L. Bartholdi, Michael Figelius, Markus Lohrey, A. Weiss
We give lower bounds on the complexity of the word problem for a large class of non-solvable infinite groups that we call strongly efficiently non-solvable groups. This class includes free groups, Grigorchuk’s group, and Thompson’s groups. We prove that these groups have an NC1-hard word problem and that for some of them (including Grigorchuk’s group and Thompson’s groups) the compressed word problem (which is equivalent to the circuit evaluation problem) is PSPACE-complete.
{"title":"Groups with ALOGTIME-hard Word Problems and PSPACE-complete Compressed Word Problems","authors":"L. Bartholdi, Michael Figelius, Markus Lohrey, A. Weiss","doi":"10.1145/3569708","DOIUrl":"https://doi.org/10.1145/3569708","url":null,"abstract":"We give lower bounds on the complexity of the word problem for a large class of non-solvable infinite groups that we call strongly efficiently non-solvable groups. This class includes free groups, Grigorchuk’s group, and Thompson’s groups. We prove that these groups have an NC1-hard word problem and that for some of them (including Grigorchuk’s group and Thompson’s groups) the compressed word problem (which is equivalent to the circuit evaluation problem) is PSPACE-complete.","PeriodicalId":44045,"journal":{"name":"ACM Transactions on Computation Theory","volume":"14 1","pages":"1 - 41"},"PeriodicalIF":0.7,"publicationDate":"2019-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49495762","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 Boolean function in n variables is weakly evasive if its decision-tree complexity is (n). By k-graphs, we mean k-uniform hypergraphs. A k-graph property on v vertices is a Boolean function on n =...
{"title":"Monotone Properties of k-Uniform Hypergraphs Are Weakly Evasive","authors":"BlackTimothy","doi":"10.1145/3313908","DOIUrl":"https://doi.org/10.1145/3313908","url":null,"abstract":"A Boolean function in n variables is weakly evasive if its decision-tree complexity is (n). By k-graphs, we mean k-uniform hypergraphs. A k-graph property on v vertices is a Boolean function on n =...","PeriodicalId":44045,"journal":{"name":"ACM Transactions on Computation Theory","volume":"8 1","pages":"1-14"},"PeriodicalIF":0.7,"publicationDate":"2019-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83065583","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}
We study limitations of polynomials computed by depth-2 circuits built over read-once formulas (ROFs). In particular: We prove a 2(n) lower bound for the sum of ROFs computing the 2n-variate polyno...
{"title":"Lower bounds for Sum and Sum of Products of Read-once Formulas","authors":"RamyaC., R. Raghavendra","doi":"10.1145/3313232","DOIUrl":"https://doi.org/10.1145/3313232","url":null,"abstract":"We study limitations of polynomials computed by depth-2 circuits built over read-once formulas (ROFs). In particular: We prove a 2(n) lower bound for the sum of ROFs computing the 2n-variate polyno...","PeriodicalId":44045,"journal":{"name":"ACM Transactions on Computation Theory","volume":"18 1","pages":"1-27"},"PeriodicalIF":0.7,"publicationDate":"2019-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1145/3313232","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64017579","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}
We study the question of algebraic rank or transcendence degree preserving homomorphisms over finite fields. This concept was first introduced by Beecken et al. [3] and exploited by them, and Agrawal et al. [2] to design algebraic independence–based identity tests using the Jacobian criterion over characteristic zero fields. An analogue of such constructions over finite characteristic fields was unknown due to the failure of the Jacobian criterion over finite characteristic fields. Building on a recent criterion of Pandey et al. [15], we construct explicit faithful maps for some natural classes of polynomials in the positive characteristic field setting, when a certain parameter called the inseparable degree of the underlying polynomials is bounded (this parameter is always 1 in fields of characteristic zero). This presents the first generalisation of some of the results of Beecken et al. [3] and Agrawal et al. [2] in the positive characteristic setting.
研究有限域上代数保秩或保超越度同态的问题。这个概念首先由Beecken等人提出，并被他们利用，Agrawal等人利用特征零域上的雅可比准则设计了基于代数独立性的恒等检验。由于雅可比准则在有限特征场上的失效，这种结构在有限特征场上的模拟是未知的。在Pandey et al.[15]的最新准则的基础上，我们构造了一些自然多项式类在正特征域设置中的显式忠实映射，当某个参数称为底层多项式的不可分度是有界的(该参数在特征为零的域中总是1)。这是Beecken et al.[3]和Agrawal et al.[2]在正特征设置下的一些结果的首次推广。
{"title":"Constructing Faithful Homomorphisms over Fields of Finite Characteristic","authors":"Prerona Chatterjee, Ramprasad Saptharishi","doi":"10.1145/3580351","DOIUrl":"https://doi.org/10.1145/3580351","url":null,"abstract":"We study the question of algebraic rank or transcendence degree preserving homomorphisms over finite fields. This concept was first introduced by Beecken et al. [3] and exploited by them, and Agrawal et al. [2] to design algebraic independence–based identity tests using the Jacobian criterion over characteristic zero fields. An analogue of such constructions over finite characteristic fields was unknown due to the failure of the Jacobian criterion over finite characteristic fields. Building on a recent criterion of Pandey et al. [15], we construct explicit faithful maps for some natural classes of polynomials in the positive characteristic field setting, when a certain parameter called the inseparable degree of the underlying polynomials is bounded (this parameter is always 1 in fields of characteristic zero). This presents the first generalisation of some of the results of Beecken et al. [3] and Agrawal et al. [2] in the positive characteristic setting.","PeriodicalId":44045,"journal":{"name":"ACM Transactions on Computation Theory","volume":"15 1","pages":"1 - 19"},"PeriodicalIF":0.7,"publicationDate":"2018-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44335778","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}
Pub Date : 2017-07-21DOI: 10.4230/LIPIcs.ICALP.2016.27
A. Feldmann, D. Marx
Given a directed graph G and a list (s1, t1), …, (sd, td) of terminal pairs, the Directed Steiner Network problem asks for a minimum-cost subgraph of G that contains a directed si → ti path for every 1 ≤ i ≤ d. The special case Directed Steiner Tree (when we ask for paths from a root r to terminals t1, …, td) 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 ti to every other tj) 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 (s1, t1), …, (sd, td) 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.
给定有向图G和终端对的列表（s1，t1），…，（sd，td），有向Steiner网络问题要求G的最小代价子图包含有向si→ 每1≤i≤d的ti路径。已知特例有向施泰纳树（当我们要求从根r到终端t1，…，td的路径时）是可通过终端数量参数化的固定参数，而特殊情况下的强连通Steiner子图（当我们要求从每一个ti到每一个其他tj的路径时）已知是由终端的数量来硬参数化的W[1]。我们系统地探索了有向Steiner问题的复杂性景观，以充分理解其他哪些特殊情况是FPT或W[1]-困难的。形式上，如果（｛mathcal｛H｝｝）是一类有向图，那么我们研究有向Steiner网络的特殊情况，其中需求的列表（s1，t1），…，（sd，td）形成了一个有向图。我们的主要结果是类（｛mathcal｛H｝｝）的完全刻画，导致固定参数可处理的特殊情况：我们证明了如果（{mathcal{H｝}）中的每个模式都具有“传递等价于具有有界数量的额外边的有界长度毛虫”的组合性质，那么问题是FPT，并且对于每一个不具有此性质的递归可枚举（｛mathcal｛H｝｝），它是W[1]-硬的。这种完全二分法统一并推广了已知结果，表明有向Steiner树是FPT[Dreyfus和Wagner，Networks 1971]，q-Root Steiner树对常数q[Ssuchý，WG 2016]是FPT，强连通Steiner子图是W[1]-hard[Guo et al.，SIAM J.Discrete Math.2011]，并且对于恒定数量的终端，有向Steiner网络在多项式时间内是可解的[Feldman和Ruhl，SIAM J.Comput.2006]，并且还揭示了以前未知的大量可处理的情况。
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