Flow-based problem solving in combinatorial optimization

Abstract

Ever since the seminal work of Ford and Fulkerson network flows belong to the most important and fundamental class of problems in combinatorial optimization. Network flow problems arise in real-world applications and in theoretical optimization problems that, at first sight, might not seem to involve networks at all. In this thesis, we consider three different problems that all have very strong connections to network flows and flow decomposition. The first part of this thesis considers the single source unsplittable flow problem, introduced by Kleinberg in 1996. In a digraph with one source and several destination nodes with associated demands, an unsplittable flow routes each demand along a single path from the common source to its destination. Given some fractional flow x that satisfies all demands, it is a natural question to ask for an unsplittable flow y that does not deviate from x by too much, i.e., y_a ~ x_a for all arcs a. Twenty-five years ago, Dinitz, Garg, and Goemans proved that there is an unsplittable flow y such that y_a <= x_a+d_max for all arcs a, where d_max denotes the maximum demand value. Our first contribution is a considerably simpler one-page proof for this classical result. Secondly, we show that there is an unsplittable flow y such that y_a >= x_a – d_max for all arcs a. Finally, we prove existence of an unsplittable flow that simultaneously satisfies the upper and lower bounds for the special case where demands are integer multiples of each other. For arbitrary demand values, we prove slightly weaker simultaneous bounds. Finding optimal subgraphs of a surface-embedded graph that satisfy certain topological properties is a basic subject in topological graph theory and an important ingredient in many algorithms. In the second part of this thesis, we study a variant of the minimum-cost circulation problem with such an additional topological constraint. Given a directed graph D cellularly embedded in a surface together with nonnegative cost c on its arcs and any integer circulation y in D, find a minimum-cost nonnegative integer circulation in D that is homologous over the integers to y. Here, a circulation x is said to be homologous over the integers to y if their difference x-y is a linear combination of facial circulations with integer coefficients, where a facial circulation is a circulation that sends one unit along the boundary of a single face. The above-mentioned problem is NP-hard for general non-orientable surfaces. We present a polynomial-time algorithm for non-orientable surfaces of fixed genus. For this purpose, we provide a characterization of homology over the integers by exploiting a flow decomposition technique. This allows us to reformulate the problem as an integer program in standard form with a constant number of quality constraints, provided that the genus is fixed. This integer program can then be efficiently solved using some general integer programming techniques. In the third part of this thesis, we consider the problem of allocating indivisible resources to players so as to maximize the minimum total value that any player receives. This problem is sometimes dubbed the Santa Claus problem, and its different variants have been subject to extensive research toward approximation algorithms over the past two decades. In the case where each player has a potentially different additive valuation function, Chakrabarty, Chuzhoy, and Khanna gave an O(n^epsilon)-approximation algorithm with polynomial run time for any constant epsilon>0 and a polylogarithmic approximation algorithm in quasi-polynomial time. We show that the same can be achieved for monotone submodular valuation functions, improving over the previously best algorithm due to Goemans, Harvey, Iwata, and Mirrokni. Similarly to Chakrabarty, Chuzhoy, and Khanna, our techniques rely on a reduction of the Santa Claus problem to a carefully designed layered flow problem. As a second step, we formulate and solve a strong linear programming relaxation of this flow problem whose fractional solution can then be turned into an integral solution via randomized rounding.