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

2011
*
SIAM journal on computing (Print)
*

We prove that, assuming

doi:10.1137/090756144
fatcat:ndcwrnfp2zbsbj5gnahqvmo73u
*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. ...##
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Beating the Random Ordering is Hard: Inapproximability of Maximum Acyclic Subgraph

2008
*
2008 49th Annual IEEE Symposium on Foundations of Computer Science
*

We prove that

doi:10.1109/focs.2008.51
dblp:conf/focs/GuruswamiMR08
fatcat:rh2dzlq2wbayfizmfdpcbdwa7e
*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. ...##
###
Cryptographic Hardness of Random Local Functions–Survey
[chapter]

2013
*
Lecture Notes in Computer Science
*

In this work, we will study

doi:10.1007/978-3-642-36594-2_33
fatcat:cnghkimszra2ljkp7ersxiwgoq
*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 ...##
###
Tight Hardness for Shortest Cycles and Paths in Sparse Graphs
[article]

2020
*
arXiv
*
pre-print

That

arXiv:1712.08147v4
fatcat:4kmi3ia3d5cmriclwvyj6oynwa
*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. ...##
###
Every Permutation CSP of arity 3 is Approximation Resistant

2009
*
2009 24th Annual IEEE Conference on Computational Complexity
*

Thus,

doi:10.1109/ccc.2009.29
dblp:conf/coco/CharikarGM09
fatcat:paxbuwhb2nam7lmxcr7ntluw5m
*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*. ...##
###
On the Hardness of Approximating Balanced Homogenous 3-Lin

2017
*
Theory of Computing
*

We prove that it

doi:10.4086/toc.2017.v013a010
dblp:journals/toc/HastadM17
fatcat:bxo25zdlffdahl642bvk5ll2ti
*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 ...##
###
Approximating Bounded Occurrence Ordering CSPs
[chapter]

2012
*
Lecture Notes in Computer Science
*

It was shown recently that without

doi:10.1007/978-3-642-32512-0_14
fatcat:xcprjhjmlbhkzdeulp542aln3e
*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] ! ...##
###
Tight Hardness for Shortest Cycles and Paths in Sparse Graphs
[chapter]

2018
*
Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms
*

That

doi:10.1137/1.9781611975031.80
dblp:conf/soda/LincolnWW18
fatcat:7boftjghqrbsxe4baz736vkjty
*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. ...##
###
Local Search is Better than Random Assignment for Bounded Occurrence Ordering k-CSPs
[article]

2013
*
arXiv
*
pre-print

We prove that

arXiv:1210.1890v2
fatcat:if5gfbunwjemrowa76qwljcqse
*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*. ...##
###
Streaming approximation resistance of every ordering CSP
[article]

2021
*
arXiv
*
pre-print

We show that for

arXiv:2105.01782v2
fatcat:es3zee5cyra4jcslyqntomrgve
*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*. ...##
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Is constraint satisfaction over two variables always easy?

2004
*
Random structures & algorithms (Print)
*

not

doi:10.1002/rsa.20026
fatcat:zs5f2vbb2jeappxsisyq5d44ga
*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). ...##
###
Is Constraint Satisfaction Over Two Variables Always Easy?
[chapter]

2002
*
Lecture Notes in Computer Science
*

not

doi:10.1007/3-540-45726-7_18
fatcat:soq24wkg5zbitk5v4costjq2s4
*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). ...##
###
On the optimality of semidefinite relaxations for average-case and generalized constraint satisfaction

2013
*
Proceedings of the 4th conference on Innovations in Theoretical Computer Science - ITCS '13
*

We show that it

doi:10.1145/2422436.2422460
dblp:conf/innovations/BarakKS13
fatcat:k3omfqtzq5h7hgx5p3qpzabgpi
*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. ...##
###
Approximating Linear Threshold Predicates
[chapter]

2010
*
Lecture Notes in Computer Science
*

In fact, it

doi:10.1007/978-3-642-15369-3_9
fatcat:xibuyeq3ord4rbagno3sxnalne
*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*. ...##
###
Approximating Linear Threshold Predicates

2012
*
ACM Transactions on Computation Theory
*

In fact, it

doi:10.1145/2141938.2141940
fatcat:llxy4l2a6rey3eaqlmptgoshbi
*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*. ...
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