Convex Relaxations for Permutation Problems

Seriation seeks to reconstruct a linear order between variables using unsorted, pairwise similarity information. It has direct applications in archeology and shotgun gene sequencing for example. We write seriation as an optimization problem by proving the equivalence between the seriation and combinatorial 2-SUM problems on similarity matrices (2-SUM is a quadratic minimization problem over permutations). The seriation problem can be solved exactly by a spectral algorithm in the noiseless case and we derive several convex relaxations for 2-SUM to improve the robustness of seriation solutions in noisy settings. These convex relaxations also allow us to impose structural constraints on the solution, hence solve semi-supervised seriation problems. We derive new approximation bounds for some of these relaxations and present numerical experiments on archeological data, Markov chains and DNA assembly from shotgun gene sequencing data.Comment: Final journal version, a few typos and references fixe

On bounding the bandwidth of graphs with symmetry

We derive a new lower bound for the bandwidth of a graph that is based on a new lower bound for the minimum cut problem. Our new semidefinite programming relaxation of the minimum cut problem is obtained by strengthening the known semidefinite programming relaxation for the quadratic assignment problem (or for the graph partition problem) by fixing two vertices in the graph; one on each side of the cut. This fixing results in several smaller subproblems that need to be solved to obtain the new bound. In order to efficiently solve these subproblems we exploit symmetry in the data; that is, both symmetry in the min-cut problem and symmetry in the graphs. To obtain upper bounds for the bandwidth of graphs with symmetry, we develop a heuristic approach based on the well-known reverse Cuthill-McKee algorithm, and that improves significantly its performance on the tested graphs. Our approaches result in the best known lower and upper bounds for the bandwidth of all graphs under consideration, i.e., Hamming graphs, 3-dimensional generalized Hamming graphs, Johnson graphs, and Kneser graphs, with up to 216 vertices