On the Complexity of Hilbert Refutations for Partition

Given a set of integers W, the Partition problem determines whether W can be divided into two disjoint subsets with equal sums. We model the Partition problem as a system of polynomial equations, and then investigate the complexity of a Hilbert's Nullstellensatz refutation, or certificate, that a given set of integers is not partitionable. We provide an explicit construction of a minimum-degree certificate, and then demonstrate that the Partition problem is equivalent to the determinant of a carefully constructed matrix called the partition matrix. In particular, we show that the determinant of the partition matrix is a polynomial that factors into an iteration over all possible partitions of W.Comment: Final versio

Near-optimal asymmetric binary matrix partitions

We study the asymmetric binary matrix partition problem that was recently introduced by Alon et al. (WINE 2013) to model the impact of asymmetric information on the revenue of the seller in take-it-or-leave-it sales. Instances of the problem consist of an $n \times m$ binary matrix $A$ and a probability distribution over its columns. A partition scheme $B=(B_1,...,B_n)$ consists of a partition $B_i$ for each row $i$ of $A$. The partition $B_i$ acts as a smoothing operator on row $i$ that distributes the expected value of each partition subset proportionally to all its entries. Given a scheme $B$ that induces a smooth matrix $A^B$, the partition value is the expected maximum column entry of $A^B$. The objective is to find a partition scheme such that the resulting partition value is maximized. We present a $9/10$-approximation algorithm for the case where the probability distribution is uniform and a $(1-1/e)$-approximation algorithm for non-uniform distributions, significantly improving results of Alon et al. Although our first algorithm is combinatorial (and very simple), the analysis is based on linear programming and duality arguments. In our second result we exploit a nice relation of the problem to submodular welfare maximization.Comment: 17 page

Semidefinite programming and eigenvalue bounds for the graph partition problem

The graph partition problem is the problem of partitioning the vertex set of a graph into a fixed number of sets of given sizes such that the sum of weights of edges joining different sets is optimized. In this paper we simplify a known matrix-lifting semidefinite programming relaxation of the graph partition problem for several classes of graphs and also show how to aggregate additional triangle and independent set constraints for graphs with symmetry. We present an eigenvalue bound for the graph partition problem of a strongly regular graph, extending a similar result for the equipartition problem. We also derive a linear programming bound of the graph partition problem for certain Johnson and Kneser graphs. Using what we call the Laplacian algebra of a graph, we derive an eigenvalue bound for the graph partition problem that is the first known closed form bound that is applicable to any graph, thereby extending a well-known result in spectral graph theory. Finally, we strengthen a known semidefinite programming relaxation of a specific quadratic assignment problem and the above-mentioned matrix-lifting semidefinite programming relaxation by adding two constraints that correspond to assigning two vertices of the graph to different parts of the partition. This strengthening performs well on highly symmetric graphs when other relaxations provide weak or trivial bounds

Soft matrix models and Chern-Simons partition functions

We study the properties of matrix models with soft confining potentials. Their precise mathematical characterization is that their weight function is not determined by its moments. We mainly rely on simple considerations based on orthogonal polynomials and the moment problem. In addition, some of these models are equivalent, by a simple mapping, to matrix models that appear in Chern-Simons theory. The models can be solved with q deformed orthogonal polynomials (Stieltjes-Wigert polynomials), and the deformation parameter turns out to be the usual $q$ parameter in Chern-Simons theory. In this way, we give a matrix model computation of the Chern-Simons partition function on $S^{3}$ and show that there are infinitely many matrix models with this partition function.Comment: 13 pages, 3 figure

RNA Folding and Large N Matrix Theory

We formulate the RNA folding problem as an $N\times N$ matrix field theory. This matrix formalism allows us to give a systematic classification of the terms in the partition function according to their topological character. The theory is set up in such a way that the limit $N\to \infty$ yields the so-called secondary structure (Hartree theory). Tertiary structure and pseudo-knots are obtained by calculating the $1/N^2$ corrections to the partition function. We propose a generalization of the Hartree recursion relation to generate the tertiary structure.Comment: 29 pages (LaTex), 13 figures (eps). Missing paragraph and figure adde