12,065 research outputs found
Lift & Project Systems Performing on the Partial-Vertex-Cover Polytope
We study integrality gap (IG) lower bounds on strong LP and SDP relaxations
derived by the Sherali-Adams (SA), Lovasz-Schrijver-SDP (LS+), and
Sherali-Adams-SDP (SA+) lift-and-project (L&P) systems for the
t-Partial-Vertex-Cover (t-PVC) problem, a variation of the classic Vertex-Cover
problem in which only t edges need to be covered. t-PVC admits a
2-approximation using various algorithmic techniques, all relying on a natural
LP relaxation. Starting from this LP relaxation, our main results assert that
for every epsilon > 0, level-Theta(n) LPs or SDPs derived by all known L&P
systems that have been used for positive algorithmic results (but the Lasserre
hierarchy) have IGs at least (1-epsilon)n/t, where n is the number of vertices
of the input graph. Our lower bounds are nearly tight.
Our results show that restricted yet powerful models of computation derived
by many L&P systems fail to witness c-approximate solutions to t-PVC for any
constant c, and for t = O(n). This is one of the very few known examples of an
intractable combinatorial optimization problem for which LP-based algorithms
induce a constant approximation ratio, still lift-and-project LP and SDP
tightenings of the same LP have unbounded IGs.
We also show that the SDP that has given the best algorithm known for t-PVC
has integrality gap n/t on instances that can be solved by the level-1 LP
relaxation derived by the LS system. This constitutes another rare phenomenon
where (even in specific instances) a static LP outperforms an SDP that has been
used for the best approximation guarantee for the problem at hand. Finally, one
of our main contributions is that we make explicit of a new and simple
methodology of constructing solutions to LP relaxations that almost trivially
satisfy constraints derived by all SDP L&P systems known to be useful for
algorithmic positive results (except the La system).Comment: 26 page
Approximation in stochastic integer programming
Approximation algorithms are the prevalent solution methods in the field of stochastic programming. Problems in this field are very hard to solve. Indeed, most of the research in this field has concentrated on designing solution methods that approximate the optimal solutions. However, efficiency in the complexity theoretical sense is usually not taken into account. Quality statements mostly remain restricted to convergence to an optimal solution without accompanying implications on the running time of the algorithms for attaining more and more accurate solutions. However, over the last twenty years also some studies on performance analysis of approximation algorithms for stochastic programming have appeared. In this direction we find both probabilistic analysis and worst-case analysis. There have been studies on performance ratios and on absolute divergence from optimality. Only recently the complexity of stochastic programming problems has been addressed, indeed confirming that these problems are harder than most combinatorial optimization problems.
Cluster Variation Method in Statistical Physics and Probabilistic Graphical Models
The cluster variation method (CVM) is a hierarchy of approximate variational
techniques for discrete (Ising--like) models in equilibrium statistical
mechanics, improving on the mean--field approximation and the Bethe--Peierls
approximation, which can be regarded as the lowest level of the CVM. In recent
years it has been applied both in statistical physics and to inference and
optimization problems formulated in terms of probabilistic graphical models.
The foundations of the CVM are briefly reviewed, and the relations with
similar techniques are discussed. The main properties of the method are
considered, with emphasis on its exactness for particular models and on its
asymptotic properties.
The problem of the minimization of the variational free energy, which arises
in the CVM, is also addressed, and recent results about both provably
convergent and message-passing algorithms are discussed.Comment: 36 pages, 17 figure
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