Iannis Tourlakis

Proving Integrality Gaps without Knowing the Linear Program

Proving integrality gaps for linear relaxations of NP optimization problems is a difficult task and usually undertaken on a case-by-case basis. We initiate a more systematic approach. We prove an integrality gap of 2−o(1) for three families of linear relaxations for vertex cover, and our methods seem relevant to other problems as well.

Towards strong nonapproximability results in the Lovasz-Schrijver hierarchy

Lovasz and Schrijver described a generic method of tightening the LP and SDP relaxation for any 0-1 optimization problem. These tightened relaxations were the basis of several celebrated approximation algorithms (such as for max-cu , max-3sat , and sparsest cut ).We prove strong inapproximability results in this model for well-known problems such as max-3sat , hypergraph vertex cover and set cover . 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 approximation algorithms for these problems.We also point out why our results are somewhat incomparable to known inapproximability results proved using PCPs, and formalize several interesting open questions.

Integrality gaps of 2 - o(1) for Vertex Cover SDPs in the Lovész-Schrijver Hierarchy

Linear and semidefinite programming are highly successful approaches for obtaining good approximations for NP-hard optimization problems. For example, breakthrough approximation algorithms for Max Cut and Sparsest Cut use semidefinite programming. Perhaps the most prominent NP-hard problem whose exact approximation factor is still unresolved is Vertex Cover. PCP-based techniques of Dinur and Safra [7] show that it is not possible to achieve a factor better than 1.36; on the other hand no known algorithm does better than the factor of 2 achieved by the simple greedy algorithm. Furthermore, there is a widespread belief that SDP technicptes are the most promising methods available for improving upon this factor of 2. Following a line of study initiated by Arora et al. [3], our aim is to show that a large family of LP and SDP based algorithms fail to produce an approximation for Vertex Cover better than 2. Lovasz and Schrijver [21] introduced the systems LS and LS+for systematically tightening LP and SDP relaxations, respectively, over many rounds. These systems naturally capture large classes of LP and SDP relaxations; indeed, LS+ captures the celebrated SDP-based algorithms for Max Cur and Sparsest Cur mentioned above. We rule out polynomial-time 2 - Omega(lfloor) approximations for Vertex Cover using LS+. In particular, we prove an integrality gap of 2 - o(lfloor)for Vertex Cover SDPs obtained by tightening the standard LP relaxation with Omega(radiclog n/ log log n) rounds of LS+. While tight integrality gaps were known for Vertex Cover in the weaker LS system [23 ], previous results did not rule out a2 - Omega(1) approximation after even two rounds of LS+.

Time Space Tradeoffs for SAT on Nonuniform Machines

The arguments used by R. Kannan (1984, Math. Systems Theory17, 29?45), L. Fortnow (1997, in “Proceedings, Twelfth Annual IEEE Conference on Computational Complexity, Ulm, Germany, 24?27 June, 1997,” pp. 52?60), and R. J. Lipton and A. Viglas (1999, in “40th Annual Symposium on Foundations of Computer Science, New York, 17?19 Oct. 1999,” pp. 459?469) are generalized and combined with an argument for diagonalizing over machines taking n bits of advice on inputs of length n to obtain the first nontrivial time?space lower bounds for SAT on nonuniform machines. In particular, we show that for any a 0, SAT cannot be computed by a random access deterministic Turing machine using na time, no(1) space, and o(n2/2??) advice nor by a random access deterministic Turing machine using n1+o(1) time, n1?? space, and n1?? advice. More generally, we show that if for some ?>0 there exists a random access deterministic Turing machine solving SAT using na time, nb space, and o(n(a+b)/2??) advice, then a?12(b2+8?b). Lower bounds for computing \overline{{\bfSAT}} on random access nondeterministic Turing machines taking sublinear advice are also obtained. Moreover, we show that SAT does not have NC1 circuits of size nl+o(1) generated by a nondeterministic log?space machine taking no(1) advice. Additionally, new separations of uniform classes are obtained. We show that for all ?>0 and all rational numbers r?1, DTISP(nr, n1??) is properly contained in NTIME(nr).

Integrality Gaps of

Linear and semidefinite programming are highly successful approaches for obtaining good approximations for NP-hard optimization problems. For example, breakthrough approximation algorithms for Max Cut and Sparsest Cut use semidefinite programming. Perhaps the most prominent NP-hard problem whose exact approximation factor is still unresolved is Vertex Cover. Probabilistically checkable proof (PCP)-based techniques of Dinur and Safra [Ann. of Math./ (2), 162 (2005), pp. 439-486] show that it is not possible to achieve a factor better than 1.36; on the other hand no known algorithm does better than the factor of 2 achieved by the simple greedy algorithm. There is a widespread belief that semidefinite programming (SDP) techniques are the most promising methods available for improving upon this factor of 2. Following a line of study initiated by Arora et al. [Theory Comput., 2 (2006), pp. 19-51], our aim is to show that a large family of linear programming (LP)- and SDP-based algorithms fail to produce an approximation for Vertex Cover better than 2. Lovasz and Schrijver [SIAM J. Optim., 1 (1991), pp. 166-190] introduced the systems

2-o(1)

LS

for Vertex Cover SDPs in the Lovász-Schrijver Hierarchy

and

Vertex cover resists SDPs tightened by local hypermetric inequalities

LS_+

Towards Strong Nonapproximability Results in the Lovász-Schrijver Hierarchy

for systematically tightening LP and SDP relaxations, respectively, over many rounds. These systems naturally capture large classes of LP and SDP relaxations; indeed,

Towards optimal integrality gaps for hypergraph vertex cover in the lovász-schrijver hierarchy

LS_+