Prasad Raghavendra

Optimal algorithms and inapproximability results for every CSP

Semidefinite Programming(SDP) is one of the strongest algorithmic techniques used in the design of approximation algorithms. In recent years, Unique Games Conjecture(UGC) has proved to be intimately connected to the limitations of Semidefinite Programming. Making this connection precise, we show the following result : If UGC is true, then for every constraint satisfaction problem(CSP) the best approximation ratio is given by a certain simple SDP. Specifically, we show a generic conversion from SDP integrality gaps to UGC hardness results for every CSP. This result holds both for maximization and minimization problems over arbitrary finite domains. Using this connection between integrality gaps and hardness results we obtain a generic polynomial-time algorithm for all CSPs. Assuming the Unique Games Conjecture, this algorithm achieves the optimal approximation ratio for every CSP. Unconditionally, for all 2-CSPs the algorithm achieves an approximation ratio equal to the integrality gap of a natural SDP used in literature. Further the algorithm achieves at least as good an approximation ratio as the best known algorithms for several problems like MaxCut, Max2Sat, MaxDiCut and Unique Games.

Lower Bounds on the Size of Semidefinite Programming Relaxations

We introduce a method for proving lower bounds on the efficacy of semidefinite programming (SDP) relaxations for combinatorial problems. In particular, we show that the cut, TSP, and stable set polytopes on n-vertex graphs are not the linear image of the feasible region of any SDP (i.e., any spectrahedron) of dimension less than 2nδ, for some constant δ > 0. This result yields the first super-polynomial lower bounds on the semidefinite extension complexity of any explicit family of polytopes. Our results follow from a general technique for proving lower bounds on the positive semidefinite rank of a matrix. To this end, we establish a close connection between arbitrary SDPs and those arising from the sum-of-squares SDP hierarchy. For approximating maximum constraint satisfaction problems, we prove that SDPs of polynomial-size are equivalent in power to those arising from degree-O(1) sum-of-squares relaxations. This result implies, for instance, that no family of polynomial-size SDP relaxations can achieve better than a 7/8-approximation for max-sat.

Integrality Gaps for Strong SDP Relaxations of UNIQUE GAMES

With the work of Khot and Vishnoi (FOCS 2005) as a starting point, we obtain integrality gaps for certain strong SDP relaxations of unique games. Specifically, we exhibit a gap instance for the basic semidefinite program strengthened by all valid linear inequalities on the inner products of up to

Beating the Random Ordering is Hard: Inapproximability of Maximum Acyclic Subgraph

\exp(\Omega(\log\log~n)^{1/4})

Rounding Semidefinite Programming Hierarchies via Global Correlation

vectors. For stronger relaxations obtained from the basic semidefinite program by

Reductions between Expansion Problems

R

Hardness of Learning Halfspaces with Noise

rounds of Sherali--Adams lift-and-project, we prove a unique games integrality gap for

Approximate Constraint Satisfaction Requires Large LP Relaxations

R = \Omega(\log\log~n)^{1/4}

Agnostic Learning of Monomials by Halfspaces Is Hard

.By composing these SDP gaps with UGC-hardness reductions, the above results imply corresponding integrality gaps for every problem for which a UGC-based hardness is known. Consequently, this work implies that including any valid constraints on up to

How to Round Any CSP

\exp(\Omega(\log\log~n)^{1/4})