An Efficient Algorithm for Bicriteria Minimum-Cost Circulation Problem

This paper is concerned with a bicriteria minimum-cost circulation problem which arises in interactive multicriteria decision making. The author presents a strongly polynomial algorithm for this problem, that is achieved by making use of the parametric characterization of optimal solutions and a strongly polynomial algorithm for the single objective minimum-cost circulation problem

A network flow approach to a common generalization of Clar and Fries numbers

Clar number and Fries number are two thoroughly investigated parameters of plane graphs emerging from mathematical chemistry to measure stability of organic molecules. We consider first a common generalization of these two concepts for bipartite plane graphs, and then extend it to a framework on general (not necessarily planar) directed graphs. The corresponding optimization problem can be transformed into a maximum weight feasible tension problem which is the linear programming dual of a minimum cost network flow (or circulation) problem. Therefore the approach gives rise to a min-max theorem and to a strongly polynomial algorithm that relies exclusively on standard network flow subroutines. In particular, we give the first network flow based algorithm for an optimal Fries structure and its variants

Strongly polynomial algorithm for a class of minimum-cost flow problems with separable convex objectives

A well-studied nonlinear extension of the minimum-cost flow problem is to minimize the objective $\sum_{ij\in E} C_{ij}(f_{ij})$ over feasible flows $f$, where on every arc $ij$ of the network, $C_{ij}$ is a convex function. We give a strongly polynomial algorithm for the case when all $C_{ij}$'s are convex quadratic functions, settling an open problem raised e.g. by Hochbaum [1994]. We also give strongly polynomial algorithms for computing market equilibria in Fisher markets with linear utilities and with spending constraint utilities, that can be formulated in this framework (see Shmyrev [2009], Devanur et al. [2011]). For the latter class this resolves an open question raised by Vazirani [2010]. The running time is $O(m^4\log m)$ for quadratic costs, $O(n^4+n^2(m+n\log n)\log n)$ for Fisher's markets with linear utilities and $O(mn^3 +m^2(m+n\log n)\log m)$ for spending constraint utilities. All these algorithms are presented in a common framework that addresses the general problem setting. Whereas it is impossible to give a strongly polynomial algorithm for the general problem even in an approximate sense (see Hochbaum [1994]), we show that assuming the existence of certain black-box oracles, one can give an algorithm using a strongly polynomial number of arithmetic operations and oracle calls only. The particular algorithms can be derived by implementing these oracles in the respective settings

A strongly polynomial algorithm for generalized flow maximization

A strongly polynomial algorithm is given for the generalized flow maximization problem. It uses a new variant of the scaling technique, called continuous scaling. The main measure of progress is that within a strongly polynomial number of steps, an arc can be identified that must be tight in every dual optimal solution, and thus can be contracted. As a consequence of the result, we also obtain a strongly polynomial algorithm for the linear feasibility problem with at most two nonzero entries per column in the constraint matrix.Comment: minor correction

Discrete Convex Functions on Graphs and Their Algorithmic Applications

The present article is an exposition of a theory of discrete convex functions on certain graph structures, developed by the author in recent years. This theory is a spin-off of discrete convex analysis by Murota, and is motivated by combinatorial dualities in multiflow problems and the complexity classification of facility location problems on graphs. We outline the theory and algorithmic applications in combinatorial optimization problems