Probability Distributions on Partially Ordered Sets and Network Interdiction Games

This article poses the following problem: Does there exist a probability distribution over subsets of a finite partially ordered set (poset), such that a set of constraints involving marginal probabilities of the poset's elements and maximal chains is satisfied? We present a combinatorial algorithm to positively resolve this question. The algorithm can be implemented in polynomial time in the special case where maximal chain probabilities are affine functions of their elements. This existence problem is relevant for the equilibrium characterization of a generic strategic interdiction game on a capacitated flow network. The game involves a routing entity that sends its flow through the network while facing path transportation costs, and an interdictor who simultaneously interdicts one or more edges while facing edge interdiction costs. Using our existence result on posets and strict complementary slackness in linear programming, we show that the Nash equilibria of this game can be fully described using primal and dual solutions of a minimum-cost circulation problem. Our analysis provides a new characterization of the critical components in the interdiction game. It also leads to a polynomial-time approach for equilibrium computation

Cube versus Torus Models for Combinatorial Optimization Problems and the Euclidean Minimum Spanning Tree Constant

For a sample of points drawn uniformly from either the d-dimensional torus or the d-cube, d > 2, we define a class of random processes with the property of being asymptotically equivalent in expectation in the two models. Examples include the traveling salesman problem (TSP), the minimum spanning tree problem (MST), etc. Application of this result helps closing down one open question: We prove that the analytical expression recently obtained by Avram and Bertsimas for the MST constant in the d-torus model is in fact valid for the traditional d-cube model. For the MST, we also extend our result and show that stronger equivalences hold. Finally we present some remarks on the possible use of the d-torus model for exploring rates of convergence for the TSP in the square

A Note on the Number of Leaves of a Euclidean Minimal Spanning Tree

We show that the number of vertices of degree k in the Euclidean minimal spanning tree through points drawn uniformly from either the d-dimensional torus or from the d-cube, d > 2, are asymptotically equivalent with probability one. Implications are discussed

A Decomposition Algorithm for Nested Resource Allocation Problems

We propose an exact polynomial algorithm for a resource allocation problem with convex costs and constraints on partial sums of resource consumptions, in the presence of either continuous or integer variables. No assumption of strict convexity or differentiability is needed. The method solves a hierarchy of resource allocation subproblems, whose solutions are used to convert constraints on sums of resources into bounds for separate variables at higher levels. The resulting time complexity for the integer problem is $O(n \log m \log (B/n))$, and the complexity of obtaining an $\epsilon$-approximate solution for the continuous case is $O(n \log m \log (B/\epsilon))$, $n$ being the number of variables, $m$ the number of ascending constraints (such that $m < n$), $\epsilon$ a desired precision, and $B$ the total resource. This algorithm attains the best-known complexity when $m = n$, and improves it when $\log m = o(\log n)$. Extensive experimental analyses are conducted with four recent algorithms on various continuous problems issued from theory and practice. The proposed method achieves a higher performance than previous algorithms, addressing all problems with up to one million variables in less than one minute on a modern computer.Comment: Working Paper -- MIT, 23 page