Dean Doron, Jack Murtagh, Salil P. Vadhan, David Zuckerman
We give a deterministic, nearly logarithmic-space algorithm for mild spectral sparsification of undirected graphs. Given a weighted, undirected graph G on n vertices described by a binary string of length N , an integer k ≤ log n , and an error parameter ϵ > 0, our algorithm runs in space (tilde{O}(klog (Ncdot w_{mathrm{max}}/w_{mathrm{min}})) ) where w max and w min are the maximum and minimum edge weights in G , and produces a weighted graph H with (tilde{O}(n^{1+2/k}/epsilon ^2) ) edges that spectrally approximates G , in the sense of Spielmen and Teng [54], up to an error of ϵ. 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 k above, and the resulting sparsity that can be achieved.
我们给出了一种用于无向图温和谱稀疏化的确定性近对数空间算法。给定一个由长度为 N 的二进制字符串描述的 n 个顶点上的加权无向图 G、一个整数 k ≤ log n 和一个误差参数 ϵ > 0，我们的算法在空间 (tilde{O}(klog (Ncdot w_mathrm{max}}/w_{mathrm{min}})) 中运行。)，其中 w max 和 w min 是 G 中最大和最小的边权重，生成的加权图 H 带有 (tilde{O}(n^{1+2/k}/epsilon ^2) )条边，在 Spielmen 和 Teng [54] 的意义上，频谱上近似于 G，误差不超过 ϵ。 我们的算法基于对 Spielman 和 Srivastava 基于有效阻力的边缘采样算法 [53] 的新的有界独立性分析，并使用了最近关于空间有界拉普拉斯求解器 [43] 的研究成果。特别是，我们证明了边缘采样算法中使用的（有界）独立性量（以上用 k 表示）与所能达到的稀疏性之间的内在权衡（通过上下限）。
{"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}
Adam Kurpisz, Samuli Leppänen, Monaldo Mastrolilli
For binary polynomial optimization problems of degree 2 d with n variables Sakaue, Takeda, Kim and Ito [SIAM J. Optim., 2017] proved that the (lceil frac{n+2d-1}{2}rceil ) th semidefinite (SDP) relaxation in the SoS/Lasserre hierarchy of SDP relaxations provides the exact optimal value. When n is an odd number, we show that their analysis is tight, i.e. we prove that (frac{n+2d-1}{2} ) levels of the SoS/Lasserre hierarchy are also necessary. Laurent [Math. Oper. Res., 2003] showed that the Sherali-Adams hierarchy requires n levels to detect the empty integer hull of a linear representation of a set with no integral points. She conjectured that the SoS/Lasserre rank for the same problem is n − 1. In this paper we disprove this conjecture and derive lower and upper bounds for the rank.
{"title":"Tight Sum-of-Squares lower bounds for binary polynomial optimization problems","authors":"Adam Kurpisz, Samuli Leppänen, Monaldo Mastrolilli","doi":"10.1145/3626106","DOIUrl":"https://doi.org/10.1145/3626106","url":null,"abstract":"For binary polynomial optimization problems of degree 2 d with n variables Sakaue, Takeda, Kim and Ito [SIAM J. Optim., 2017] proved that the (lceil frac{n+2d-1}{2}rceil ) th semidefinite (SDP) relaxation in the SoS/Lasserre hierarchy of SDP relaxations provides the exact optimal value. When n is an odd number, we show that their analysis is tight, i.e. we prove that (frac{n+2d-1}{2} ) levels of the SoS/Lasserre hierarchy are also necessary. Laurent [Math. Oper. Res., 2003] showed that the Sherali-Adams hierarchy requires n levels to detect the empty integer hull of a linear representation of a set with no integral points. She conjectured that the SoS/Lasserre rank for the same problem is n − 1. In this paper we disprove this conjecture and derive lower and upper bounds for the rank.","PeriodicalId":44045,"journal":{"name":"ACM Transactions on Computation Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135944277","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}
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 .
{"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}
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.).
{"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}
Quantum computers may achieve speedups over their classical counterparts for solving linear algebra problems. However, in some cases – such as for low-rank matrices – dequantised algorithms demonstrate that there cannot be an exponential quantum speedup. In this work, we show that quantum computers have provable polynomial and exponential speedups in terms of communication complexity for some fundamental linear algebra problems if there is no restriction on the rank. We mainly focus on solving linear regression and Hamiltonian simulation. In the quantum case, the task is to prepare the quantum state of the result. To allow for a fair comparison, in the classical case, the task is to sample from the result. We investigate these two problems in two-party and multiparty models, propose near-optimal quantum protocols and prove quantum/classical lower bounds. In this process, we propose an efficient quantum protocol for quantum singular value transformation, which is a powerful technique for designing quantum algorithms. We feel this will be helpful in developing efficient quantum protocols for many other problems.
{"title":"Quantum communication complexity of linear regression","authors":"Ashley Montanaro, Changpeng Shao","doi":"10.1145/3625225","DOIUrl":"https://doi.org/10.1145/3625225","url":null,"abstract":"Quantum computers may achieve speedups over their classical counterparts for solving linear algebra problems. However, in some cases – such as for low-rank matrices – dequantised algorithms demonstrate that there cannot be an exponential quantum speedup. In this work, we show that quantum computers have provable polynomial and exponential speedups in terms of communication complexity for some fundamental linear algebra problems if there is no restriction on the rank. We mainly focus on solving linear regression and Hamiltonian simulation. In the quantum case, the task is to prepare the quantum state of the result. To allow for a fair comparison, in the classical case, the task is to sample from the result. We investigate these two problems in two-party and multiparty models, propose near-optimal quantum protocols and prove quantum/classical lower bounds. In this process, we propose an efficient quantum protocol for quantum singular value transformation, which is a powerful technique for designing quantum algorithms. We feel this will be helpful in developing efficient quantum protocols for many other problems.","PeriodicalId":44045,"journal":{"name":"ACM Transactions on Computation Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136062302","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}
Hubie Chen, Bart M. P. Jansen, Karolina Okrasa, Astrid Pieterse, Paweł Rzążewski
We investigate the List H -Coloring problem, the generalization of graph coloring that asks whether an input graph G admits a homomorphism to the undirected graph H (possibly with loops), such that each vertex v ∈ V ( G ) is mapped to a vertex on its list L ( v )⊆ V ( H ). An important result by Feder, Hell, and Huang [JGT 2003] states that List H -Coloring is polynomial-time solvable if H is a so-called bi-arc graph , and NP-complete otherwise. We investigate the NP-complete cases of the problem from the perspective of polynomial-time sparsification: can an n -vertex instance be efficiently reduced to an equivalent instance of bitsize (mathcal {O}(n^{2-varepsilon }) ) for some ε > 0? We prove that if H is not a bi-arc graph, then List H -Coloring does not admit such a sparsification algorithm unless ({mathsf {NP subseteq coNP/poly}} ) . Our proofs combine techniques from kernelization lower bounds with a study of the structure of graphs H which are not bi- graphs.
{"title":"Sparsification Lower Bounds for List <i>H</i> -Coloring","authors":"Hubie Chen, Bart M. P. Jansen, Karolina Okrasa, Astrid Pieterse, Paweł Rzążewski","doi":"10.1145/3612938","DOIUrl":"https://doi.org/10.1145/3612938","url":null,"abstract":"We investigate the List H -Coloring problem, the generalization of graph coloring that asks whether an input graph G admits a homomorphism to the undirected graph H (possibly with loops), such that each vertex v ∈ V ( G ) is mapped to a vertex on its list L ( v )⊆ V ( H ). An important result by Feder, Hell, and Huang [JGT 2003] states that List H -Coloring is polynomial-time solvable if H is a so-called bi-arc graph , and NP-complete otherwise. We investigate the NP-complete cases of the problem from the perspective of polynomial-time sparsification: can an n -vertex instance be efficiently reduced to an equivalent instance of bitsize (mathcal {O}(n^{2-varepsilon }) ) for some ε > 0? We prove that if H is not a bi-arc graph, then List H -Coloring does not admit such a sparsification algorithm unless ({mathsf {NP subseteq coNP/poly}} ) . Our proofs combine techniques from kernelization lower bounds with a study of the structure of graphs H which are not bi- graphs.","PeriodicalId":44045,"journal":{"name":"ACM Transactions on Computation Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135353860","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}
In this paper, we study the classic Weighted 3-Set k-Packing problem: given a universe U, a family (mathcal {S} ) of subsets of size 3 of U, a weight function (w : {mathcal {S}} rightarrow mathbb {R} ) , (W in mathbb {R} ) and a parameter (k in mathbb {N} ) , the objective is to decide if there is a subfamily ({mathcal {S}}^{prime } subseteq {mathcal {S}} ) of k disjoint sets and total weight at least W. We present a deterministic parameterized algorithm for this problem that runs in time O*(8.097k), where O* hides factors polynomial in the input size. This substantially improves upon the previously best deterministic algorithm for Weighted 3-Set k-Packing, which runs in time O*(12.155k) [SIDMA 2015], and was also the best deterministic algorithm for the unweighted version of this problem. Our algorithm is based on a novel application of the method of representative sets that might be of independent interest.
本文研究了经典的加权3集k- packing问题:给定一个域U，一个U的大小为3的子集族(mathcal {S} )，一个权函数(w : {mathcal {S}} rightarrow mathbb {R} ), (W in mathbb {R} )和一个参数(k in mathbb {N} )，目的是确定是否存在k个不相交集的子族({mathcal {S}}^{prime } subseteq {mathcal {S}} )，并且总权值至少为w。我们给出了一个确定性参数化算法，该算法运行时间为O*(8.097k)，其中O*隐藏了输入大小中的因子多项式。这大大改进了之前加权3集k-Packing的最佳确定性算法，该算法运行时间为O*(12.155k) [SIDMA 2015]，也是该问题的非加权版本的最佳确定性算法。我们的算法是基于代表集方法的一种新的应用，这可能是独立的兴趣。
{"title":"Forgetfulness Can Make You Faster: An O*(8.097k)-Time Algorithm for Weighted 3-Set k-Packing","authors":"M. Zehavi","doi":"10.1145/3599722","DOIUrl":"https://doi.org/10.1145/3599722","url":null,"abstract":"In this paper, we study the classic Weighted 3-Set k-Packing problem: given a universe U, a family (mathcal {S} ) of subsets of size 3 of U, a weight function (w : {mathcal {S}} rightarrow mathbb {R} ) , (W in mathbb {R} ) and a parameter (k in mathbb {N} ) , the objective is to decide if there is a subfamily ({mathcal {S}}^{prime } subseteq {mathcal {S}} ) of k disjoint sets and total weight at least W. We present a deterministic parameterized algorithm for this problem that runs in time O*(8.097k), where O* hides factors polynomial in the input size. This substantially improves upon the previously best deterministic algorithm for Weighted 3-Set k-Packing, which runs in time O*(12.155k) [SIDMA 2015], and was also the best deterministic algorithm for the unweighted version of this problem. Our algorithm is based on a novel application of the method of representative sets that might be of independent interest.","PeriodicalId":44045,"journal":{"name":"ACM Transactions on Computation Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45260280","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}
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.
给定一个有向图G和一个末端对的列表(s1, t1)，…，(s1, t1)，有向斯坦纳网络问题要求求G的一个最小代价子图，该子图在每1≤i≤d时包含一条有向路径s1→t1。特殊情况下的有向斯坦纳树(当我们要求从根r到终端t1，…，t d的路径时)已知是由终端数量参数化的固定参数可处理的，而特殊情况下的强连通斯坦纳子图(当我们要求从每t i到每其他t j的路径时)已知是W[1]-由终端数量硬参数化的。我们系统地探索有向斯坦纳问题的复杂性景观，以充分了解哪些其他特殊情况是FPT或W[1]-困难的。形式上，如果({mathcal {H}} )是一类有向图，那么我们看有向斯坦纳网络的特殊情况，其中需求的列表(s 1, t 1)，…，(s d, t d)构成一个有向图，该有向图是({mathcal {H}} )的成员。我们的主要结果是对类({mathcal {H}} )的完整描述，导致固定参数可处理的特殊情况:我们表明，如果({mathcal {H}} )中的每个模式都具有“传递等效于具有有限数量的额外边的有限长度的毛虫”的组合性质，那么问题是FPT，并且对于每个递归可枚举的({mathcal {H}} )都很难不具有此性质。这一完全二分类统一并推广了已知的结果，表明有向斯坦纳树是FPT [Dreyfus and Wagner, Networks 1971]， q -根斯坦纳树是常数q的FPT [Suchý， WG 2016]，强连通斯坦纳子图是W[1]-hard [Guo等，SIAM J. Discrete Math. 2011]，有向斯坦纳网络在常数终端数的多项式时间内可解[Feldman and Ruhl, SIAM J. Comput. 2006]，而且还揭示了以前不为人知的一大片可处理病例。
{"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}
CCCatalytic branching programs (catalytic bps) compute the same n-bit boolean function f at multiple entry points that need to be remembered at the exit nodes of the bp. When a doubly exponential number of entry points is allowed, linear amortized catalytic bp size is known to be achievable for any f. Here a method is introduced that produces a catalytic bp out of a reversible bp and a deterministic dag-like communication protocol. In a multiplicity range as low as linear, approximating a threshold is shown possible at linear amortized cost. In the same low range, computing (texttt {Maj} ) and (texttt {Mod} ) are shown possible at a cost that beats the brute force repetition of the best known bp for these functions by a polylog factor. In the exponential range, the method yields O(nlog n) amortized cost for any symmetric function.
{"title":"Catalytic branching programs from groups and general protocols","authors":"Hugo Côté, P. McKenzie","doi":"10.1145/3583085","DOIUrl":"https://doi.org/10.1145/3583085","url":null,"abstract":"CCCatalytic branching programs (catalytic bps) compute the same n-bit boolean function f at multiple entry points that need to be remembered at the exit nodes of the bp. When a doubly exponential number of entry points is allowed, linear amortized catalytic bp size is known to be achievable for any f. Here a method is introduced that produces a catalytic bp out of a reversible bp and a deterministic dag-like communication protocol. In a multiplicity range as low as linear, approximating a threshold is shown possible at linear amortized cost. In the same low range, computing (texttt {Maj} ) and (texttt {Mod} ) are shown possible at a cost that beats the brute force repetition of the best known bp for these functions by a polylog factor. In the exponential range, the method yields O(nlog n) amortized cost for any symmetric function.","PeriodicalId":44045,"journal":{"name":"ACM Transactions on Computation Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49359455","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}
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.
{"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}
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