The graph homomorphism problem is a canonical $NP$-complete problem.
It generalizes various other well-studied problems such as graph
coloring and finding cliques. To get a better insight into a
combinatorial problem, one often studies relaxations of the problem.
We define fractional homomorphisms and pseudo-homomorphisms ... more >>>

We design a $0.795$ approximation algorithm for the Max-Bisection problem
restricted to regular graphs. In the case of three regular graphs our
results imply an approximation ratio of $0.834$.

Abstract. It is known that random k-SAT formulas with at least
(2^k*ln2)*n random clauses are unsatisfiable with high probability. This
result is simply obtained by bounding the expected number of satisfy-
ing assignments of a random k-SAT instance by an expression tending
to 0 when n, the number of variables ... more >>>

We present a c.k/2^k approximation algorithm for the Max k-CSP problem (where c > 0.44 is an absolute constant). This result improves the previously best known algorithm by Hast, which has an approximation guarantee of Omega(k/(2^k log k)). Our approximation guarantee matches the upper bound of Samorodnitsky and Trevisan up ... more >>>

In this note we present an approximation algorithm for MAX 2SAT that given a (1 - epsilon) satisfiable instance finds an assignment of variables satisfying a 1 - O(sqrt{epsilon}) fraction of all constraints. This result is optimal assuming the Unique Games Conjecture.

The best previously known result, due to Zwick, ... more >>>

We study semidefinite programming relaxations of Vertex Cover arising from
repeated applications of the LS+ ``lift-and-project'' method of Lovasz and
Schrijver starting from the standard linear programming relaxation.

Goemans and Kleinberg prove that after one round of LS+ the integrality
gap remains arbitrarily close to 2. Charikar proves an integrality ... more >>>

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).

We study the approximability of the \maxcsp problem over non-boolean domains, more specifically over $\{0,1,\ldots,q-1\}$ for some integer $q$. We obtain a approximation algorithm that achieves a ratio of $C(q) \cdot k/q^k$ for some constant $C(q)$ depending only on $q$. Further, we extend the techniques of Samorodnitsky and Trevisan to ... more >>>

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 >>>

We study the question of whether the value of mathematical programs such as
linear and semidefinite programming hierarchies on a graph $G$, is preserved
when taking a small random subgraph $G'$ of $G$. We show that the value of the
Goemans-Williamson (1995) semidefinite program (SDP) for \maxcut of $G'$ is
more >>>

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 >>>

The Unique Games conjecture (UGC) has emerged in recent years as the starting point for several optimal inapproximability results. While for none of these results a reverse reduction to Unique Games is known, the assumption of bijective projections in the Label Cover instance seems critical in these proofs. In this ... more >>>

We present an approximation scheme for optimizing certain Quadratic Integer Programming problems with positive semidefinite objective functions and global linear constraints. This framework includes well known graph problems such as Minimum graph bisection, Edge expansion, Uniform sparsest cut, and Small Set expansion, as well as the Unique Games problem. These ... more >>>