Yury Makarychev

O(√log n) approximation algorithms for min UnCut, min 2CNF deletion, and directed cut problems

We give O(√log n)-approximation algorithms for the MIN UNCUT, MIN 2CNF DELETION, DIRECTED BALANCED SEPERATOR, and DIRECTED SPARSEST CUT problems. The previously best known algorithms give an O(log n)-approximation for MIN UNCUT [9], DIRECTED BALANCED SEPERATOR [17], DIRECTED SPARSEST CUT [17], and an O(log n log log n)-approximation for MIN 2CNF DELETION [14].We also show that the integrality gap of an SDP relaxation of the MINIMUM MULTICUT problem is Ω(log n).

Near-optimal algorithms for unique games

Unique games are constraint satisfaction problems that can be viewed as a generalization of Max-Cut to a larger domain size. The Unique Games Conjecture states that it is hard to distinguish between instances of unique games where almost all constraints are satisfiable and those where almost none are satisfiable. It has been shown to imply a number of inapproximability results for fundamental problems that seem difficult to obtain by more standard complexity assumptions. Thus, proving or refuting this conjecture is an important goal. We present significantly improved approximation algorithms for unique games. For instances with domain size k where the optimal solution satisfies 1-ε fraction of all constraints, our algorithms satisfy roughly k-ε/(2-ε) and 1- O(√εlog k) fraction of all constraints. Our algorithms are based on rounding a natural semidefinite programming relaxation for the problem and their performance almost matches the integrality gap of this relaxation. Our results are near optimal if the Unique Games Conjecture is true, i.e. any improvement (beyond low order terms) would refute the conjecture.

Integrality gaps for Sherali-Adams relaxations

We prove strong lower bounds on integrality gaps of Sherali-Adams relaxations for MAX CUT, Vertex Cover, Sparsest Cut and other problems. Our constructions show gaps for Sherali-Adams relaxations that survive nδ rounds of lift and project. For MAX CUT and Vertex Cover, these show that even nδ rounds of Sherali-Adams do not yield a better than 2-ε approximation. The main combinatorial challenge in constructing these gap examples is the construction of a fractional solution that is far from an integer solution, but yet admits consistent distributions of local solutions for all small subsets of variables. Satisfying this consistency requirement is one of the major hurdles to constructing Sherali-Adams gap examples. We present a modular recipe for achieving this, building on previous work on metrics with a local-global structure. We develop a conceptually simple geometric approach to constructing Sherali-Adams gap examples via constructions of consistent local SDP solutions. This geometric approach is surprisingly versatile. We construct Sherali-Adams gap examples for Unique Games based on our construction for MAX CUT together with a parallel repetition like procedure. This in turn allows us to obtain Sherali-Adams gap examples for any problem that has a Unique Games based hardness result (with some additional conditions on the reduction from Unique Games). Using this, we construct 2-ε gap examples for Maximum Acyclic Subgraph that rules out any family of linear constraints with support at most nδ.

How to Play Unique Games Using Embeddings

In this paper we present a new approximation algorithm for unique games. For a unique game with n vertices and k states (labels), if a (1 - epsiv) fraction of all constraints is satisfiable, the algorithm finds an assignment satisfying a 1 - O(epsiv radic(log n log k)) fraction of all constraints. To this end, we introduce new embedding techniques for rounding semidefinite relaxations of problems with large domain size

The Grothendieck Constant is Strictly Smaller than Krivine's Bound

The classical Grothendieck constant, denoted

Metric Extension Operators, Vertex Sparsifiers and Lipschitz Extendability

K_G

Approximation algorithms for semi-random partitioning problems

, is equal to the integrality gap of the natural semi definite relaxation of the problem of computing

Directed metrics and directed graph partitioning problems

\max \{\sum_{i=1}^m\sum_{j=1}^n a_{ij} \epsilon_i\delta_j: \{\epsilon_i\}_{i=1}^m,\{\delta_j\}_{j=1}^n\subseteq \{-1,1\}\},