs.CCC.2019.27},
  annote =	{Keywords: Circuit Complexity, Minimum Circuit Size Problem, Kolmogorov Complexity}
}
Document
Optimal Separation and Strong Direct Sum for Randomized Query Complexity

Authors: Eric Blais and Joshua Brody

Published in: LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)


Abstract
We establish two results regarding the query complexity of bounded-error randomized algorithms. Bounded-error separation theorem. There exists a total function f : {0,1}^n -> {0,1} whose epsilon-error randomized query complexity satisfies overline{R}_epsilon(f) = Omega(R(f) * log 1/epsilon). Strong direct sum theorem. For every function f and every k >= 2, the randomized query complexity of computing k instances of f simultaneously satisfies overline{R}_epsilon(f^k) = Theta(k * overline{R}_{epsilon/k}(f)). As a consequence of our two main results, we obtain an optimal superlinear direct-sum-type theorem for randomized query complexity: there exists a function f for which R(f^k) = Theta(k log k * R(f)). This answers an open question of Drucker (2012). Combining this result with the query-to-communication complexity lifting theorem of Göös, Pitassi, and Watson (2017), this also shows that there is a total function whose public-coin randomized communication complexity satisfies R^{cc}(f^k) = Theta(k log k * R^{cc}(f)), answering a question of Feder, Kushilevitz, Naor, and Nisan (1995).

Cite as

Eric Blais and Joshua Brody. Optimal Separation and Strong Direct Sum for Randomized Query Complexity. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 29:1-29:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{blais_et_al:LIPIcs.CCC.2019.29,
  author =	{Blais, Eric and Brody, Joshua},
  title =	{{Optimal Separation and Strong Direct Sum for Randomized Query Complexity}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{29:1--29:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-116-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{137},
  editor =	{Shpilka, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.29},
  URN =		{urn:nbn:de:0030-drops-108511},
  doi =		{10.4230/LIPIcs.CCC.2019.29},
  annote =	{Keywords: Decision trees, query complexity, communication complexity}
}
Document
Parity Helps to Compute Majority

Authors: Igor Carboni Oliveira, Rahul Santhanam, and Srikanth Srinivasan

Published in: LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)


Abstract
We study the complexity of computing symmetric and threshold functions by constant-depth circuits with Parity gates, also known as AC^0[oplus] circuits. Razborov [Alexander A. Razborov, 1987] and Smolensky [Roman Smolensky, 1987; Roman Smolensky, 1993] showed that Majority requires depth-d AC^0[oplus] circuits of size 2^{Omega(n^{1/2(d-1)})}. By using a divide-and-conquer approach, it is easy to show that Majority can be computed with depth-d AC^0[oplus] circuits of size 2^{O~(n^{1/(d-1)})}. This gap between upper and lower bounds has stood for nearly three decades. Somewhat surprisingly, we show that neither the upper bound nor the lower bound above is tight for large d. We show for d >= 5 that any symmetric function can be computed with depth-d AC^0[oplus] circuits of size exp(O~(n^{2/3 * 1/(d-4)})). Our upper bound extends to threshold functions (with a constant additive loss in the denominator of the double exponent). We improve the Razborov-Smolensky lower bound to show that for d >= 3 Majority requires depth-d AC^0[oplus] circuits of size 2^{Omega(n^{1/(2d-4)})}. For depths d <= 4, we are able to refine our techniques to get almost-optimal bounds: the depth-3 AC^0[oplus] circuit size of Majority is 2^{Theta~(n^{1/2})}, while its depth-4 AC^0[oplus] circuit size is 2^{Theta~(n^{1/4})}.

Cite as

Igor Carboni Oliveira, Rahul Santhanam, and Srikanth Srinivasan. Parity Helps to Compute Majority. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 23:1-23:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{oliveira_et_al:LIPIcs.CCC.2019.23,
  author =	{Oliveira, Igor Carboni and Santhanam, Rahul and Srinivasan, Srikanth},
  title =	{{Parity Helps to Compute Majority}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{23:1--23:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-116-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{137},
  editor =	{Shpilka, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.23},
  URN =		{urn:nbn:de:0030-drops-108453},
  doi =		{10.4230/LIPIcs.CCC.2019.23},
  annote =	{Keywords: Computational Complexity, Boolean Circuits, Lower Bounds, Parity, Majority}
}
Document
Fourier Bounds and Pseudorandom Generators for Product Tests

Authors: Chin Ho Lee

Published in: LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)


Abstract
We study the Fourier spectrum of functions f : {0,1}^{mk} -> {-1,0,1} which can be written as a product of k Boolean functions f_i on disjoint m-bit inputs. We prove that for every positive integer d, sum_{S subseteq [mk]: |S|=d} |hat{f_S}| = O(min{m, sqrt{m log(2k)}})^d . Our upper bounds are tight up to a constant factor in the O(*). Our proof uses Schur-convexity, and builds on a new "level-d inequality" that bounds above sum_{|S|=d} hat{f_S}^2 for any [0,1]-valued function f in terms of its expectation, which may be of independent interest. As a result, we construct pseudorandom generators for such functions with seed length O~(m + log(k/epsilon)), which is optimal up to polynomial factors in log m, log log k and log log(1/epsilon). Our generator in particular works for the well-studied class of combinatorial rectangles, where in addition we allow the bits to be read in any order. Even for this special case, previous generators have an extra O~(log(1/epsilon)) factor in their seed lengths. We also extend our results to functions f_i whose range is [-1,1].

Cite as

Chin Ho Lee. Fourier Bounds and Pseudorandom Generators for Product Tests. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 7:1-7:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{lee:LIPIcs.CCC.2019.7,
  author =	{Lee, Chin Ho},
  title =	{{Fourier Bounds and Pseudorandom Generators for Product Tests}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{7:1--7:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-116-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{137},
  editor =	{Shpilka, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.7},
  URN =		{urn:nbn:de:0030-drops-108296},
  doi =		{10.4230/LIPIcs.CCC.2019.7},
  annote =	{Keywords: bounded independence plus noise, Fourier spectrum, product test, pseudorandom generators}
}
Document
Typically-Correct Derandomization for Small Time and Space

Authors: William M. Hoza

Published in: LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)


Abstract
Suppose a language L can be decided by a bounded-error randomized algorithm that runs in space S and time n * poly(S). We give a randomized algorithm for L that still runs in space O(S) and time n * poly(S) that uses only O(S) random bits; our algorithm has a low failure probability on all but a negligible fraction of inputs of each length. As an immediate corollary, there is a deterministic algorithm for L that runs in space O(S) and succeeds on all but a negligible fraction of inputs of each length. We also give several other complexity-theoretic applications of our technique.

Cite as

William M. Hoza. Typically-Correct Derandomization for Small Time and Space. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 9:1-9:39, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{hoza:LIPIcs.CCC.2019.9,
  author =	{Hoza, William M.},
  title =	{{Typically-Correct Derandomization for Small Time and Space}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{9:1--9:39},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-116-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{137},
  editor =	{Shpilka, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.9},
  URN =		{urn:nbn:de:0030-drops-108317},
  doi =		{10.4230/LIPIcs.CCC.2019.9},
  annote =	{Keywords: Derandomization, pseudorandomness, space complexity}
}
Document
Lower Bounds for Combinatorial Algorithms for Boolean Matrix Multiplication

Authors: Debarati Das, Michal Koucký, and Michael Saks

Published in: LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)


Abstract
In this paper we propose models of combinatorial algorithms for the Boolean Matrix Multiplication (BMM), and prove lower bounds on computing BMM in these models. First, we give a relatively relaxed combinatorial model which is an extension of the model by Angluin (1976), and we prove that the time required by any algorithm for the BMM is at least Omega(n^3 / 2^{O( sqrt{ log n })}). Subsequently, we propose a more general model capable of simulating the "Four Russian Algorithm". We prove a lower bound of Omega(n^{7/3} / 2^{O(sqrt{ log n })}) for the BMM under this model. We use a special class of graphs, called (r,t)-graphs, originally discovered by Rusza and Szemeredi (1978), along with randomization, to construct matrices that are hard instances for our combinatorial models.

Cite as

Debarati Das, Michal Koucký, and Michael Saks. Lower Bounds for Combinatorial Algorithms for Boolean Matrix Multiplication. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 23:1-23:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)


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@InProceedings{das_et_al:LIPIcs.STACS.2018.23,
  author =	{Das, Debarati and Kouck\'{y}, Michal and Saks, Michael},
  title =	{{Lower Bounds for Combinatorial Algorithms for Boolean Matrix Multiplication}},
  booktitle =	{35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
  pages =	{23:1--23:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-062-0},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{96},
  editor =	{Niedermeier, Rolf and Vall\'{e}e, Brigitte},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.23},
  URN =		{urn:nbn:de:0030-drops-85050},
  doi =		{10.4230/LIPIcs.STACS.2018.23},
  annote =	{Keywords: Lower bounds, Combinatorial algorithm, Boolean matrix multiplication}
}
Document
Circuit Size Lower Bounds and #SAT Upper Bounds Through a General Framework

Authors: Alexander Golovnev, Alexander S. Kulikov, Alexander V. Smal, and Suguru Tamaki

Published in: LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)


Abstract
Most of the known lower bounds for binary Boolean circuits with unrestricted depth are proved by the gate elimination method. The most efficient known algorithms for the #SAT problem on binary Boolean circuits use similar case analyses to the ones in gate elimination. Chen and Kabanets recently showed that the known case analyses can also be used to prove average case circuit lower bounds, that is, lower bounds on the size of approximations of an explicit function. In this paper, we provide a general framework for proving worst/average case lower bounds for circuits and upper bounds for #SAT that is built on ideas of Chen and Kabanets. A proof in such a framework goes as follows. One starts by fixing three parameters: a class of circuits, a circuit complexity measure, and a set of allowed substitutions. The main ingredient of a proof goes as follows: by going through a number of cases, one shows that for any circuit from the given class, one can find an allowed substitution such that the given measure of the circuit reduces by a sufficient amount. This case analysis immediately implies an upper bound for #SAT. To~obtain worst/average case circuit complexity lower bounds one needs to present an explicit construction of a function that is a disperser/extractor for the class of sources defined by the set of substitutions under consideration. We show that many known proofs (of circuit size lower bounds and upper bounds for #SAT) fall into this framework. Using this framework, we prove the following new bounds: average case lower bounds of 3.24n and 2.59n for circuits over U_2 and B_2, respectively (though the lower bound for the basis B_2 is given for a quadratic disperser whose explicit construction is not currently known), and faster than 2^n #SAT-algorithms for circuits over U_2 and B_2 of size at most 3.24n and 2.99n, respectively. Here by B_2 we mean the set of all bivariate Boolean functions, and by U_2 the set of all bivariate Boolean functions except for parity and its complement.

Cite as

Alexander Golovnev, Alexander S. Kulikov, Alexander V. Smal, and Suguru Tamaki. Circuit Size Lower Bounds and #SAT Upper Bounds Through a General Framework. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 45:1-45:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


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@InProceedings{golovnev_et_al:LIPIcs.MFCS.2016.45,
  author =	{Golovnev, Alexander and Kulikov, Alexander S. and Smal, Alexander V. and Tamaki, Suguru},
  title =	{{Circuit Size Lower Bounds and #SAT Upper Bounds Through a General Framework}},
  booktitle =	{41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)},
  pages =	{45:1--45:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-016-3},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{58},
  editor =	{Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.45},
  URN =		{urn:nbn:de:0030-drops-64588},
  doi =		{10.4230/LIPIcs.MFCS.2016.45},
  annote =	{Keywords: circuit complexity, lower bounds, exponential time algorithms, satisfiability}
}
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