ACM Transactions on Computation Theory最新文献

Small-Space Spectral Sparsification via Bounded-Independence Sampling 通过有界独立性采样实现小空间谱稀疏化

IF 0.7

ACM Transactions on Computation Theory Pub Date : 2024-01-19 DOI: 10.1145/3637034

Dean Doron, Jack Murtagh, Salil P. Vadhan, David Zuckerman

{"title":"Small-Space Spectral Sparsification via Bounded-Independence Sampling","authors":"Dean Doron, Jack Murtagh, Salil P. Vadhan, David Zuckerman","doi":"10.1145/3637034","DOIUrl":"https://doi.org/10.1145/3637034","url":null,"abstract":"\u0000 We give a deterministic, nearly logarithmic-space algorithm for mild spectral sparsification of undirected graphs. Given a weighted, undirected graph\u0000 G\u0000 on\u0000 n\u0000 vertices described by a binary string of length\u0000 N\u0000 , an integer\u0000 k\u0000 ≤ log \u0000 n\u0000 , and an error parameter ϵ > 0, our algorithm runs in space\u0000 \u0000 (tilde{O}(klog (Ncdot w_{mathrm{max}}/w_{mathrm{min}})) )\u0000 \u0000 where\u0000 w\u0000 max\u0000 and\u0000 w\u0000 min\u0000 are the maximum and minimum edge weights in\u0000 G\u0000 , and produces a weighted graph\u0000 H\u0000 with\u0000 \u0000 (tilde{O}(n^{1+2/k}/epsilon ^2) )\u0000 \u0000 edges that spectrally approximates\u0000 G\u0000 , in the sense of Spielmen and Teng [54], up to an error of ϵ.\u0000 \u0000 \u0000 Our algorithm is based on a new bounded-independence analysis of Spielman and Srivastava’s effective resistance based edge sampling algorithm [53] and uses results from recent work on space-bounded Laplacian solvers [43]. In particular, we demonstrate an inherent tradeoff (via upper and lower bounds) between the amount of (bounded) independence used in the edge sampling algorithm, denoted by\u0000 k\u0000 above, and the resulting sparsity that can be achieved.\u0000","PeriodicalId":44045,"journal":{"name":"ACM Transactions on Computation Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139613971","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}

引用次数: 0

Tight Sum-of-Squares lower bounds for binary polynomial optimization problems 二元多项式优化问题的紧平方和下界

ACM Transactions on Computation Theory Pub Date : 2023-10-17 DOI: 10.1145/3626106

Adam Kurpisz, Samuli Leppänen, Monaldo Mastrolilli

引用次数: 16

Optimal Polynomial-time Compression for Boolean Max CSP 布尔Max CSP的最优多项式时间压缩

ACM Transactions on Computation Theory Pub Date : 2023-09-26 DOI: 10.1145/3624704

Bart M.P. Jansen, Michał Włodarczyk

{"title":"Optimal Polynomial-time Compression for Boolean Max CSP","authors":"Bart M.P. Jansen, Michał Włodarczyk","doi":"10.1145/3624704","DOIUrl":"https://doi.org/10.1145/3624704","url":null,"abstract":"In the Boolean maximum constraint satisfaction problem – Max CSP ( Γ ) – one is given a collection of weighted applications of constraints from a finite constraint language Γ , over a common set of variables, and the goal is to assign Boolean values to the variables so that the total weight of satisfied constraints is maximized. There exists a concise dichotomy theorem providing a criterion on Γ for the problem to be polynomial-time solvable and stating that otherwise it becomes NP-hard. We study the NP-hard cases through the lens of kernelization and provide a complete characterization of Max CSP ( Γ ) with respect to the optimal compression size. Namely, we prove that Max CSP ( Γ ) parameterized by the number of variables n is either polynomial-time solvable, or there exists an integer d ≥ 2 depending on Γ , such that: (1) An instance of Max CSP ( Γ ) can be compressed into an equivalent instance with (mathcal {O}(n^dlog n) ) bits in polynomial time, (2) Max CSP( Γ ) does not admit such a compression to (mathcal {O}(n^{d-varepsilon }) ) bits unless NP⊆co-NP/poly. Our reductions are based on interpreting constraints as multilinear polynomials combined with the framework of ‘constraint implementations’, formerly used in the context of APX-hardness. As another application of our reductions, we reveal tight connections between optimal running times for solving Max CSP( Γ ) . More precisely, we show that obtaining a running time of the form (mathcal {O}(2^{(1-varepsilon)n}) ) for particular classes of Max CSP s is as hard as breaching this barrier for Max d - SAT for some d .","PeriodicalId":44045,"journal":{"name":"ACM Transactions on Computation Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134903421","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}

引用次数: 1

On p -Group Isomorphism: search-to-decision, counting-to-decision, and nilpotency class reductions via tensors 论p群同构:通过张量的搜索-决策、计数-决策和幂零类约简

ACM Transactions on Computation Theory Pub Date : 2023-09-24 DOI: 10.1145/3625308

Joshua A. Grochow, Youming Qiao

{"title":"On <i>p</i> -Group Isomorphism: search-to-decision, counting-to-decision, and nilpotency class reductions via tensors","authors":"Joshua A. Grochow, Youming Qiao","doi":"10.1145/3625308","DOIUrl":"https://doi.org/10.1145/3625308","url":null,"abstract":"In this paper we study some classical complexity-theoretic questions regarding G roup I somorphism (G p I). We focus on p -groups (groups of prime power order) with odd p , which are believed to be a bottleneck case for G p I, and work in the model of matrix groups over finite fields. Our main results are as follows. • Although search-to-decision and counting-to-decision reductions have been known for over four decades for G raph I somorphism (GI), they had remained open for G p I, explicitly asked by Arvind & Torán (Bull. EATCS, 2005). Extending methods from T ensor I somorphism (Grochow & Qiao, ITCS 2021), we show moderately exponential-time such reductions within p -groups of class 2 and exponent p . • D espite the widely held belief that p -groups of class 2 and exponent p are the hardest cases of GpI, there was no reduction to these groups from any larger class of groups. Again using methods from Tensor Isomorphism (ibid.), we show the first such reduction, namely from isomorphism testing of p -groups of “small” class and exponent p to those of class two and exponent p . For the first results, our main innovation is to develop linear-algebraic analogues of classical graph coloring gadgets, a key technique in studying the structural complexity of GI . Unlike the graph coloring gadgets, which support restricting to various subgroups of the symmetric group, the problems we study require restricting to various subgroups of the general linear group, which entails significantly different and more complicated gadgets. The analysis of one of our gadgets relies on a classical result from group theory regarding random generation of classical groups (Kantor & Lubotzky, Geom. Dedicata, 1990). For the nilpotency class reduction, we combine a runtime analysis of the Lazard correspondence with T ensor I somorphism -completeness results (Grochow & Qiao, ibid.).","PeriodicalId":44045,"journal":{"name":"ACM Transactions on Computation Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135925672","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}

引用次数: 0

Quantum communication complexity of linear regression 量子通信复杂度线性回归

ACM Transactions on Computation Theory Pub Date : 2023-09-22 DOI: 10.1145/3625225

Ashley Montanaro, Changpeng Shao

引用次数: 8

Sparsification Lower Bounds for List H -Coloring 列表H -着色的稀疏化下界

ACM Transactions on Computation Theory Pub Date : 2023-09-15 DOI: 10.1145/3612938

Hubie Chen, Bart M. P. Jansen, Karolina Okrasa, Astrid Pieterse, Paweł Rzążewski

引用次数: 1

Forgetfulness Can Make You Faster: An O*(8.097k)-Time Algorithm for Weighted 3-Set k-Packing 遗忘可以让你更快:一个O*(8.097k)时间的加权3集k-包装算法

IF 0.7

ACM Transactions on Computation Theory Pub Date : 2023-08-16 DOI: 10.1145/3599722

M. Zehavi

引用次数: 0

The Complexity Landscape of Fixed-Parameter Directed Steiner Network Problems 定参数有向Steiner网络问题的复杂性格局

ACM Transactions on Computation Theory Pub Date : 2023-06-06 DOI: 10.1145/3580376

Andreas Emil Feldmann, Daniel Marx

{"title":"The Complexity Landscape of Fixed-Parameter Directed Steiner Network Problems","authors":"Andreas Emil Feldmann, Daniel Marx","doi":"10.1145/3580376","DOIUrl":"https://doi.org/10.1145/3580376","url":null,"abstract":"Given a directed graph G and a list ( s 1 , t 1 ), …, ( s d , t d ) of terminal pairs, the Directed Steiner Network problem asks for a minimum-cost subgraph of G that contains a directed s i → t i path for every 1 ≤ i ≤ d . The special case Directed Steiner Tree (when we ask for paths from a root r to terminals t 1 , …, t d ) is known to be fixed-parameter tractable parameterized by the number of terminals, while the special case Strongly Connected Steiner Subgraph (when we ask for a path from every t i to every other t j ) is known to be W[1]-hard parameterized by the number of terminals. We systematically explore the complexity landscape of directed Steiner problems to fully understand which other special cases are FPT or W[1]-hard. Formally, if ({mathcal {H}} ) is a class of directed graphs, then we look at the special case of Directed Steiner Network where the list ( s 1 , t 1 ), …, ( s d , t d ) of demands form a directed graph that is a member of ({mathcal {H}} ) . Our main result is a complete characterization of the classes ({mathcal {H}} ) resulting in fixed-parameter tractable special cases: we show that if every pattern in ({mathcal {H}} ) has the combinatorial property of being “transitively equivalent to a bounded-length caterpillar with a bounded number of extra edges,” then the problem is FPT, and it is W[1]-hard for every recursively enumerable ({mathcal {H}} ) not having this property. This complete dichotomy unifies and generalizes the known results showing that Directed Steiner Tree is FPT [Dreyfus and Wagner, Networks 1971], q -Root Steiner Tree is FPT for constant q [Suchý, WG 2016], Strongly Connected Steiner Subgraph is W[1]-hard [Guo et al., SIAM J. Discrete Math. 2011], and Directed Steiner Network is solvable in polynomial-time for constant number of terminals [Feldman and Ruhl, SIAM J. Comput. 2006], and moreover reveals a large continent of tractable cases that were not known before.","PeriodicalId":44045,"journal":{"name":"ACM Transactions on Computation Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135494130","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}

引用次数: 0

Catalytic branching programs from groups and general protocols 组和通用协议的催化分支程序

IF 0.7

ACM Transactions on Computation Theory Pub Date : 2023-05-17 DOI: 10.1145/3583085

Hugo Côté, P. McKenzie

引用次数: 0

Linearly Ordered Colourings of Hypergraphs 超图的线性有序着色

IF 0.7

ACM Transactions on Computation Theory Pub Date : 2022-04-12 DOI: 10.1145/3570909

Tamio-Vesa Nakajima, Stanislav Živný

{"title":"Linearly Ordered Colourings of Hypergraphs","authors":"Tamio-Vesa Nakajima, Stanislav Živný","doi":"10.1145/3570909","DOIUrl":"https://doi.org/10.1145/3570909","url":null,"abstract":"A linearly ordered (LO) k-colouring of an r-uniform hypergraph assigns an integer from {1, ... , k } to every vertex so that, in every edge, the (multi)set of colours has a unique maximum. Equivalently, for r = 3, if two vertices in an edge are assigned the same colour, then the third vertex is assigned a larger colour (as opposed to a different colour, as in classic non-monochromatic colouring). Barto, Battistelli, and Berg [STACS’21] studied LO colourings on 3-uniform hypergraphs in the context of promise constraint satisfaction problems (PCSPs). We show two results. First, given a 3-uniform hypergraph that admits an LO 2-colouring, one can find in polynomial time an LO k-colouring with ( k=O(sqrt [3]{n log log n / log n} ) . Second, given an r-uniform hypergraph that admits an LO 2-colouring, we establish NP-hardness of finding an LO k-colouring for every constant uniformity r≥k+2. In fact, we determine relationships between polymorphism minions for all uniformities r≥ 3, which reveals a key difference between r< k+2 and r≥ k+2 and which may be of independent interest. Using the algebraic approach to PCSPs, we actually show a more general result establishing NP-hardness of finding an LO k-colouring for LO ℓ-colourable r-uniform hypergraphs for 2 ≤ ℓ ≤ k and r ≥ k - ℓ + 4.","PeriodicalId":44045,"journal":{"name":"ACM Transactions on Computation Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47582306","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}

引用次数: 8