path-constrained network flows

This thesis focuses on approximation algorithms and complexity assessments concerning network flows. It deals with various network flow problems with path restrictions. These restrictions cover the number of paths that are used to route commodities as well as the amount of flow that is routed along a single path or the path's length. Concerning the first restriction we study the unsplittable flow problem-a generalization of the NP-hard edge-disjoint paths problem. Given a network with commodities that must be routed from their sources to their sinks, the unsplittable flow problem forbids each commodity to use more than one path. For this problem we prove a new lower bound on the performance guarantee of randomized rounding which so far belongs to the best approximation algorithms known for this problem. Further, we present an interesting relation between unsplittable flows and classical (splittable) multicommodity flows in the case that all commodities share a common source: Each single source multicommodity flow can be represented as a convex combination of unsplittable flows of congestion at most 2. Further, we combine different path restrictions from the ones mentioned above. In the k-splittable flow problem with path capacities, we study the NP-hard problem that each commodity may be sent along a limited number of paths while the flow value of each path is bounded. This yields a generalization of the unsplittable flow problem, but we show how one can obtain the same asymptotic approximation ratios. For the length-bounded k-splittable flow problem, we consider the single commodity case and develop a constant factor approximation algorithm. A crucial characteristic of network flows occurring in real-world applications is flow variation over time and the fact that flow does not travel instantaneously through a network but requires a certain amount of time to travel through each arc. Both characteristics are captured by "flows over time" which specify a flow rate for each arc and each point in time. We consider the quickest single commodity k-splittable flow problem and give a constant factor approximation algorithm for it. So far only results for k-splittable flows as well as for length-bounded flows and flows over time have been known, but nothing was known for combinations of them. Bounding the flow value of each path is also interesting in the classical maximum s-t-flow problem. We study the case that each path may carry at most one unit of flow and prove that this restriction makes the maximum s-t-flow problem strongly NP-hard. In contrast to the classical maximum s-t-flow problem, the fractional and the integral problem diverge strongly with the new restriction. For the integral problem, we even prove APX-hardness. We develop an FPTAS for the fractional problem and an O(log m)-approximation algorithm for the integral one. (Here, m is the number of arcs in the network under consideration.) Similar results emerge for the multicommodity case. For the objective to find a maximum integral multicommodity flow our asymptotic approximation ratio of O(m^{0.5}) is proven to be best possible, unless P = NP

Dynamic unsplittable flows with path-change penalties: new formulations and solution schemes for large instances

In this work, we consider the dynamic unsplittable flow problem. This variation of the unsplittable flow problem has received little attention so far. The unsplittable flow problem is an NP-hard extension of the multi-commodity flow problem where each commodity sends its flow on only one path. In its dynamic version, this problem features several time steps and a penalty is paid when a commodity changes its path from one time step to the next. We present several mixed-integer linear programming formulations for this problem and compare the strength of their linear relaxation. These formulations are embedded in several solvers which are extensively compared on small to large instances. One of these formulations must be solved through a column generation process whose pricing problem is more difficult than those used in classical flow problems. We present limitations of the pricing schemes proposed in earlier works and describe two new schemes with a better worst-case complexity. Overall, this work lays a strong algorithmic baseline for the resolution of the dynamic unsplittable flow problem, proposes original formulations, and discusses the compared advantages of each, thus hopefully contributing a step towards a better understanding of this problem for both OR researchers and practical applications

Network flow problems and congestion games : complexity and approximation results

This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2006.Includes bibliographical references (p. 155-164).(cont.) We first address the complexity of finding an optimal minimum cost solution to a congestion game. We consider both network and general congestion games, and we examine several variants of the problem concerning the structure of the game and its associated cost functions. Many of the problem variants are NP-hard, though we do identify several versions of the games that are solvable in polynomial time. We then investigate existence and the price of anarchy of pure Nash equilibria in k-splittable congestion games with linear costs. A k-splittable congestion game is one in which each player may split its flow on at most k different paths. We identify conditions for the existence of equilibria by providing a series of potential functions. For the price of anarchy, we show an asymptotic lower bound of 2.4 for unweighted k-splittable congestion games and 2.401 for weighted k-splittable congestion games, and an upper bound of 2.618 in both cases.In this thesis we examine four network flow problems arising in the study of transportation, communication, and water networks. The first of these problems is the Integer Equal Flow problem, a network flow variant in which some arcs are restricted to carry equal amounts of flow. Our main contribution is that this problem is not approximable within a factor of 2n(1-epsilon]), for any fixed [epsilon] > 0, where n is the number of nodes in the graph. We extend this result to a number of variants on the size and structure of the arc sets. We next study the Pup Matching problem, a truck routing problem where two commodities ('pups') traversing an arc together in the network incur the arc cost only once. We propose a tighter integer programming formulation for this problem, and we address practical problems that arise with implementing such integer programming solutions. Additionally, we provide approximation and exact algorithms for special cases of the problem where the number of pups is fixed or the total cost in the network is bounded. Our final two problems are on the topic of congestion games, which were introduced in the area of communications networks.by Carol Meyers.Ph.D