Understanding the Correlation Gap For Matchings

Given a set of vertices V with |V| = n, a weight vector w in (R^+ cup {0})^{binom{V}{2}}, and a probability vector x In [0, 1]^{binom{V}{2}} in the matching polytope, we study the quantity (E_{G}[ nu_w(G)])/(sum_(u, v) in binom{V}{2} w_{u, v} x_{u, v}) where G is a random graph where each edge e with weight w_e appears with probability x_e independently, and let nu_w(G) denotes the weight of the maximum matching of G. This quantity is closely related to correlation gap and contention resolution schemes, which are important tools in the design of approximation algorithms, algorithmic game theory, and stochastic optimization. We provide lower bounds for the above quantity for general and bipartite graphs, and for weighted and unweighted settings. The best known upper bound is 0.54 by Karp and Sipser, and the best lower bound is 0.4. We show that it is at least 0.47 for unweighted bipartite graphs, at least 0.45 for weighted bipartite graphs, and at least 0.43 for weighted general graphs. To achieve our results, we construct local distribution schemes on the dual which may be of independent interest

Simultaneous identification of specifically interacting paralogs and inter-protein contacts by Direct-Coupling Analysis

Understanding protein-protein interactions is central to our understanding of almost all complex biological processes. Computational tools exploiting rapidly growing genomic databases to characterize protein-protein interactions are urgently needed. Such methods should connect multiple scales from evolutionary conserved interactions between families of homologous proteins, over the identification of specifically interacting proteins in the case of multiple paralogs inside a species, down to the prediction of residues being in physical contact across interaction interfaces. Statistical inference methods detecting residue-residue coevolution have recently triggered considerable progress in using sequence data for quaternary protein structure prediction; they require, however, large joint alignments of homologous protein pairs known to interact. The generation of such alignments is a complex computational task on its own; application of coevolutionary modeling has in turn been restricted to proteins without paralogs, or to bacterial systems with the corresponding coding genes being co-localized in operons. Here we show that the Direct-Coupling Analysis of residue coevolution can be extended to connect the different scales, and simultaneously to match interacting paralogs, to identify inter-protein residue-residue contacts and to discriminate interacting from noninteracting families in a multiprotein system. Our results extend the potential applications of coevolutionary analysis far beyond cases treatable so far.Comment: Main Text 19 pages Supp. Inf. 16 page

On the Configuration LP for Maximum Budgeted Allocation

We study the Maximum Budgeted Allocation problem, i.e., the problem of selling a set of $m$ indivisible goods to $n$ players, each with a separate budget, such that we maximize the collected revenue. Since the natural assignment LP is known to have an integrality gap of $\frac{3}{4}$, which matches the best known approximation algorithms, our main focus is to improve our understanding of the stronger configuration LP relaxation. In this direction, we prove that the integrality gap of the configuration LP is strictly better than $\frac{3}{4}$, and provide corresponding polynomial time roundings, in the following restrictions of the problem: (i) the Restricted Budgeted Allocation problem, in which all the players have the same budget and every item has the same value for any player it can be sold to, and (ii) the graph MBA problem, in which an item can be assigned to at most 2 players. Finally, we improve the best known upper bound on the integrality gap for the general case from $\frac{5}{6}$ to $2\sqrt{2}-2\approx 0.828$ and also prove hardness of approximation results for both cases.Comment: 29 pages, 4 figures. To appear in the 17th Conference on Integer Programming and Combinatorial Optimization (IPCO), 201

Lift-and-Round to Improve Weighted Completion Time on Unrelated Machines

We consider the problem of scheduling jobs on unrelated machines so as to minimize the sum of weighted completion times. Our main result is a $(3/2-c)$-approximation algorithm for some fixed $c>0$, improving upon the long-standing bound of 3/2 (independently due to Skutella, Journal of the ACM, 2001, and Sethuraman & Squillante, SODA, 1999). To do this, we first introduce a new lift-and-project based SDP relaxation for the problem. This is necessary as the previous convex programming relaxations have an integrality gap of $3/2$. Second, we give a new general bipartite-rounding procedure that produces an assignment with certain strong negative correlation properties.Comment: 21 pages, 4 figure

Quantum Crystals and Spin Chains

In this note, we discuss the quantum version of the melting crystal corner in one, two, and three dimensions, generalizing the treatment for the quantum dimer model. Using a mapping to spin chains we find that the two--dimensional case (growth of random partitions) is integrable and leads directly to the Hamiltonian of the Heisenberg XXZ ferromagnet. The three--dimensional case of the melting crystal corner is described in terms of a system of coupled XXZ spin chains. We give a conjecture for its mass gap and analyze the system numerically.Comment: 34 pages, 26 picture

Belief-Propagation for Weighted b-Matchings on Arbitrary Graphs and its Relation to Linear Programs with Integer Solutions

We consider the general problem of finding the minimum weight \bm-matching on arbitrary graphs. We prove that, whenever the linear programming (LP) relaxation of the problem has no fractional solutions, then the belief propagation (BP) algorithm converges to the correct solution. We also show that when the LP relaxation has a fractional solution then the BP algorithm can be used to solve the LP relaxation. Our proof is based on the notion of graph covers and extends the analysis of (Bayati-Shah-Sharma 2005 and Huang-Jebara 2007}. These results are notable in the following regards: (1) It is one of a very small number of proofs showing correctness of BP without any constraint on the graph structure. (2) Variants of the proof work for both synchronous and asynchronous BP; it is the first proof of convergence and correctness of an asynchronous BP algorithm for a combinatorial optimization problem.Comment: 28 pages, 2 figures. Submitted to SIAM journal on Discrete Mathematics on March 19, 2009; accepted for publication (in revised form) August 30, 2010; published electronically July 1, 201

Random multi-index matching problems

The multi-index matching problem (MIMP) generalizes the well known matching problem by going from pairs to d-uplets. We use the cavity method from statistical physics to analyze its properties when the costs of the d-uplets are random. At low temperatures we find for d>2 a frozen glassy phase with vanishing entropy. We also investigate some properties of small samples by enumerating the lowest cost matchings to compare with our theoretical predictions.Comment: 22 pages, 16 figure