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Subexponential LPs Approximate Max-Cut [article]

Samuel B. Hopkins, Tselil Schramm, Luca Trevisan
2020 arXiv   pre-print
Our result provides a surprising converse to known lower bounds against all linear programming relaxations of Max-Cut, and hence resolves the extension complexity of approximate Max-Cut for approximation  ...  Our results separate the power of Sherali-Adams versus Lovász-Schrijver hierarchies for approximating Max-Cut, since it is known that (1/2+ε) approximation of Max Cut requires Ω_ε (n) rounds in the Lovász-Schrijver  ...  For a variety of 2-CSPs, including Max-2-Lin and Max-k-Cut, we show that Sherali-Adams LPs of subexponential size obtain nontrivial worst-case approximations.  ... 
arXiv:1911.10304v2 fatcat:a6zdnblhmrb2fgw5lc7rt7r42m

Approximating Unique Games Using Low Diameter Graph Decomposition [article]

Vedat Levi Alev, Lap Chi Lau
2017 arXiv   pre-print
A corollary is an improved approximation algorithm for the MaxCut problem for K_r-minor free graphs.  ...  We design approximation algorithms for Unique Games when the constraint graph admits good low diameter graph decomposition.  ...  For example, there is a (1 − ε)-approximation algorithm for Max-Cut with running time 2 1/ε · n Or(1) for K r -minor free graphs.  ... 
arXiv:1702.06969v5 fatcat:f2gxofddxnduvavidrtsvk2zp4

Combinatorial structure and randomized subexponential algorithms for infinite games

Henrik Björklund, Sergei Vorobyov
2005 Theoretical Computer Science  
In this setting, we suggest randomized subexponential algorithms appropriate for RLG-and PRLG-function optimization.  ...  We show that the subexponential algorithms for combinatorial linear programming, due to Kalai and Matoušek, Sharir, Welzl, can be adapted for optimizing the RLG-and PRLG-functions.  ...  Otherwise, a 0-weight cycle enforced by MAX and cutting MIN from the sink gives distance +∞, but does not yield a positive mean.  ... 
doi:10.1016/j.tcs.2005.07.041 fatcat:u3gsujjxkjflzfu5ndcr6cgqru

Sherali--Adams Strikes Back [article]

Ryan O'Donnell, Tselil Schramm
2018 arXiv   pre-print
We show that the (c n/Δ)-round Sherali--Adams linear programming hierarchy certifies that the maximum cut in such a G is at most 50.1% (in fact, at most 12 + 2^-Ω(c)).  ...  We also thank the Schloss Dagstuhl Leibniz Center for Informatics (and more specifically the organizers of the CSP Complexity and Approximability workshop), as well as the Casa Mathemática Oaxaca (and  ...  organizers of the Proof Complexity and Beyond workshop), the Simons Institute (and the organizers of the Optimization semester program), and the Banff International Research Station (and the organizers of the Approximation  ... 
arXiv:1812.09967v1 fatcat:smqmhumghbghvksjejhsje2k7q

Sub-exponential Approximation Schemes for CSPs: from Dense to Almost Sparse [article]

Dimitris Fotakis and Michael Lampis and Vangelis Th. Paschos
2015 arXiv   pre-print
The framework is quite general and includes classical optimization problems, such as , Max-DICUT, , and (with a slight extension) k- Densest Subgraph, as special cases.  ...  Even for there exists r<1 such that for all δ' > δ >0, instances with n^1+δ edges cannot be approximated within a ratio better than r in time 2^n^1-δ'.  ...  Approximating Max-CUT in Almost Sparse Graphs In this section, we apply our approach to Max-CUT, which serves as a convenient example and allows us to present the intuition and the main ideas.  ... 
arXiv:1507.04391v1 fatcat:ij2lb77eevfh3j2432pcnbc3k4

Page 2240 of Mathematical Reviews Vol. , Issue 2003C [page]

2003 Mathematical Reviews  
, Uriel (IL-WEIZMC; Rehovot) ; Schechtman, Gideon (IL-WEIZMC; Rehovot) On the optimality of the random hyperplane rounding technique for MAX CUT.  ...  Summary: “MAX CUT is the problem of partitioning the vertices of a graph into two sets, maximizing the number of edges joining these sets. This problem is NP-hard.  ... 

Affine reductions for LPs and SDPs [article]

Gábor Braun, Sebastian Pokutta, Daniel Zink
2016 arXiv   pre-print
We define a reduction mechanism for LP and SDP formulations that degrades approximation factors in a controlled fashion.  ...  Moreover, using our reduction framework we are able to reproduce various results for CSPs from [arXiv:1309.0563] via simple reductions from Max-2-XOR.  ...  The authors would like to thank James Lee for the helpful discussions regarding max-CSPs.  ... 
arXiv:1410.8816v5 fatcat:tqklld6tnbbmvokxxag2l6sjem

Towards strong nonapproximability results in the Lovasz-Schrijver hierarchy

Mikhail Alekhnovich, Sanjeev Arora, Iannis Tourlakis
2005 Proceedings of the thirty-seventh annual ACM symposium on Theory of computing - STOC '05  
These tightened relaxations were the basis of several celebrated approximation algorithms (such as for MAX-CUT, MAX-3SAT, and SPARSEST CUT).  ...  We show that the relaxations produced by as many as Ω(n) rounds of the LS + procedure do not allow nontrivial approximation, thus ruling out the possibility that the LS + approach gives even slightly subexponential  ...  Acknowledgements The authors would like to thank Toniann Pitassi and Joshua Buresh-Oppenheim for suggesting the problem of proving integrality gaps for MAX-3SAT as well as for many helpful discussions  ... 
doi:10.1145/1060590.1060634 dblp:conf/stoc/AlekhnovichAT05 fatcat:kahlffldvjblthacbkazonyi3a

Towards Strong Nonapproximability Results in the Lovász-Schrijver Hierarchy

Mikhail Alekhnovich, Sanjeev Arora, Iannis Tourlakis
2011 Computational Complexity  
These tightened relaxations were the basis of several celebrated approximation algorithms (such as for MAX-CUT, MAX-3SAT, and SPARSEST CUT).  ...  We show that the relaxations produced by as many as Ω(n) rounds of the LS + procedure do not allow nontrivial approximation, thus ruling out the possibility that the LS + approach gives even slightly subexponential  ...  Acknowledgements The authors would like to thank Toniann Pitassi and Joshua Buresh-Oppenheim for suggesting the problem of proving integrality gaps for MAX-3SAT as well as for many helpful discussions  ... 
doi:10.1007/s00037-011-0027-z fatcat:fz4lo7jksrf4xphhnzt6d7zlze

Maximum Quadratic Assignment Problem: Reduction from Maximum Label Cover and LP-based Approximation Algorithm [article]

Konstantin Makarychev and Rajsekar Manokaran and Maxim Sviridenko
2014 arXiv   pre-print
Our result also implies that Approximate Graph Isomorphism is not robust and is in fact, 1 - ϵ vs ϵ hard assuming the Unique Games Conjecture.  ...  Then, we present an O(√(n))-approximation algorithm for the problem based on rounding of the linear programming relaxation often used in the state of the art exact algorithms.  ...  Since for a random permutation ν I the maximum is at least C r.alg LP * / √ n, we get u∈V G v∈V G w G (u, v)w H (ϕ(u), ν(v)) ≥ C r.alg LP * √ n . (6) We now use the greedy deterministic MAX CUT approximation  ... 
arXiv:1403.7721v1 fatcat:k3k4zelnazd55h2lakc4sibxfa

Sherali - Adams Strikes Back

Ryan O'Donnell, Tselil Schramm, Michael Wagner
2019 Computational Complexity Conference  
Our results stand in contrast to the conventional beliefs that linear programming hierarchies perform poorly for max-cut and other CSPs, and that eigenvalue/SDP methods are needed for effective refutation  ...  We show that the exp c log n log ∆ -round Sherali-Adams linear programming hierarchy certifies that the maximum cut in such a G is at most 50.1% (in fact, at most 1 2 + 2 −Ω(c) ).  ...  We also thank the Schloss Dagstuhl Leibniz Center for Informatics (and more specifically the organizers of the CSP Complexity and Approximability workshop), as well as the Casa Mathemática  ... 
doi:10.4230/lipics.ccc.2019.8 dblp:conf/coco/ODonnellS19 fatcat:y4qmkupv6ngq3cuvgpmpp2556q

Page 2001 of Mathematical Reviews Vol. , Issue 98C [page]

1998 Mathematical Reviews  
It is shown how these so-called feasible region reduction cuts are found starting with valid inequalities for the original LP relaxation of the plant location problem or with a system of inequalities and  ...  2001 90C Mathematical programming “In this note, we derive the first polynomial-time approximation scheme for this problem.  ... 

LP Solutions of Vectorial Integer Subset Sums – Cryptanalysis of Galbraith's Binary Matrix LWE [chapter]

Gottfried Herold, Alexander May
2017 Lecture Notes in Computer Science  
We also show under a mild assumption that instances with m ≤ 2n can be broken in polynomial time via LP relaxation.  ...  If S r < √ w, i.e. f max − f min < w, then adding such a new inequality always makes the solution space of the LP relaxation smaller.  ...  If the score S is small, we expect that the ILP solver can find lots of such good cuts, possibly even cuts with w = 1.  ... 
doi:10.1007/978-3-662-54365-8_1 fatcat:4ar73tgenfhqpcvqqcqo3ggei4

The label cut problem with respect to path length and label frequency

Peng Zhang, Bin Fu
2016 Theoretical Computer Science  
max − 1 < c < l max , (ii) a combinatorial l max -approximation algorithm for Label s-t Cut, and (iii) a polynomial time exact algorithm for Global Label Cut with f max bounded from above.  ...  We show that l max = 2 and f max = 2 are two complexity thresholds for Label s-t Cut.  ...  This means a minimum global label cut is an f max -approximate cut, where by α-approximate cut we mean a cut whose capacity is at most α times the capacity of a minimum cut.  ... 
doi:10.1016/j.tcs.2016.08.006 fatcat:2lvdwbzounhpplvtcfkt4zcf5e

Page 6166 of Mathematical Reviews Vol. , Issue 99i [page]

1999 Mathematical Reviews  
A perikernel % is an analytic function defined on the cut domain Ap = (x. x a) \ Xo, with Lp = {(z,z’) | (z—z’)? € R,}. To such a perikernel % one associates a kernel K on the sphere S“~!  ...  The main result is the following: If H is symmetric, i.e., H = y,(H), then H C [m,M], where m = min(supp(u)), M = max(supp(z)).  ... 
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