Pub Date : 2013-11-13DOI: 10.1007/978-3-662-43948-7_26
Andrew M. Childs, David Gosset, Zak Webb
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Pub Date : 2013-01-01DOI: 10.4086/toc.2013.v009a015
N. Alon, Shachar Lovett
A family of permutations in Sn is k-wise independent if a uniform permutation chosen from the family maps any sequence of k distinct elements to any sequence of k distinct elements with equal probability. Efficient constructions of k-wise independent permutations are known for k = 2 and k = 3 based on multiply transitive permutation groups but are unknown for k≥ 4. In fact, it is known that there are no nontrivial subgroups of Sn for n≥ 25 which are 4-wise independent (“4-transitive”). Faced with this obstacle, research has turned towards constructing almost k-wise independent families, where small errors are allowed. Constructions of almost k-wise independent families of permutations, with optimal size up to polynomial factors, have been achieved by several authors. Motivated by this problem, we give several results relating almost k-wise and k-wise distributions over permutations. ∗An earlier version of this paper appeared in the Proceedings of the 16th International Workshop on Randomization and Computation (RANDOM ’12), pages 350–361, 2012. †Supported in part by an ERC advanced grant and by NSF grant DMS-0835373. ‡Supported by NSF grant DMS-0835373. ACM Classification: G.3 AMS Classification: 68W20,68Q25
{"title":"Almost k-Wise vs. k-Wise Independent Permutations, and Uniformity for General Group Actions","authors":"N. Alon, Shachar Lovett","doi":"10.4086/toc.2013.v009a015","DOIUrl":"https://doi.org/10.4086/toc.2013.v009a015","url":null,"abstract":"A family of permutations in Sn is k-wise independent if a uniform permutation chosen from the family maps any sequence of k distinct elements to any sequence of k distinct elements with equal probability. Efficient constructions of k-wise independent permutations are known for k = 2 and k = 3 based on multiply transitive permutation groups but are unknown for k≥ 4. In fact, it is known that there are no nontrivial subgroups of Sn for n≥ 25 which are 4-wise independent (“4-transitive”). Faced with this obstacle, research has turned towards constructing almost k-wise independent families, where small errors are allowed. Constructions of almost k-wise independent families of permutations, with optimal size up to polynomial factors, have been achieved by several authors. Motivated by this problem, we give several results relating almost k-wise and k-wise distributions over permutations. ∗An earlier version of this paper appeared in the Proceedings of the 16th International Workshop on Randomization and Computation (RANDOM ’12), pages 350–361, 2012. †Supported in part by an ERC advanced grant and by NSF grant DMS-0835373. ‡Supported by NSF grant DMS-0835373. ACM Classification: G.3 AMS Classification: 68W20,68Q25","PeriodicalId":55992,"journal":{"name":"Theory of Computing","volume":"70 1","pages":"350-361"},"PeriodicalIF":1.0,"publicationDate":"2013-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84466639","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2012-06-15DOI: 10.4086/toc.2013.v009a024
O. Svensson
Assuming the Unique Games Conjecture, we show strong inapproximability results for two natural vertex deletion problems on directed graphs: for any integer k ≥ 2 and arbitrary small e > 0, the Feedback Vertex Set problem and the DAG Vertex Deletion problem are inapproximable within a factor k − e even on graphs where the vertices can be almost partitioned into k solutions. This gives a more structured and therefore stronger UGC-based hardness result for the Feedback Vertex Set problem that is also simpler (albeit using the “It Ain’t Over Till It’s Over” theorem) than the previous hardness result.
假设唯一游戏猜想，我们证明了有向图上的两个自然顶点删除问题的强不可逼近性结果:对于任意整数k≥2和任意小e > 0，反馈顶点集问题和DAG顶点删除问题在k−e因子内是不可逼近的，即使在顶点几乎可以划分为k个解的图上也是如此。这为反馈顶点集问题提供了一个更结构化、更强大的基于ugc的硬度结果，它也比之前的硬度结果更简单(尽管使用了“It Ain 't Over until It 's Over”定理)。
{"title":"Hardness of Vertex Deletion and Project Scheduling","authors":"O. Svensson","doi":"10.4086/toc.2013.v009a024","DOIUrl":"https://doi.org/10.4086/toc.2013.v009a024","url":null,"abstract":"Assuming the Unique Games Conjecture, we show strong inapproximability results for two natural vertex deletion problems on directed graphs: for any integer k ≥ 2 and arbitrary small e > 0, the Feedback Vertex Set problem and the DAG Vertex Deletion problem are inapproximable within a factor k − e even on graphs where the vertices can be almost partitioned into k solutions. This gives a more structured and therefore stronger UGC-based hardness result for the Feedback Vertex Set problem that is also simpler (albeit using the “It Ain’t Over Till It’s Over” theorem) than the previous hardness result.","PeriodicalId":55992,"journal":{"name":"Theory of Computing","volume":"116 1","pages":"301-312"},"PeriodicalIF":1.0,"publicationDate":"2012-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87779833","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2011-04-19DOI: 10.4086/toc.2013.v009a003
Per Austrin, M. Braverman, E. Chlamtáč
In recent work of Hazan and Krauthgamer (SICOMP 2011), it was shown that finding an e-approximate Nash equilibrium with nearoptimal value in a two-player game is as hard as finding a hidden clique of size O(log n) in the random graph G(n, 1/2). This raises the question of whether a similar intractability holds for approximate Nash equilibrium without such constraints. We give evidence that the constraint of near-optimal value makes the problem distinctly harder: a simple algorithm finds an optimal 1/2 -approximate equilibrium, while finding strictly better than 1/2 -approximate equilibria is as hard as the Hidden Clique problem. This is in contrast to the unconstrained problem where more sophisticated algorithms, achieving better approximations, are known. Unlike general Nash equilibrium, which is in PPAD, optimal (maximum value) Nash equilibrium is NP-hard. We proceed to show that optimal Nash equilibrium is just one of several known NP-hard problems related to Nash equilibrium, all of which have approximate variants which are as hard as finding a planted clique. In particular, we show this for approximate variants of the following problems: finding a Nash equilibrium with value greater than η (for any η > 0, even when the best Nash equilibrium has value 1 - η), finding a second Nash equilibrium, and finding a Nash equilibrium with small support. Finally, we consider the complexity of approximate pure Bayes Nash equilibria in two-player games. Here we show that for general Bayesian games the problem is NP-hard. For the special case where the distribution over types is uniform, we give a quasi-polynomial time algorithm matched by a hardness result based on the Hidden Clique problem.
{"title":"Inapproximability of NP-Complete Variants of Nash Equilibrium","authors":"Per Austrin, M. Braverman, E. Chlamtáč","doi":"10.4086/toc.2013.v009a003","DOIUrl":"https://doi.org/10.4086/toc.2013.v009a003","url":null,"abstract":"In recent work of Hazan and Krauthgamer (SICOMP 2011), it was shown that finding an e-approximate Nash equilibrium with nearoptimal value in a two-player game is as hard as finding a hidden clique of size O(log n) in the random graph G(n, 1/2). This raises the question of whether a similar intractability holds for approximate Nash equilibrium without such constraints. We give evidence that the constraint of near-optimal value makes the problem distinctly harder: a simple algorithm finds an optimal 1/2 -approximate equilibrium, while finding strictly better than 1/2 -approximate equilibria is as hard as the Hidden Clique problem. This is in contrast to the unconstrained problem where more sophisticated algorithms, achieving better approximations, are known. Unlike general Nash equilibrium, which is in PPAD, optimal (maximum value) Nash equilibrium is NP-hard. We proceed to show that optimal Nash equilibrium is just one of several known NP-hard problems related to Nash equilibrium, all of which have approximate variants which are as hard as finding a planted clique. In particular, we show this for approximate variants of the following problems: finding a Nash equilibrium with value greater than η (for any η > 0, even when the best Nash equilibrium has value 1 - η), finding a second Nash equilibrium, and finding a Nash equilibrium with small support. Finally, we consider the complexity of approximate pure Bayes Nash equilibria in two-player games. Here we show that for general Bayesian games the problem is NP-hard. For the special case where the distribution over types is uniform, we give a quasi-polynomial time algorithm matched by a hardness result based on the Hidden Clique problem.","PeriodicalId":55992,"journal":{"name":"Theory of Computing","volume":"25 6","pages":"13-25"},"PeriodicalIF":1.0,"publicationDate":"2011-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72458688","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2007-02-22DOI: 10.4086/toc.2009.v005a006
I. Newman, Y. Rabinovich
Hard metrics are the class of extremal metrics with respect to embedding into Euclidean Spaces: their distortion is as bad as it possibly gets, which is Ω(log n). Besides being very interesting objects akin to expanders and good codes, with rich structure of independent interest, such metrics are important for obtaining lower bounds in Combinatorial Optimization, e.g., on the value of MinCut/MaxFlow ratio for multicommodity flows. For more than a decade, a single family of hard metrics was known (see [10,3]). Recently, a different such family was found (see [8]), causing a certain excitement among the researchers in the area. In this paper we present another construction of hard metrics, different from [10,3], and more general yet clearer and simpler than [8]. Our results naturally extend to NEG and to l1.
{"title":"Hard Metrics from Cayley Graphs of Abelian Groups","authors":"I. Newman, Y. Rabinovich","doi":"10.4086/toc.2009.v005a006","DOIUrl":"https://doi.org/10.4086/toc.2009.v005a006","url":null,"abstract":"Hard metrics are the class of extremal metrics with respect to embedding into Euclidean Spaces: their distortion is as bad as it possibly gets, which is Ω(log n). Besides being very interesting objects akin to expanders and good codes, with rich structure of independent interest, such metrics are important for obtaining lower bounds in Combinatorial Optimization, e.g., on the value of MinCut/MaxFlow ratio for multicommodity flows. \u0000 \u0000For more than a decade, a single family of hard metrics was known (see [10,3]). Recently, a different such family was found (see [8]), causing a certain excitement among the researchers in the area. \u0000 \u0000In this paper we present another construction of hard metrics, different from [10,3], and more general yet clearer and simpler than [8]. Our results naturally extend to NEG and to l1.","PeriodicalId":55992,"journal":{"name":"Theory of Computing","volume":"42 1","pages":"157-162"},"PeriodicalIF":1.0,"publicationDate":"2007-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84595338","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2005-08-22DOI: 10.4086/toc.2006.v002a008
J. C. Jackson, R. Servedio
We study the average-case learnability of DNF formulas in the model of learning from uniformly distributed random examples. We define a natural model of random monotone DNF formulas and give an efficient algorithm which with high probability can learn, for any fixed constant γ>0, a random t-term monotone DNF for any t = O(n2−γ). We also define a model of random nonmonotone DNF and give an efficient algorithm which with high probability can learn a random t-term DNF for any t=O(n3/2−γ). These are the first known algorithms that can successfully learn a broad class of polynomial-size DNF in a reasonable average-case model of learning from random examples.
{"title":"On Learning Random DNF Formulas Under the Uniform Distribution","authors":"J. C. Jackson, R. Servedio","doi":"10.4086/toc.2006.v002a008","DOIUrl":"https://doi.org/10.4086/toc.2006.v002a008","url":null,"abstract":"We study the average-case learnability of DNF formulas in the model of learning from uniformly distributed random examples. We define a natural model of random monotone DNF formulas and give an efficient algorithm which with high probability can learn, for any fixed constant γ>0, a random t-term monotone DNF for any t = O(n2−γ). We also define a model of random nonmonotone DNF and give an efficient algorithm which with high probability can learn a random t-term DNF for any t=O(n3/2−γ). These are the first known algorithms that can successfully learn a broad class of polynomial-size DNF in a reasonable average-case model of learning from random examples.","PeriodicalId":55992,"journal":{"name":"Theory of Computing","volume":"26 1","pages":"342-353"},"PeriodicalIF":1.0,"publicationDate":"2005-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85157744","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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