L. Bartholdi, Michael Figelius, Markus Lohrey, A. Weiss
{"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}
{"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}
{"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}
{"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}
{"title":"The Complexity Landscape of Fixed-Parameter Directed Steiner Network Problems","authors":"A. Feldmann, D. Marx","doi":"10.4230/LIPIcs.ICALP.2016.27","DOIUrl":"https://doi.org/10.4230/LIPIcs.ICALP.2016.27","url":null,"abstract":"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.","PeriodicalId":44045,"journal":{"name":"ACM Transactions on Computation Theory","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2017-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43112911","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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