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Beating the Random Ordering Is Hard: Every Ordering CSP Is Approximation Resistant

Venkatesan Guruswami, Johan HÅstad, Rajsekar Manokaran, Prasad Raghavendra, Moses Charikar
2011 SIAM journal on computing (Print)  
We prove that, assuming the Unique Games conjecture (UGC), every problem in the class of ordering constraint satisfaction problems (OCSPs) where each constraint has constant arity is approximation resistant  ...  Conditioned on the UGC, for every C > 0, it is NP-hard to find a C-approximation to the FAS problem.  ...  We would like to thank Alantha Newman for pointing out the similarity between MAS and CSPs, and for suggesting the problem. We would like to thank Sanjeev Arora for many fruitful discussions.  ... 
doi:10.1137/090756144 fatcat:ndcwrnfp2zbsbj5gnahqvmo73u

Beating the Random Ordering is Hard: Inapproximability of Maximum Acyclic Subgraph

Venkatesan Guruswami, Rajsekar Manokaran, Prasad Raghavendra
2008 2008 49th Annual IEEE Symposium on Foundations of Computer Science  
We prove that approximating the Max Acyclic Subgraph problem within a factor better than 1/2 is Unique-Games hard.  ...  The existence of a ρ-approximation algorithm for ρ > 1/2 has been a basic open problem for a while. Our result is the first tight inapproximability result for an ordering problem.  ...  Acknowledgments We would like to thank Alantha Newman for pointing out the similarity between Maximum Acyclic Subgraph and CSPs, and suggesting the problem.  ... 
doi:10.1109/focs.2008.51 dblp:conf/focs/GuruswamiMR08 fatcat:rh2dzlq2wbayfizmfdpcbdwa7e

Cryptographic Hardness of Random Local Functions–Survey [chapter]

Benny Applebaum
2013 Lecture Notes in Computer Science  
In this work, we will study the cryptographic hardness of random local functions.  ...  A natural way to obtain local cryptographic constructions is to use random local functions in which each output bit is computed by applying some fixed d-ary predicate P to a randomly chosen d-size subset  ...  The author is also grateful to Boaz Barak, Andrej Bogdanov, Uri Feige, Oded Goldreich, Yuval Ishai, Eyal Kushilevitz, Ryan O'Donnell, Alon Rosen, Dan Vilenchik, and Avi Wigderson for stimulating discussions  ... 
doi:10.1007/978-3-642-36594-2_33 fatcat:cnghkimszra2ljkp7ersxiwgoq

Tight Hardness for Shortest Cycles and Paths in Sparse Graphs [article]

Andrea Lincoln, Virginia Vassilevska Williams, Ryan Williams
2020 arXiv   pre-print
That is, we prove hardness for a variety of sparse graph problems from the hardness of a dense graph problem.  ...  Starting from the hypothesis that the minimum weight (2ℓ+1)-Clique problem in edge weighted graphs requires n^2ℓ+1-o(1) time, we prove that for all sparsities of the form m = Θ(n^1+1/ℓ), there is no O(  ...  We would like to thank the anonymous reviewers whose suggestions we implemented. We thank Pawel Gawrychowski for pointing out a typo in a previous version of the paper.  ... 
arXiv:1712.08147v4 fatcat:4kmi3ia3d5cmriclwvyj6oynwa

Every Permutation CSP of arity 3 is Approximation Resistant

Moses Charikar, Venkatesan Guruswami, Rajsekar Manokaran
2009 2009 24th Annual IEEE Conference on Computational Complexity  
Thus, every permutation CSP of arity up to 3 resists approximation beyond the trivial random ordering threshold.  ...  Building on this work, in this paper we show that for every permCSP of arity 3, beating the random ordering is Unique-Games hard.  ...  Building on this work, in this paper we show that for every permCSP of arity 3, beating the random ordering is Unique-Games hard.  ... 
doi:10.1109/ccc.2009.29 dblp:conf/coco/CharikarGM09 fatcat:paxbuwhb2nam7lmxcr7ntluw5m

On the Hardness of Approximating Balanced Homogenous 3-Lin

Johan Håstad, Rajsekar Manokaran
2017 Theory of Computing  
We prove that it is hard to distinguish systems where there is a balanced assignment that satisfies a fraction 1 − ε of the equations from systems where the best balanced assignment satisfies a fraction  ...  The key for the improvement is to replace long codes used by Holmerin and Khot by the low-degree long code.  ...  We are grateful to three referees for a very careful reading of a preliminary version of the paper and in particular for pointing out a difference in two probability distributions claimed to be the same  ... 
doi:10.4086/toc.2017.v013a010 dblp:journals/toc/HastadM17 fatcat:bxo25zdlffdahl642bvk5ll2ti

Approximating Bounded Occurrence Ordering CSPs [chapter]

Venkatesan Guruswami, Yuan Zhou
2012 Lecture Notes in Computer Science  
It was shown recently that without the bounded occurrence restriction, for every ordering CSP it is Unique Games-hard to beat the naive random ordering algorithm.  ...  algorithm, in the sense that one can beat (by an additive constant) the approximation ratio achieved by the naive algorithm that simply picks a random assignment.  ...  This hardness result was generalized to all ordering CSPs of arity 3 [4] , and later to higher arities, showing that every ordering CSP is approximation resistant (under the UGC) [3] !  ... 
doi:10.1007/978-3-642-32512-0_14 fatcat:xcprjhjmlbhkzdeulp542aln3e

Tight Hardness for Shortest Cycles and Paths in Sparse Graphs [chapter]

Andrea Lincoln, Virginia Vassilevska Williams, Ryan Williams
2018 Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms  
That is, we prove hardness for a variety of sparse graph problems from the hardness of a dense graph problem.  ...  Starting from the hypothesis that the minimum weight (2 + 1)-Clique problem in edge weighted graphs requires n 2 +1−o(1) time, we prove that for all sparsities of the form m = Θ(n 1+1/ ), there is no O  ...  We would like to thank the anonymous reviewers whose suggestions we implemented.  ... 
doi:10.1137/1.9781611975031.80 dblp:conf/soda/LincolnWW18 fatcat:7boftjghqrbsxe4baz736vkjty

Local Search is Better than Random Assignment for Bounded Occurrence Ordering k-CSPs [article]

Konstantin Makarychev
2013 arXiv   pre-print
We prove that the Bounded Occurrence Ordering k-CSP Problem is not approximation resistant.  ...  The question whether bounded occurrence ordering k-CSPs are approximation resistant was raised by Guruswami and Zhou (APPROX 2012) who recently showed that bounded occurrence 3-CSPs and "monotone" k-CSPs  ...  Håstad (1997) showed that for some CSPs e.g., 3LIN-2 and E3-SAT, beating the approximation ratio of the random assignment algorithm (by any positive constant ε) is NP-hard.  ... 
arXiv:1210.1890v2 fatcat:if5gfbunwjemrowa76qwljcqse

Streaming approximation resistance of every ordering CSP [article]

Noah Singer and Madhu Sudan and Santhoshini Velusamy
2021 arXiv   pre-print
We show that for every Π, OCSP(Π) is approximation-resistant to o(n)-space streaming algorithms. This space bound is tight up to polylogarithmic factors.  ...  In the case of MAS our result shows that for every ϵ>0, MAS is not 1/2+ϵ-approximable in o(n) space. The previous best inapproximability result only ruled out a 3/4-approximation in o(√(n)) space.  ...  (Π) is approximation-resistant.  ... 
arXiv:2105.01782v2 fatcat:es3zee5cyra4jcslyqntomrgve

Is constraint satisfaction over two variables always easy?

Lars Engebretsen, Venkatesan Guruswami
2004 Random structures & algorithms (Print)  
not approximation resistant, and, secondly, that the Not-All-Equal Sat problem over domain size d with three variables per constraint, is approximation resistant, for every d ≥ 3.  ...  ACM 42(6):1115-1145), we know that every Boolean 2-CSP has a non-trivial approximation algorithm whose performance ratio is better than that obtained by picking a random assignment to the variables.  ...  We reduce Max E3-Sat(d), which is hard to approximate within (1 − d −3 + dε) with perfect completeness by Theorem 4 applied to G = Z d combined with the discussion in § 2, to Max E2-Sat(d).  ... 
doi:10.1002/rsa.20026 fatcat:zs5f2vbb2jeappxsisyq5d44ga

Is Constraint Satisfaction Over Two Variables Always Easy? [chapter]

Lars Engebretsen, Venkatesan Guruswami
2002 Lecture Notes in Computer Science  
not approximation resistant, and, secondly, that the Not-All-Equal Sat problem over domain size d with three variables per constraint, is approximation resistant, for every d ≥ 3.  ...  ACM 42(6):1115-1145), we know that every Boolean 2-CSP has a non-trivial approximation algorithm whose performance ratio is better than that obtained by picking a random assignment to the variables.  ...  We reduce Max E3-Sat(d), which is hard to approximate within (1 − d −3 + dε) with perfect completeness by Theorem 4 applied to G = Z d combined with the discussion in § 2, to Max E2-Sat(d).  ... 
doi:10.1007/3-540-45726-7_18 fatcat:soq24wkg5zbitk5v4costjq2s4

On the optimality of semidefinite relaxations for average-case and generalized constraint satisfaction

Boaz Barak, Guy Kindler, David Steurer
2013 Proceedings of the 4th conference on Innovations in Theoretical Computer Science - ITCS '13  
We show that it is NP-hard to approximate the Generalized CSP for P better than this guarantee.  ...  Raghavendra (2008) showed that it is Unique Games-hard to approximate the CSP for P better than this guarantee.  ...  In particular he showed that for every CSP it is Unique Games-hard to beat the approximation guarantee achieved by a simple semidefinite programming relaxation known as Basic Sdp.  ... 
doi:10.1145/2422436.2422460 dblp:conf/innovations/BarakKS13 fatcat:k3omfqtzq5h7hgx5p3qpzabgpi

Approximating Linear Threshold Predicates [chapter]

Mahdi Cheraghchi, Johan Håstad, Marcus Isaksson, Ola Svensson
2010 Lecture Notes in Computer Science  
In fact, it is not easy to guess whether there exists a homogeneous linear threshold predicate that is approximation resistant or not.  ...  The focus of this paper is to identify and study the approximation curve of a class of threshold predicates that allow for non-trivial approximation.  ...  Almost all Max-CSPs of interest turn out to be NP-hard and the main focus is that of efficient approximability.  ... 
doi:10.1007/978-3-642-15369-3_9 fatcat:xibuyeq3ord4rbagno3sxnalne

Approximating Linear Threshold Predicates

Mahdi Cheraghchi, Johan Håstad, Marcus Isaksson, Ola Svensson
2012 ACM Transactions on Computation Theory  
In fact, it is not easy to guess whether there exists a homogeneous linear threshold predicate that is approximation resistant or not.  ...  The focus of this paper is to identify and study the approximation curve of a class of threshold predicates that allow for non-trivial approximation.  ...  Almost all Max-CSPs of interest turn out to be NP-hard and the main focus is that of efficient approximability.  ... 
doi:10.1145/2141938.2141940 fatcat:llxy4l2a6rey3eaqlmptgoshbi
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