A Combinatorial Polynomial Algorithm for the Linear Arrow-Debreu Market

We present the first combinatorial polynomial time algorithm for computing the equilibrium of the Arrow-Debreu market model with linear utilities.Comment: Preliminary version in ICALP 201

Finding Short Paths on Polytopes by the Shadow Vertex Algorithm

We show that the shadow vertex algorithm can be used to compute a short path between a given pair of vertices of a polytope P = {x : Ax \leq b} along the edges of P, where A \in R^{m \times n} is a real-valued matrix. Both, the length of the path and the running time of the algorithm, are polynomial in m, n, and a parameter 1/delta that is a measure for the flatness of the vertices of P. For integer matrices A \in Z^{m \times n} we show a connection between delta and the largest absolute value Delta of any sub-determinant of A, yielding a bound of O(Delta^4 m n^4) for the length of the computed path. This bound is expressed in the same parameter Delta as the recent non-constructive bound of O(Delta^2 n^4 \log (n Delta)) by Bonifas et al. For the special case of totally unimodular matrices, the length of the computed path simplifies to O(m n^4), which significantly improves the previously best known constructive bound of O(m^{16} n^3 \log^3(mn)) by Dyer and Frieze

The cyclic-routing UAV problem is PSPACE-complete

© 2015, Springer-Verlag Berlin Heidelberg. Consider a finite set of targets, with each target assigned a relative deadline, and each pair of targets assigned a fixed transit flight time. Given a flock of identical UAVs, can one ensure that every target is repeatedly visited by some UAV at intervals of duration at most the target’s relative deadline? The Cyclic-Routing UAV Problem (cr-uav) is the question of whether this task has a solution. This problem can straightforwardly be solved in PSPACE by modelling it as a network of timed automata. The special case of there being a single UAV is claimed to be NP-complete in the literature. In this paper, we show that the cr-uav Problem is in fact PSPACE-complete even in the single-UAV case

Integer Programming: Optimization and Evaluation Are Equivalent

Link to conference publication published by Springer: http://dx.doi.org/10.1007/978-3-642-03367-4We show that if one can find the optimal value of an integer linear programming problem in polynomial time, then one can find an optimal solution in polynomial time. We also present a proper generalization to (general) integer programs and to local search problems of the well-known result that optimization and augmentation are equivalent for 0/1-integer programs. Among other things, our results imply that PLS-complete problems cannot have “near-exact” neighborhoods, unless PLS = P.United States. Office of Naval Research (ONR grant N00014-01208-1-0029

An update on the Hirsch conjecture

The Hirsch conjecture was posed in 1957 in a letter from Warren M. Hirsch to George Dantzig. It states that the graph of a d-dimensional polytope with n facets cannot have diameter greater than n - d. Despite being one of the most fundamental, basic and old problems in polytope theory, what we know is quite scarce. Most notably, no polynomial upper bound is known for the diameters that are conjectured to be linear. In contrast, very few polytopes are known where the bound $n-d$ is attained. This paper collects known results and remarks both on the positive and on the negative side of the conjecture. Some proofs are included, but only those that we hope are accessible to a general mathematical audience without introducing too many technicalities.Comment: 28 pages, 6 figures. Many proofs have been taken out from version 2 and put into the appendix arXiv:0912.423

Greedy D-Approximation Algorithm for Covering with Arbitrary Constraints and Submodular Cost

This paper describes a simple greedy D-approximation algorithm for any covering problem whose objective function is submodular and non-decreasing, and whose feasible region can be expressed as the intersection of arbitrary (closed upwards) covering constraints, each of which constrains at most D variables of the problem. (A simple example is Vertex Cover, with D = 2.) The algorithm generalizes previous approximation algorithms for fundamental covering problems and online paging and caching problems