This article studies property testing for NP optimization problems with parameter k under the general graph model with an augmentation of random edge sampling capability. It is shown that a variety of such problems, including k-Vertex Cover, k-Feedback Vertex Set, k-Multicut, k-Path-Free, and k-Dominating Set, are constant-query testable if k is constant. It should be noted that the first four problems are fixed parameter tractable (FPT) and it turns out that algorithmic techniques for their FPT algorithms (branch-and-bound search, color coding, etc.) are also useful for our testers. k-Dominating Set is W[2]-hard, but we can still test the property with a constant number of queries, since the definition of ε-farness makes the problem trivial for non-sparse graphs that are the source of hardness for the original optimization problem. We also consider k-Odd Cycle Transversal, which is another well-known FPT problem, but we only give a sublinear-query tester when k is a constant.
{"title":"Parameterized Testability","authors":"K. Iwama, Yuichi Yoshida","doi":"10.1145/3155294","DOIUrl":"https://doi.org/10.1145/3155294","url":null,"abstract":"This article studies property testing for NP optimization problems with parameter k under the general graph model with an augmentation of random edge sampling capability. It is shown that a variety of such problems, including k-Vertex Cover, k-Feedback Vertex Set, k-Multicut, k-Path-Free, and k-Dominating Set, are constant-query testable if k is constant. It should be noted that the first four problems are fixed parameter tractable (FPT) and it turns out that algorithmic techniques for their FPT algorithms (branch-and-bound search, color coding, etc.) are also useful for our testers. k-Dominating Set is W[2]-hard, but we can still test the property with a constant number of queries, since the definition of ε-farness makes the problem trivial for non-sparse graphs that are the source of hardness for the original optimization problem. We also consider k-Odd Cycle Transversal, which is another well-known FPT problem, but we only give a sublinear-query tester when k is a constant.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121024590","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}
R. Pallavoor, Sofya Raskhodnikova, Nithin M. Varma
We investigate the parameters in terms of which the complexity of sublinear-time algorithms should be expressed. Our goal is to find input parameters that are tailored to the combinatorics of the specific problem being studied and design algorithms that run faster when these parameters are small. This direction enables us to surpass the (worst-case) lower bounds, expressed in terms of the input size, for several problems. Our aim is to develop a similar level of understanding of the complexity of sublinear-time algorithms to the one that was enabled by research in parameterized complexity for classical algorithms. Specifically, we focus on testing properties of functions. By parameterizing the query complexity in terms of the size r of the image of the input function, we obtain testers for monotonicity and convexity of functions of the form f:[n]→ R with query complexity O (log r), with no dependence on n. The result for monotonicity circumvents the Ω (log n) lower bound by Fischer (Inf. Comput. 2004) for this problem. We present several other parameterized testers, providing compelling evidence that expressing the query complexity of property testers in terms of the input size is not always the best choice.
{"title":"Parameterized Property Testing of Functions","authors":"R. Pallavoor, Sofya Raskhodnikova, Nithin M. Varma","doi":"10.1145/3155296","DOIUrl":"https://doi.org/10.1145/3155296","url":null,"abstract":"We investigate the parameters in terms of which the complexity of sublinear-time algorithms should be expressed. Our goal is to find input parameters that are tailored to the combinatorics of the specific problem being studied and design algorithms that run faster when these parameters are small. This direction enables us to surpass the (worst-case) lower bounds, expressed in terms of the input size, for several problems. Our aim is to develop a similar level of understanding of the complexity of sublinear-time algorithms to the one that was enabled by research in parameterized complexity for classical algorithms. Specifically, we focus on testing properties of functions. By parameterizing the query complexity in terms of the size r of the image of the input function, we obtain testers for monotonicity and convexity of functions of the form f:[n]→ R with query complexity O (log r), with no dependence on n. The result for monotonicity circumvents the Ω (log n) lower bound by Fischer (Inf. Comput. 2004) for this problem. We present several other parameterized testers, providing compelling evidence that expressing the query complexity of property testers in terms of the input size is not always the best choice.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115850966","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 major goal in complexity theory is to understand the communication complexity of number-on-the-forehead problems f:({0, 1} n)k → {0, 1} with k > log n parties. We study the problems of inner product and set disjointness and determine their randomized communication complexity for every k ≥ log n, showing in both cases that Θ(1 + ⌈log n⌉/ log ⌈1 + k/ log n⌉) bits are necessary and sufficient. In particular, these problems admit constant-cost protocols if and only if the number of parties is k ≥ nε for some constant ε > 0.
复杂性理论的一个主要目标是理解数-额问题f:({0,1} n)k→{0,1}与k > log n方的通信复杂性。我们研究了每k≥log n时的内积和集合不连通问题，并确定了它们的随机通信复杂度，证明了在这两种情况下Θ(1 +≤≤log n /≤≤log 1 +≤k/≤log n /)位是充分必要的。特别地，当且仅当对于某常数ε >，这些问题承认常数成本协议。
{"title":"Inner Product and Set Disjointness","authors":"V. Podolskii, Alexander A. Sherstov","doi":"10.1145/3428671","DOIUrl":"https://doi.org/10.1145/3428671","url":null,"abstract":"A major goal in complexity theory is to understand the communication complexity of number-on-the-forehead problems f:({0, 1} n)k → {0, 1} with k > log n parties. We study the problems of inner product and set disjointness and determine their randomized communication complexity for every k ≥ log n, showing in both cases that Θ(1 + ⌈log n⌉/ log ⌈1 + k/ log n⌉) bits are necessary and sufficient. In particular, these problems admit constant-cost protocols if and only if the number of parties is k ≥ nε for some constant ε > 0.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130077370","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 : 2017-10-08DOI: 10.1145/3313904 10.1007/978-3-319-71924-5_26
G. Schoenebeck, Biaoshuai Tao
We consider the problem of maximizing the spread of influence in a social network by choosing a fixed number of initial seeds, formally referred to as the influence maximization problem. It admits a (1−1/e)-factor approximation algorithm if the influence function is submodular. Otherwise, in the worst case, the problem is NP-hard to approximate to within a factor of N1−ϵ. This article studies whether this worst-case hardness result can be circumvented by making assumptions about either the underlying network topology or the cascade model. All our assumptions are motivated by many real-life social network cascades. First, we present strong inapproximability results for a very restricted class of networks called the (stochastic) hierarchical blockmodel, a special case of the well-studied (stochastic) blockmodel in which relationships between blocks admit a tree structure. We also provide a dynamic-programming-based polynomial time algorithm, which optimally computes a directed variant of the influence maximization problem on hierarchical blockmodel networks. Our algorithm indicates that the inapproximability result is due to the bidirectionality of influence between agent-blocks. Second, we present strong inapproximability results for a class of influence functions that are “almost” submodular, called 2-quasi-submodular. Our inapproximability results hold even for any 2-quasi-submodular f fixed in advance. This result also indicates that the “threshold” between submodularity and nonsubmodularity is sharp, regarding the approximability of influence maximization.
{"title":"Beyond Worst-case (In)approximability of Nonsubmodular Influence Maximization","authors":"G. Schoenebeck, Biaoshuai Tao","doi":"10.1145/3313904 10.1007/978-3-319-71924-5_26","DOIUrl":"https://doi.org/10.1145/3313904 10.1007/978-3-319-71924-5_26","url":null,"abstract":"We consider the problem of maximizing the spread of influence in a social network by choosing a fixed number of initial seeds, formally referred to as the influence maximization problem. It admits a (1−1/e)-factor approximation algorithm if the influence function is submodular. Otherwise, in the worst case, the problem is NP-hard to approximate to within a factor of N1−ϵ. This article studies whether this worst-case hardness result can be circumvented by making assumptions about either the underlying network topology or the cascade model. All our assumptions are motivated by many real-life social network cascades. First, we present strong inapproximability results for a very restricted class of networks called the (stochastic) hierarchical blockmodel, a special case of the well-studied (stochastic) blockmodel in which relationships between blocks admit a tree structure. We also provide a dynamic-programming-based polynomial time algorithm, which optimally computes a directed variant of the influence maximization problem on hierarchical blockmodel networks. Our algorithm indicates that the inapproximability result is due to the bidirectionality of influence between agent-blocks. Second, we present strong inapproximability results for a class of influence functions that are “almost” submodular, called 2-quasi-submodular. Our inapproximability results hold even for any 2-quasi-submodular f fixed in advance. This result also indicates that the “threshold” between submodularity and nonsubmodularity is sharp, regarding the approximability of influence maximization.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116001421","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}
The Surjective H-Colouring problem is to test if a given graph allows a vertex-surjective homomorphism to a fixed graph H. The complexity of this problem has been well studied for undirected (partially) reflexive graphs. We introduce endo-triviality, the property of a structure that all of its endomorphisms that do not have range of size 1 are automorphisms, as a means to obtain complexity-theoretic classifications of Surjective H-Colouring in the case of reflexive digraphs. Chen (2014) proved, in the setting of constraint satisfaction problems, that Surjective H-Colouring is NP-complete if H has the property that all of its polymorphisms are essentially unary. We give the first concrete application of his result by showing that every endo-trivial reflexive digraph H has this property. We then use the concept of endo-triviality to prove, as our main result, a dichotomy for Surjective H-Colouring when H is a reflexive tournament: if H is transitive, then Surjective H-Colouring is in NL; otherwise, it is NP-complete. By combining this result with some known and new results, we obtain a complexity classification for Surjective H-Colouring when H is a partially reflexive digraph of size at most 3.
{"title":"Surjective H-Colouring over Reflexive Digraphs","authors":"B. Larose, B. Martin, D. Paulusma","doi":"10.1145/3282431","DOIUrl":"https://doi.org/10.1145/3282431","url":null,"abstract":"The Surjective H-Colouring problem is to test if a given graph allows a vertex-surjective homomorphism to a fixed graph H. The complexity of this problem has been well studied for undirected (partially) reflexive graphs. We introduce endo-triviality, the property of a structure that all of its endomorphisms that do not have range of size 1 are automorphisms, as a means to obtain complexity-theoretic classifications of Surjective H-Colouring in the case of reflexive digraphs. Chen (2014) proved, in the setting of constraint satisfaction problems, that Surjective H-Colouring is NP-complete if H has the property that all of its polymorphisms are essentially unary. We give the first concrete application of his result by showing that every endo-trivial reflexive digraph H has this property. We then use the concept of endo-triviality to prove, as our main result, a dichotomy for Surjective H-Colouring when H is a reflexive tournament: if H is transitive, then Surjective H-Colouring is in NL; otherwise, it is NP-complete. By combining this result with some known and new results, we obtain a complexity classification for Surjective H-Colouring when H is a partially reflexive digraph of size at most 3.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"214 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116161717","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. Ambainis, M. Kokainis, Krisjanis Prusis, J. Vihrovs
We show that all known classical adversary lower bounds on randomized query complexity are equivalent for total functions and are equal to the fractional block sensitivity fbs(f). That includes the Kolmogorov complexity bound of Laplante and Magniez and the earlier relational adversary bound of Aaronson. This equivalence also implies that for total functions, the relational adversary is equivalent to a simpler lower bound, which we call rank-1 relational adversary. For partial functions, we show unbounded separations between fbs(f) and other adversary bounds, as well as between the adversary bounds themselves. We also show that, for partial functions, fractional block sensitivity cannot give lower bounds larger than √ n ⋅ bs(f), where n is the number of variables and bs(f) is the block sensitivity. Then, we exhibit a partial function f that matches this upper bound, fbs(f) = Ω (√ n ⋅ bs(f)).
{"title":"All Classical Adversary Methods Are Equivalent for Total Functions","authors":"A. Ambainis, M. Kokainis, Krisjanis Prusis, J. Vihrovs","doi":"10.1145/3442357","DOIUrl":"https://doi.org/10.1145/3442357","url":null,"abstract":"We show that all known classical adversary lower bounds on randomized query complexity are equivalent for total functions and are equal to the fractional block sensitivity fbs(f). That includes the Kolmogorov complexity bound of Laplante and Magniez and the earlier relational adversary bound of Aaronson. This equivalence also implies that for total functions, the relational adversary is equivalent to a simpler lower bound, which we call rank-1 relational adversary. For partial functions, we show unbounded separations between fbs(f) and other adversary bounds, as well as between the adversary bounds themselves. We also show that, for partial functions, fractional block sensitivity cannot give lower bounds larger than √ n ⋅ bs(f), where n is the number of variables and bs(f) is the block sensitivity. Then, we exhibit a partial function f that matches this upper bound, fbs(f) = Ω (√ n ⋅ bs(f)).","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125294433","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 construct a pseudorandom generator that fools known-order read-k oblivious branching programs and, more generally, any linear length oblivious branching program. For polynomial width branching programs, the seed lengths in our constructions are Õ(n1−1/2k−1) (for the read-k case) and O(n/ log log n) (for the linear length case). Previously, the best construction for these models required seed length (1 − Ω(1))n.
{"title":"Pseudorandom Bits for Oblivious Branching Programs","authors":"R. Gurjar, Ben lee Volk","doi":"10.1145/3378663","DOIUrl":"https://doi.org/10.1145/3378663","url":null,"abstract":"We construct a pseudorandom generator that fools known-order read-k oblivious branching programs and, more generally, any linear length oblivious branching program. For polynomial width branching programs, the seed lengths in our constructions are Õ(n1−1/2k−1) (for the read-k case) and O(n/ log log n) (for the linear length case). Previously, the best construction for these models required seed length (1 − Ω(1))n.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126416241","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 G(V, E) (|V| = n) and a vertex s ∈ V, a weighting scheme (W : E ↦ Z+) is called a min-unique (resp. max-unique) weighting scheme if, for any vertex v of the graph G, there is a unique path of minimum (resp. maximum) weight from s to v, where weight of a path is the sum of the weights assigned to the edges. Instead, if the number of paths of minimum (resp. maximum) weight is bounded by nc for some constant c, then the weighting scheme is called a min-poly (resp. max-poly) weighting scheme. In this article, we propose an unambiguous nondeterministic log-space (UL) algorithm for the problem of testing reachability graphs augmented with a min-poly weighting scheme. This improves the result in Reinhardt and Allender [2000], in which a UL algorithm was given for the case when the weighting scheme is min-unique. Our main technique involves triple inductive counting and generalizes the techniques of Immerman [1988], Szelepcsényi [1988], and Reinhardt and Allender [2000], combined with a hashing technique due to Fredman et al. [1984] (also used in Garvin et al. [2014]). We combine this with a complementary unambiguous verification method to give the desired UL algorithm. At the other end of the spectrum, we propose a UL algorithm for testing reachability in layered DAGs augmented with max-poly weighting schemes. To achieve this, we first reduce reachability in layered DAGs to the longest path problem for DAGs with a unique source, such that the reduction also preserves the max-unique and max-poly properties of the graph. Using our techniques, we generalize the double inductive counting method in Limaye et al. [2009], in which the UL algorithm was given for the longest path problem on DAGs with a unique sink and augmented with a max-unique weighting scheme. An important consequence of our results is that, to show NL = UL, it suffices to design log-space computable min-poly (or max-poly) weighting schemes for layered DAGs.
对于一个图G(V, E) (|V| = n)和一个顶点s∈V，一个加权方案(W: E∈Z+)称为最小唯一(resp。如果对于图G的任意顶点v，存在唯一的最小路径(resp。最大)从s到v的权值，其中路径的权值是分配给边的权值之和。相反，如果最小路径数(对应的路径数)。对于某常数c，最大)权值以nc为界，则加权方案称为最小多边形。Max-poly)加权方案。在本文中，我们提出了一种无二义性的非确定性对数空间(UL)算法，用于测试最小多加权增广的可达图问题。这改进了Reinhardt和Allender[2000]的结果，其中给出了加权方案为最小唯一的情况下的UL算法。我们的主要技术涉及三重感应计数，并推广了Immerman[1988]、szelepcs[1988]和Reinhardt and Allender[2000]的技术，并结合了Fredman等人[1984]的哈希技术(Garvin等人[2014]也使用了哈希技术)。我们将其与互补的无二义性验证方法相结合，以给出所需的UL算法。在频谱的另一端，我们提出了一种UL算法，用于测试具有最大多加权方案的分层dag的可达性。为了实现这一点，我们首先将分层dag中的可达性简化为具有唯一源的dag的最长路径问题，这样的简化也保留了图的最大唯一和最大多边形属性。利用我们的技术，我们推广了Limaye等人[2009]中的双感应计数方法，其中给出了具有唯一sink的dag上最长路径问题的UL算法，并用最大唯一加权方案进行了扩充。我们的结果的一个重要结论是，为了显示NL = UL，它足以为分层dag设计对数空间可计算的最小聚(或最大聚)加权方案。
{"title":"Min/Max-Poly Weighting Schemes and the NL versus UL Problem","authors":"Anant Dhayal, Jayalal Sarma, Saurabh Sawlani","doi":"10.1145/3070902","DOIUrl":"https://doi.org/10.1145/3070902","url":null,"abstract":"For a graph G(V, E) (|V| = n) and a vertex s ∈ V, a weighting scheme (W : E ↦ Z+) is called a min-unique (resp. max-unique) weighting scheme if, for any vertex v of the graph G, there is a unique path of minimum (resp. maximum) weight from s to v, where weight of a path is the sum of the weights assigned to the edges. Instead, if the number of paths of minimum (resp. maximum) weight is bounded by nc for some constant c, then the weighting scheme is called a min-poly (resp. max-poly) weighting scheme. In this article, we propose an unambiguous nondeterministic log-space (UL) algorithm for the problem of testing reachability graphs augmented with a min-poly weighting scheme. This improves the result in Reinhardt and Allender [2000], in which a UL algorithm was given for the case when the weighting scheme is min-unique. Our main technique involves triple inductive counting and generalizes the techniques of Immerman [1988], Szelepcsényi [1988], and Reinhardt and Allender [2000], combined with a hashing technique due to Fredman et al. [1984] (also used in Garvin et al. [2014]). We combine this with a complementary unambiguous verification method to give the desired UL algorithm. At the other end of the spectrum, we propose a UL algorithm for testing reachability in layered DAGs augmented with max-poly weighting schemes. To achieve this, we first reduce reachability in layered DAGs to the longest path problem for DAGs with a unique source, such that the reduction also preserves the max-unique and max-poly properties of the graph. Using our techniques, we generalize the double inductive counting method in Limaye et al. [2009], in which the UL algorithm was given for the longest path problem on DAGs with a unique sink and augmented with a max-unique weighting scheme. An important consequence of our results is that, to show NL = UL, it suffices to design log-space computable min-poly (or max-poly) weighting schemes for layered DAGs.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"63 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124427959","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 present a general framework for balancing expressions (terms) in the form of so-called tree straight-line programs. The latter can be seen as circuits over the free term algebra extended by contexts (terms with a hole) and the operations, which insert terms/contexts into contexts. In Ref. [16], it was shown that one can compute for a given term of size n in logspace a tree straight-line program of depth O(log n) and size O(n/ log n). In the present article, it is shown that the conversion can be done in DLOGTIME-uniform TC0. This allows reducing the term evaluation problem over an arbitrary algebra A to the term evaluation problem over a derived two-sorted algebra F (A). Three applications are presented: (i) an alternative proof for a recent result by Krebs et al. [25] on the expression evaluation problem is given; (ii) it is shown that expressions for an arbitrary (possibly non-commutative) semiring can be transformed in DLOGTIME-uniform TC0 into equivalent circuits of logarithmic depth and size O(n/ log n); and, (iii) a corresponding result for regular expressions is shown.
{"title":"A Universal Tree Balancing Theorem","authors":"Moses Ganardi, Markus Lohrey","doi":"10.1145/3278158","DOIUrl":"https://doi.org/10.1145/3278158","url":null,"abstract":"We present a general framework for balancing expressions (terms) in the form of so-called tree straight-line programs. The latter can be seen as circuits over the free term algebra extended by contexts (terms with a hole) and the operations, which insert terms/contexts into contexts. In Ref. [16], it was shown that one can compute for a given term of size n in logspace a tree straight-line program of depth O(log n) and size O(n/ log n). In the present article, it is shown that the conversion can be done in DLOGTIME-uniform TC0. This allows reducing the term evaluation problem over an arbitrary algebra A to the term evaluation problem over a derived two-sorted algebra F (A). Three applications are presented: (i) an alternative proof for a recent result by Krebs et al. [25] on the expression evaluation problem is given; (ii) it is shown that expressions for an arbitrary (possibly non-commutative) semiring can be transformed in DLOGTIME-uniform TC0 into equivalent circuits of logarithmic depth and size O(n/ log n); and, (iii) a corresponding result for regular expressions is shown.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134575991","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 sampling procedure for a distribution P over {0, 1}ℓ is a function C: {0, 1}n → {0, 1}ℓ such that the distribution C(Un) (obtained by applying C on the uniform distribution Un) is the “desired distribution” P. Let n > r ≥ ℓ = nΩ(1). An ε-nb-PRG (defined by Dubrov and Ishai [2006]) is a function G: {0, 1}r → {0, 1}n such that for every C: {0, 1}n → {0, 1}ℓ in some class of “interesting sampling procedures,” C′(Ur) = C(G(Ur)) is ε-close to C(Un) in statistical distance. We construct poly-time computable nb-PRGs with r = O(ℓ) for poly-size circuits relying on the assumption that there exists β > 0 and a problem L in E = DTIME(2O(n)) such that for every large enough n, nondeterministic circuits of size 2βn that have NP-gates cannot solve L on inputs of lengthn. This assumption is a scaled nonuniform analog of (the widely believed) EXP ≠ Σ2P, and similar assumptions appear in various contexts in derandomization. Previous nb-PRGs of Dubrov and Ishai have r = Ω(ℓ2) and are based on very strong cryptographic assumptions or, alternatively, on nonstandard assumptions regarding incompressibility of functions on random inputs. When restricting to poly-size circuits C: {0, 1}n → {0, 1}ℓ with Shannon entropy H(C(Un)) ⩽ k, for ℓ > k = nΩ(1), our nb-PRGs have r = O(k). The nb-PRGs of Dubrov and Ishai use seed length r = Ω(k2) and require that the probability distribution of C(Un) is efficiently computable. Our nb-PRGs follow from a notion of “conditional PRGs,” which may be of independent interest. These are PRGs where G(Ur) remains pseudorandom even when conditioned on a “large” event {A(G(Ur)) = 1}, for an arbitrary poly-size circuit A. A related notion was considered by Shaltiel and Umans [2005] in a different setting, and our proofs use ideas from that paper, as well as ideas of Dubrov and Ishai. We also give an unconditional construction of poly-time computable nb-PRGs for poly(n)-size, depth d circuits C: {0, 1}n → {0, 1}ℓ with r = O(ℓ · log d + O(1)n). This improves upon the previous work of Dubrov and Ishai that has r ≥ ℓ2. This result follows by adapting a recent PRG construction of Trevisan and Xue [2013] to the case of nb-PRGs. We also show that this PRG can be implemented by a uniform family of constant-depth circuits with slightly increased seed length.
{"title":"Pseudorandom Generators with Optimal Seed Length for Non-Boolean Poly-Size Circuits","authors":"Sergei Artemenko, Ronen Shaltiel","doi":"10.1145/3018057","DOIUrl":"https://doi.org/10.1145/3018057","url":null,"abstract":"A sampling procedure for a distribution P over {0, 1}ℓ is a function C: {0, 1}n → {0, 1}ℓ such that the distribution C(Un) (obtained by applying C on the uniform distribution Un) is the “desired distribution” P. Let n > r ≥ ℓ = nΩ(1). An ε-nb-PRG (defined by Dubrov and Ishai [2006]) is a function G: {0, 1}r → {0, 1}n such that for every C: {0, 1}n → {0, 1}ℓ in some class of “interesting sampling procedures,” C′(Ur) = C(G(Ur)) is ε-close to C(Un) in statistical distance. We construct poly-time computable nb-PRGs with r = O(ℓ) for poly-size circuits relying on the assumption that there exists β > 0 and a problem L in E = DTIME(2O(n)) such that for every large enough n, nondeterministic circuits of size 2βn that have NP-gates cannot solve L on inputs of lengthn. This assumption is a scaled nonuniform analog of (the widely believed) EXP ≠ Σ2P, and similar assumptions appear in various contexts in derandomization. Previous nb-PRGs of Dubrov and Ishai have r = Ω(ℓ2) and are based on very strong cryptographic assumptions or, alternatively, on nonstandard assumptions regarding incompressibility of functions on random inputs. When restricting to poly-size circuits C: {0, 1}n → {0, 1}ℓ with Shannon entropy H(C(Un)) ⩽ k, for ℓ > k = nΩ(1), our nb-PRGs have r = O(k). The nb-PRGs of Dubrov and Ishai use seed length r = Ω(k2) and require that the probability distribution of C(Un) is efficiently computable. Our nb-PRGs follow from a notion of “conditional PRGs,” which may be of independent interest. These are PRGs where G(Ur) remains pseudorandom even when conditioned on a “large” event {A(G(Ur)) = 1}, for an arbitrary poly-size circuit A. A related notion was considered by Shaltiel and Umans [2005] in a different setting, and our proofs use ideas from that paper, as well as ideas of Dubrov and Ishai. We also give an unconditional construction of poly-time computable nb-PRGs for poly(n)-size, depth d circuits C: {0, 1}n → {0, 1}ℓ with r = O(ℓ · log d + O(1)n). This improves upon the previous work of Dubrov and Ishai that has r ≥ ℓ2. This result follows by adapting a recent PRG construction of Trevisan and Xue [2013] to the case of nb-PRGs. We also show that this PRG can be implemented by a uniform family of constant-depth circuits with slightly increased seed length.","PeriodicalId":198744,"journal":{"name":"ACM Transactions on Computation Theory (TOCT)","volume":"111 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124699520","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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