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Electronic Colloquium on Computational Complexity

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Reports tagged with Integrality gaps:
TR06-152 | 6th December 2006
Konstantinos Georgiou, Avner Magen, Iannis Tourlakis

Tight integrality gaps for Vertex Cover SDPs in the Lovasz-Schrijver hierarchy

We prove that the integrality gap after tightening the standard LP relaxation for Vertex Cover with Omega(sqrt(log n/log log n)) rounds of the SDP LS+ system is 2-o(1).

more >>>

TR08-104 | 23rd November 2008
Madhur Tulsiani

CSP Gaps and Reductions in the Lasserre Hierarchy

We study integrality gaps for SDP relaxations of constraint satisfaction problems, in the hierarchy of SDPs defined by Lasserre. Schoenebeck recently showed the first integrality gaps for these
problems, showing that for MAX k-XOR, the ratio of the SDP optimum to the integer optimum may be as large as ... more >>>

TR09-061 | 2nd July 2009
Konstantinos Georgiou, Avner Magen, Madhur Tulsiani

Optimal Sherali-Adams Gaps from Pairwise Independence

This work considers the problem of approximating fixed predicate constraint satisfaction problems (MAX k-CSP(P)). We show that if the set of assignments accepted by $P$ contains the support of a balanced pairwise independent distribution over the domain of the inputs, then such a problem on $n$ variables cannot be approximated ... more >>>

TR10-169 | 10th November 2010
Siavosh Benabbas, Konstantinos Georgiou, Avner Magen

The Sherali-Adams System Applied to Vertex Cover: Why Borsuk Graphs Fool Strong LPs and some Tight Integrality Gaps for SDPs

Revisions: 2

We study the performance of the Sherali-Adams system for VERTEX COVER on graphs with vector
chromatic number $2+\epsilon$. We are able to construct solutions for LPs derived by any number of Sherali-Adams tightenings by introducing a new tool to establish Local-Global Discrepancy. When restricted to
$O(1/ \epsilon)$ tightenings we show ... more >>>

TR12-003 | 13th December 2011
Pratik Worah

Rank Bounds for a Hierarchy of Lov\'{a}sz and Schrijver

Lov\'{a}sz and Schrijver introduced several lift and project methods for $0$-$1$ integer programs, now collectively known as Lov\'{a}sz-Schrijver ($LS$) hierarchies. Several lower bounds have since been proven for the rank of various linear programming relaxations in the $LS$ and $LS_+$ hierarchies. In this paper we investigate rank bounds in the ... more >>>

TR12-105 | 17th August 2012
Madhur Tulsiani, Pratik Worah

$LS_+$ Lower Bounds from Pairwise Independence

We consider the complexity of LS$_+$ refutations of unsatisfiable instances of Constraint Satisfaction Problems (CSPs) when the underlying predicate supports a pairwise independent distribution on its satisfying assignments. This is the most general condition on the predicates under which the corresponding MAX-CSP problem is known to be approximation resistant.

We ... more >>>

TR12-110 | 4th September 2012
Siu On Chan

Approximation Resistance from Pairwise Independent Subgroups

We show optimal (up to constant factor) NP-hardness for Max-k-CSP over any domain,
whenever k is larger than the domain size. This follows from our main result concerning predicates
over abelian groups. We show that a predicate is approximation resistant if it contains a subgroup that
is ... more >>>

TR12-151 | 6th November 2012
Subhash Khot, Madhur Tulsiani, Pratik Worah

The Complexity of Somewhat Approximation Resistant Predicates

Revisions: 1

A boolean predicate $f:\{0,1\}^k\to\{0,1\}$ is said to be {\em somewhat approximation resistant} if for some constant $\tau > \frac{|f^{-1}(1)|}{2^k}$, given a $\tau$-satisfiable instance of the MAX-$k$-CSP$(f)$ problem, it is NP-hard to find an assignment that {\it strictly beats} the naive algorithm that outputs a uniformly random assignment. Let $\tau(f)$ denote ... more >>>

TR13-017 | 23rd January 2013
Pratik Worah

A Short Excursion into Semi-Algebraic Hierarchies

This brief survey gives a (roughly) self-contained overview of some complexity theoretic results about semi-algebraic proof systems and related hierarchies and the strong connections between them. The article is not intended to be a detailed survey on "Lift and Project" type optimization hierarchies (cf. Chlamtac and Tulsiani) or related proof ... more >>>

TR13-075 | 23rd May 2013
Subhash Khot, Madhur Tulsiani, Pratik Worah

A Characterization of Strong Approximation Resistance

For a predicate $f:\{-1,1\}^k \mapsto \{0,1\}$ with $\rho(f) = \frac{|f^{-1}(1)|}{2^k}$, we call the predicate strongly approximation resistant if given a near-satisfiable instance of CSP$(f)$, it is computationally hard to find an assignment such that the fraction of constraints satisfied is outside the range $[\rho(f)-\Omega(1), \rho(f)+\Omega(1)]$.

We present a characterization of ... more >>>

TR13-146 | 20th October 2013
Subhash Khot, Madhur Tulsiani, Pratik Worah

A Characterization of Approximation Resistance

Revisions: 1

A predicate $f:\{-1,1\}^k \mapsto \{0,1\}$ with $\rho(f) = \frac{|f^{-1}(1)|}{2^k}$ is called {\it approximation resistant} if given a near-satisfiable instance of CSP$(f)$, it is computationally hard to find an assignment that satisfies at least $\rho(f)+\Omega(1)$ fraction of the constraints.

We present a complete characterization of approximation resistant predicates under the ... more >>>

TR20-167 | 9th November 2020
Venkatesan Guruswami, Sai Sandeep

Approximate Hypergraph Vertex Cover and generalized Tuza's conjecture

A famous conjecture of Tuza states that the minimum number of edges needed to cover all the triangles in a graph is at most twice the maximum number of edge-disjoint triangles. This conjecture was couched in a broader setting by Aharoni and Zerbib who proposed a hypergraph version of this ... more >>>

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