This note proves the existence of a quadratic GF(2) map p: {0, 1}n → {0, 1} such that no constant-depth circuit of size poly(n) can sample the distribution (u, p(u)) for uniform u.
{"title":"Quadratic Maps Are Hard to Sample","authors":"Emanuele Viola","doi":"10.1145/2934308","DOIUrl":"https://doi.org/10.1145/2934308","url":null,"abstract":"This note proves the existence of a quadratic GF(2) map p: {0, 1}n → {0, 1} such that no constant-depth circuit of size poly(n) can sample the distribution (u, p(u)) for uniform u.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127388295","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}
This article analyzes to what extent it is possible to efficiently reduce the number of clauses in NP-hard satisfiability problems without changing the answer. Upper and lower bounds are established using the concept of kernelization. Existing results show that if NP ⊈ coNP/poly, no efficient preprocessing algorithm can reduce n-variable instances of cnf-sat with d literals per clause to equivalent instances with O(nd−ε ) bits for any ε > 0. For the Not-All-Equal sat problem, a compression to size Õ(nd−1) exists. We put these results in a common framework by analyzing the compressibility of CSPs with a binary domain. We characterize constraint types based on the minimum degree of multivariate polynomials whose roots correspond to the satisfying assignments, obtaining (nearly) matching upper and lower bounds in several settings. Our lower bounds show that not just the number of constraints, but also the encoding size of individual constraints plays an important role. For example, for Exact Satisfiability with unbounded clause length it is possible to efficiently reduce the number of constraints to n+1, yet no polynomial-time algorithm can reduce to an equivalent instance with O(n2−ε ) bits for any ε > 0, unless NP ⊆ coNP/poly.
{"title":"Optimal Sparsification for Some Binary CSPs Using Low-Degree Polynomials","authors":"B. Jansen, A. Pieterse","doi":"10.1145/3349618","DOIUrl":"https://doi.org/10.1145/3349618","url":null,"abstract":"This article analyzes to what extent it is possible to efficiently reduce the number of clauses in NP-hard satisfiability problems without changing the answer. Upper and lower bounds are established using the concept of kernelization. Existing results show that if NP ⊈ coNP/poly, no efficient preprocessing algorithm can reduce n-variable instances of cnf-sat with d literals per clause to equivalent instances with O(nd−ε ) bits for any ε > 0. For the Not-All-Equal sat problem, a compression to size Õ(nd−1) exists. We put these results in a common framework by analyzing the compressibility of CSPs with a binary domain. We characterize constraint types based on the minimum degree of multivariate polynomials whose roots correspond to the satisfying assignments, obtaining (nearly) matching upper and lower bounds in several settings. Our lower bounds show that not just the number of constraints, but also the encoding size of individual constraints plays an important role. For example, for Exact Satisfiability with unbounded clause length it is possible to efficiently reduce the number of constraints to n+1, yet no polynomial-time algorithm can reduce to an equivalent instance with O(n2−ε ) bits for any ε > 0, unless NP ⊆ coNP/poly.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126829328","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}
I. Bliznets, Marek Cygan, Pawel Komosa, Michal Pilipczuk
The H-free Edge Deletion problem asks, for a given graph G and integer k, whether it is possible to delete at most k edges from G to make it H-free—that is, not containing H as an induced subgraph. The H-free Edge Completion problem is defined similarly, but we add edges instead of deleting them. The study of these two problem families has recently been the subject of intensive studies from the point of view of parameterized complexity and kernelization. In particular, it was shown that the problems do not admit polynomial kernels (under plausible complexity assumptions) for almost all graphs H, with several important exceptions occurring when the class of H-free graphs exhibits some structural properties. In this work, we complement the parameterized study of edge modification problems to H-free graphs by considering their approximability. We prove that whenever H is 3-connected and has at least two nonedges, then both H-free Edge Deletion and H-free Edge Completion are very hard to approximate: they do not admit poly(OPT)-approximation in polynomial time, unless P=NP, or even in time subexponential in OPT, unless the exponential time hypothesis fails. The assumption of the existence of two nonedges appears to be important: we show that whenever H is a complete graph without one edge, then H-free Edge Deletion is tightly connected to the Min Horn Deletion problem, whose approximability is still open. Finally, in an attempt to extend our hardness results beyond 3-connected graphs, we consider the cases of H being a path or a cycle, and we achieve an almost complete dichotomy there.
{"title":"Hardness of Approximation for H-free Edge Modification Problems","authors":"I. Bliznets, Marek Cygan, Pawel Komosa, Michal Pilipczuk","doi":"10.1145/3196834","DOIUrl":"https://doi.org/10.1145/3196834","url":null,"abstract":"The H-free Edge Deletion problem asks, for a given graph G and integer k, whether it is possible to delete at most k edges from G to make it H-free—that is, not containing H as an induced subgraph. The H-free Edge Completion problem is defined similarly, but we add edges instead of deleting them. The study of these two problem families has recently been the subject of intensive studies from the point of view of parameterized complexity and kernelization. In particular, it was shown that the problems do not admit polynomial kernels (under plausible complexity assumptions) for almost all graphs H, with several important exceptions occurring when the class of H-free graphs exhibits some structural properties. In this work, we complement the parameterized study of edge modification problems to H-free graphs by considering their approximability. We prove that whenever H is 3-connected and has at least two nonedges, then both H-free Edge Deletion and H-free Edge Completion are very hard to approximate: they do not admit poly(OPT)-approximation in polynomial time, unless P=NP, or even in time subexponential in OPT, unless the exponential time hypothesis fails. The assumption of the existence of two nonedges appears to be important: we show that whenever H is a complete graph without one edge, then H-free Edge Deletion is tightly connected to the Min Horn Deletion problem, whose approximability is still open. Finally, in an attempt to extend our hardness results beyond 3-connected graphs, we consider the cases of H being a path or a cycle, and we achieve an almost complete dichotomy there.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114296091","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}
V. Arvind, Frank Fuhlbrück, J. Köbler, Sebastian Kuhnert, Gaurav Rattan
In this paper we study the algorithmic complexity of the following problems: (1) Given a vertex-colored graph X = (V,E,c), compute a minimum cardinality set of vertices S⊆ V such that no nontrivial automorphism of X fixes all vertices in S. A closely related problem is computing a minimum base S for a permutation group G ≤ Sym(n) given by generators, i.e., a minimum cardinality subset S ⊆ [n] such that no nontrivial permutation in G fixes all elements of S. Our focus is mainly on the parameterized complexity of these problems. We show that when k=|S| is treated as parameter, then both problems are MINI[1]-hard. For the dual problems, where k = n – |S| is the parameter, we give FPT algorithms.(2) A notion related to fixing is individualization, which is a useful technique combined with the Weisfeiler-Leman procedure in algorithms for Graph Isomorphism. We explore the complexity of individualization: the problem of computing the minimum number of vertices we need to individualize in a given graph such that color refinement results in a graph with useful structural properties in the context of Graph Isomorphism and the Weisfeiler-Leman procedure.
{"title":"The Parameterized Complexity of Fixing Number and Vertex Individualization in Graphs","authors":"V. Arvind, Frank Fuhlbrück, J. Köbler, Sebastian Kuhnert, Gaurav Rattan","doi":"10.1145/3558077","DOIUrl":"https://doi.org/10.1145/3558077","url":null,"abstract":"In this paper we study the algorithmic complexity of the following problems: (1) Given a vertex-colored graph X = (V,E,c), compute a minimum cardinality set of vertices S⊆ V such that no nontrivial automorphism of X fixes all vertices in S. A closely related problem is computing a minimum base S for a permutation group G ≤ Sym(n) given by generators, i.e., a minimum cardinality subset S ⊆ [n] such that no nontrivial permutation in G fixes all elements of S. Our focus is mainly on the parameterized complexity of these problems. We show that when k=|S| is treated as parameter, then both problems are MINI[1]-hard. For the dual problems, where k = n – |S| is the parameter, we give FPT algorithms.(2) A notion related to fixing is individualization, which is a useful technique combined with the Weisfeiler-Leman procedure in algorithms for Graph Isomorphism. We explore the complexity of individualization: the problem of computing the minimum number of vertices we need to individualize in a given graph such that color refinement results in a graph with useful structural properties in the context of Graph Isomorphism and the Weisfeiler-Leman procedure.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"606 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116074475","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 projective dimension, a graph parameter, denoted by pd(G) for a bipartite graph G, introduced by Pudlák and Rödl (1992). For a Boolean function f (on n bits), Pudlák and Rödl associated a bipartite graph Gf and showed that size of the optimal branching program computing f, denoted by bpsize(f), is at least pd(Gf) (also denoted by pd(f)). Hence, proving lower bounds for pd(f) implies lower bounds for bpsize(f). Despite several attempts (Pudlák and Rödl (1992), Rónyai et al. (2000)), proving super-linear lower bounds for projective dimension of explicit families of graphs has remained elusive. We observe that there exist a Boolean function f for which the gap between the pd(f) and bpsize(f)) is 2Ω(n). Motivated by the argument in Pudlák and Rödl (1992), we define two variants of projective dimension: projective dimension with intersection dimension 1, denoted by upd(f), and bitwise decomposable projective dimension, denoted by bitpdim(f). We show the following results: (a) We observe that there exist a Boolean function f for which the gap between upd(f) and bpsize(f) is 2Ω(n). In contrast, we also show that the bitwise decomposable projective dimension characterizes size of the branching program up to a polynomial factor. That is, there exists a constant c > 0 and for any function f, bitpdim(f)/6 ≤ bpsize(f) ≤ (bitpdim(f))c. (b) We introduce a new candidate family of functions f for showing super-polynomial lower bounds for bitpdim(f). As our main result, for this family of functions, we demonstrate gaps between pd(f) and the above two new measures for f: pd(f) = O(√n) upd(f) = Ω (n) bitpdim(f) = Ω (n1.5 / log n). We adapt Nechiporuk’s techniques for our linear algebraic setting to prove the best-known bpsize lower bounds for bitpdim. Motivated by this linear algebraic setting of our main result, we derive exponential lower bounds for two restricted variants of pd(f) and upd(f) by observing that they are exactly equal to well-studied graph parameters—bipartite clique cover number and bipartite partition number, respectively.
{"title":"Characterization and Lower Bounds for Branching Program Size using Projective Dimension","authors":"K. Dinesh, Sajin Koroth, Jayalal Sarma","doi":"10.1145/3305274","DOIUrl":"https://doi.org/10.1145/3305274","url":null,"abstract":"We study projective dimension, a graph parameter, denoted by pd(G) for a bipartite graph G, introduced by Pudlák and Rödl (1992). For a Boolean function f (on n bits), Pudlák and Rödl associated a bipartite graph Gf and showed that size of the optimal branching program computing f, denoted by bpsize(f), is at least pd(Gf) (also denoted by pd(f)). Hence, proving lower bounds for pd(f) implies lower bounds for bpsize(f). Despite several attempts (Pudlák and Rödl (1992), Rónyai et al. (2000)), proving super-linear lower bounds for projective dimension of explicit families of graphs has remained elusive. We observe that there exist a Boolean function f for which the gap between the pd(f) and bpsize(f)) is 2Ω(n). Motivated by the argument in Pudlák and Rödl (1992), we define two variants of projective dimension: projective dimension with intersection dimension 1, denoted by upd(f), and bitwise decomposable projective dimension, denoted by bitpdim(f). We show the following results: (a) We observe that there exist a Boolean function f for which the gap between upd(f) and bpsize(f) is 2Ω(n). In contrast, we also show that the bitwise decomposable projective dimension characterizes size of the branching program up to a polynomial factor. That is, there exists a constant c > 0 and for any function f, bitpdim(f)/6 ≤ bpsize(f) ≤ (bitpdim(f))c. (b) We introduce a new candidate family of functions f for showing super-polynomial lower bounds for bitpdim(f). As our main result, for this family of functions, we demonstrate gaps between pd(f) and the above two new measures for f: pd(f) = O(√n) upd(f) = Ω (n) bitpdim(f) = Ω (n1.5 / log n). We adapt Nechiporuk’s techniques for our linear algebraic setting to prove the best-known bpsize lower bounds for bitpdim. Motivated by this linear algebraic setting of our main result, we derive exponential lower bounds for two restricted variants of pd(f) and upd(f) by observing that they are exactly equal to well-studied graph parameters—bipartite clique cover number and bipartite partition number, respectively.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125807594","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 examine the computational complexity of approximately counting the list H-colorings of a graph. We discover a natural graph-theoretic trichotomy based on the structure of the graph H. If H is an irreflexive bipartite graph or a reflexive complete graph, then counting list H-colorings is trivially in polynomial time. Otherwise, if H is an irreflexive bipartite permutation graph or a reflexive proper interval graph, then approximately counting list H-colorings is equivalent to #BIS, the problem of approximately counting independent sets in a bipartite graph. This is a well-studied problem that is believed to be of intermediate complexity—it is believed that it does not have an FPRAS, but that it is not as difficult as approximating the most difficult counting problems in #P. For every other graph H, approximately counting list H-colorings is complete for #P with respect to approximation-preserving reductions (so there is no FPRAS unless NP = RP). Two pleasing features of the trichotomy are (1) it has a natural formulation in terms of hereditary graph classes, and (2) the proof is largely self-contained and does not require any universal algebra (unlike similar dichotomies in the weighted case). We are able to extend the hardness results to the bounded-degree setting, showing that all hardness results apply to input graphs with maximum degree at most 6.
{"title":"A Complexity Trichotomy for Approximately Counting List H-Colorings","authors":"Andreas Galanis, L. A. Goldberg, M. Jerrum","doi":"10.1145/3037381","DOIUrl":"https://doi.org/10.1145/3037381","url":null,"abstract":"We examine the computational complexity of approximately counting the list H-colorings of a graph. We discover a natural graph-theoretic trichotomy based on the structure of the graph H. If H is an irreflexive bipartite graph or a reflexive complete graph, then counting list H-colorings is trivially in polynomial time. Otherwise, if H is an irreflexive bipartite permutation graph or a reflexive proper interval graph, then approximately counting list H-colorings is equivalent to #BIS, the problem of approximately counting independent sets in a bipartite graph. This is a well-studied problem that is believed to be of intermediate complexity—it is believed that it does not have an FPRAS, but that it is not as difficult as approximating the most difficult counting problems in #P. For every other graph H, approximately counting list H-colorings is complete for #P with respect to approximation-preserving reductions (so there is no FPRAS unless NP = RP). Two pleasing features of the trichotomy are (1) it has a natural formulation in terms of hereditary graph classes, and (2) the proof is largely self-contained and does not require any universal algebra (unlike similar dichotomies in the weighted case). We are able to extend the hardness results to the bounded-degree setting, showing that all hardness results apply to input graphs with maximum degree at most 6.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"249 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114479749","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}
Robert Bredereck, P. Faliszewski, R. Niedermeier, Nimrod Talmon
Given an election, a preferred candidate p, and a budget, the SHIFT BRIBERY problem asks whether p can win the election after shifting p higher in some voters’ preference orders. Of course, shifting comes at a price (depending on the voter and on the extent of the shift) and one must not exceed the given budget. We study the (parameterized) computational complexity of SHIFT BRIBERY for multiwinner voting rules where winning the election means to be part of some winning committee. We focus on the well-established SNTV, Bloc, k-Borda, and Chamberlin-Courant rules, as well as on approximate variants of the Chamberlin-Courant rule. We show that SHIFT BRIBERY tends to be harder in the multiwinner setting than in the single-winner one by showing settings where SHIFT BRIBERY is computationally easy in the single-winner cases, but is hard (and hard to approximate) in the multiwinner ones.
{"title":"Complexity of Shift Bribery in Committee Elections","authors":"Robert Bredereck, P. Faliszewski, R. Niedermeier, Nimrod Talmon","doi":"10.1145/3470647","DOIUrl":"https://doi.org/10.1145/3470647","url":null,"abstract":"Given an election, a preferred candidate p, and a budget, the SHIFT BRIBERY problem asks whether p can win the election after shifting p higher in some voters’ preference orders. Of course, shifting comes at a price (depending on the voter and on the extent of the shift) and one must not exceed the given budget. We study the (parameterized) computational complexity of SHIFT BRIBERY for multiwinner voting rules where winning the election means to be part of some winning committee. We focus on the well-established SNTV, Bloc, k-Borda, and Chamberlin-Courant rules, as well as on approximate variants of the Chamberlin-Courant rule. We show that SHIFT BRIBERY tends to be harder in the multiwinner setting than in the single-winner one by showing settings where SHIFT BRIBERY is computationally easy in the single-winner cases, but is hard (and hard to approximate) in the multiwinner ones.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125331257","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}
Affine-invariant codes are codes whose coordinates form a vector space over a finite field and which are invariant under affine transformations of the coordinate space. They form a natural, well-studied class of codes; they include popular codes such as Reed-Muller and Reed-Solomon. A particularly appealing feature of affine-invariant codes is that they seem well suited to admit local correctors and testers. In this work, we give lower bounds on the length of locally correctable and locally testable affine-invariant codes with constant query complexity. We show that if a code C ⊂ ΣKn is an r-query affine invariant locally correctable code (LCC), where K is a finite field and Σ is a finite alphabet, then the number of codewords in C is at most exp(OK,r,|Σ|(nr−1)). Also, we show that if C ⊂ ΣKn is an r-query affine invariant locally testable code (LTC), then the number of codewords in C is at most exp(OK,r,|Σ|(nr−2)). The dependence on n in these bounds is tight for constant-query LCCs/LTCs, since Guo, Kopparty, and Sudan (ITCS’13) constructed affine-invariant codes via lifting that have the same asymptotic tradeoffs. Note that our result holds for non-linear codes, whereas previously, Ben-Sasson and Sudan (RANDOM’11) assumed linearity to derive similar results. Our analysis uses higher-order Fourier analysis. In particular, we show that the codewords corresponding to an affine-invariant LCC/LTC must be far from each other with respect to Gowers norm of an appropriate order. This then allows us to bound the number of codewords, using known decomposition theorems, which approximate any bounded function in terms of a finite number of low-degree non-classical polynomials, up to a small error in the Gowers norm.
{"title":"Lower Bounds for Constant Query Affine-Invariant LCCs and LTCs","authors":"Arnab Bhattacharyya, Sivakanth Gopi","doi":"10.1145/3016802","DOIUrl":"https://doi.org/10.1145/3016802","url":null,"abstract":"Affine-invariant codes are codes whose coordinates form a vector space over a finite field and which are invariant under affine transformations of the coordinate space. They form a natural, well-studied class of codes; they include popular codes such as Reed-Muller and Reed-Solomon. A particularly appealing feature of affine-invariant codes is that they seem well suited to admit local correctors and testers. In this work, we give lower bounds on the length of locally correctable and locally testable affine-invariant codes with constant query complexity. We show that if a code C ⊂ ΣKn is an r-query affine invariant locally correctable code (LCC), where K is a finite field and Σ is a finite alphabet, then the number of codewords in C is at most exp(OK,r,|Σ|(nr−1)). Also, we show that if C ⊂ ΣKn is an r-query affine invariant locally testable code (LTC), then the number of codewords in C is at most exp(OK,r,|Σ|(nr−2)). The dependence on n in these bounds is tight for constant-query LCCs/LTCs, since Guo, Kopparty, and Sudan (ITCS’13) constructed affine-invariant codes via lifting that have the same asymptotic tradeoffs. Note that our result holds for non-linear codes, whereas previously, Ben-Sasson and Sudan (RANDOM’11) assumed linearity to derive similar results. Our analysis uses higher-order Fourier analysis. In particular, we show that the codewords corresponding to an affine-invariant LCC/LTC must be far from each other with respect to Gowers norm of an appropriate order. This then allows us to bound the number of codewords, using known decomposition theorems, which approximate any bounded function in terms of a finite number of low-degree non-classical polynomials, up to a small error in the Gowers norm.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114602727","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}
Matthew W. Anderson, Michael A. Forbes, Ramprasad Saptharishi, Amir Shpilka, Ben lee Volk
Read-k oblivious algebraic branching programs are a natural generalization of the well-studied model of read-once oblivious algebraic branching program (ABP). In this work, we give an exponential lower bound of exp (n/kO(k)) on the width of any read-k oblivious ABP computing some explicit multilinear polynomial f that is computed by a polynomial-size depth-3 circuit. We also study the polynomial identity testing (PIT) problem for this model and obtain a white-box subexponential-time PIT algorithm. The algorithm runs in time 2Õ(n1−1/2k−1) and needs white box access only to know the order in which the variables appear in the ABP.
{"title":"Identity Testing and Lower Bounds for Read-k Oblivious Algebraic Branching Programs","authors":"Matthew W. Anderson, Michael A. Forbes, Ramprasad Saptharishi, Amir Shpilka, Ben lee Volk","doi":"10.1145/3170709","DOIUrl":"https://doi.org/10.1145/3170709","url":null,"abstract":"Read-k oblivious algebraic branching programs are a natural generalization of the well-studied model of read-once oblivious algebraic branching program (ABP). In this work, we give an exponential lower bound of exp (n/kO(k)) on the width of any read-k oblivious ABP computing some explicit multilinear polynomial f that is computed by a polynomial-size depth-3 circuit. We also study the polynomial identity testing (PIT) problem for this model and obtain a white-box subexponential-time PIT algorithm. The algorithm runs in time 2Õ(n1−1/2k−1) and needs white box access only to know the order in which the variables appear in the ABP.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"75 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125802509","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}
For a graph class Π, the Π-Vertex Deletion problem has as input an undirected graph G = (V,E) and an integer k and asks whether there is a set of at most k vertices that can be deleted from G such that the resulting graph is a member of Π. By a classic result of Lewis and Yannakakis [17], Π-Vertex Deletion is NP-hard for all hereditary properties Π. We adapt the original NP-hardness construction to show that under the exponential time hypothesis (ETH), tight complexity results can be obtained. We show that Π-Vertex Deletion does not admit a 2o(n)-time algorithm where n is the number of vertices in G. We also obtain a dichotomy for running time bounds that include the number m of edges in the input graph. On the one hand, if Π contains all edgeless graphs, then there is no 2o(n+m)-time algorithm for Π-Vertex Deletion. On the other hand, if there is a fixed edgeless graph that is not contained in Π and containment in Π can be determined in 2O(n) time or 2o(m) time, then Π-Vertex Deletion can be solved in 2O(√m)+O(n) or 2o(m)+O(n) time, respectively. We also consider restrictions on the domain of the input graph G. For example, we obtain that Π-Vertex Deletion cannot be solved in 2o(√n) time if G is planar and Π is hereditary and contains and excludes infinitely many planar graphs. Finally, we provide similar results for the problem variant where the deleted vertex set has to induce a connected graph.
{"title":"Tight Running Time Lower Bounds for Vertex Deletion Problems","authors":"Christian Komusiewicz","doi":"10.1145/3186589","DOIUrl":"https://doi.org/10.1145/3186589","url":null,"abstract":"For a graph class Π, the Π-Vertex Deletion problem has as input an undirected graph G = (V,E) and an integer k and asks whether there is a set of at most k vertices that can be deleted from G such that the resulting graph is a member of Π. By a classic result of Lewis and Yannakakis [17], Π-Vertex Deletion is NP-hard for all hereditary properties Π. We adapt the original NP-hardness construction to show that under the exponential time hypothesis (ETH), tight complexity results can be obtained. We show that Π-Vertex Deletion does not admit a 2o(n)-time algorithm where n is the number of vertices in G. We also obtain a dichotomy for running time bounds that include the number m of edges in the input graph. On the one hand, if Π contains all edgeless graphs, then there is no 2o(n+m)-time algorithm for Π-Vertex Deletion. On the other hand, if there is a fixed edgeless graph that is not contained in Π and containment in Π can be determined in 2O(n) time or 2o(m) time, then Π-Vertex Deletion can be solved in 2O(√m)+O(n) or 2o(m)+O(n) time, respectively. We also consider restrictions on the domain of the input graph G. For example, we obtain that Π-Vertex Deletion cannot be solved in 2o(√n) time if G is planar and Π is hereditary and contains and excludes infinitely many planar graphs. Finally, we provide similar results for the problem variant where the deleted vertex set has to induce a connected graph.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129030857","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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