Stability vs. optimality in selfish ring routing

We study the asymmetric atomic selfish routing in ring networks, which has diverse practical applications in network design and analysis. We are concerned with minimizing the maximum latency of source-destination node-pairs over links with linear latencies. We obtain the first constant upper bound on the price of anarchy and significantly improve the existing upper bounds on the price of stability. Moreover, we show that any optimal solution is a good approximate Nash equilibrium. Finally, we present better performance analysis and fast implementation of pseudo-polynomial algorithms for computing approximate Nash equilibria

Atomic dynamic flow games : adaptive versus nonadaptive agents

We propose a game model for selfish routing of atomic agents, who compete for use of a network to travel from their origins to a common destination as fast as possible. We follow a frequently used rule that the latency an agent experiences on each edge is a constant transit time plus a variable waiting time in a queue. A key feature that differentiates our model from related ones is an edge-based tie-breaking rule for prioritizing agents in queueing when they reach an edge at the same time. We study both nonadaptive agents (each choosing a one-off origin-destination path simultaneously at the very beginning) and adaptive ones (each making an online decision at every nonterminal vertex they reach as to which next edge to take). On the one hand, we constructively prove that a (pure) Nash equilibrium (NE) always exists for nonadaptive agents, and show that every NE is weakly Pareto optimal and globally first-in-first-out. We present efficient algorithms for finding an NE and best responses of nonadaptive agents. On the other hand, we are among the first to consider adaptive atomic agents, for which we show that a subgame perfect equilibrium (SPE) always exists, and that each NE outcome for nonadaptive agents is an SPE outcome for adaptive agents, but not vice versa

Ranking tournaments with no errors II: Minimax relation

A tournament T=(V,A) is called cycle Mengerian (CM) if it satisfies the minimax relation on packing and covering cycles, for every nonnegative integral weight function defined on A. The purpose of this series of two papers is to show that a tournament is CM iff it contains none of four Möbius ladders as a subgraph; such a tournament is referred to as Möbius-free. In the first paper we have given a structural description of all Möbius-free tournaments, and have proved that every CM tournament is Möbius-free. In this second paper we establish the converse by using our structural theorems and linear programming approach

Ranking tournaments with no errors I: Structural description

In this series of two papers we examine the classical problem of ranking a set of players on the basis of a set of pairwise comparisons arising from a sports tournament, with the objective of minimizing the total number of upsets, where an upset occurs if a higher ranked player was actually defeated by a lower ranked player. This problem can be rephrased as the so-called minimum feedback arc set problem on tournaments, which arises in a rich variety of applications and has been a subject of extensive research. In this series we study this NP-hard problem using structure-driven and linear programming approaches. Let T=(V,A) be a tournament with a nonnegative integral weight w(e) on each arc e. A subset F of arcs is called a feedback arc set if T\F contains no cycles (directed). A collection C of cycles (with repetition allowed) is called a cycle packing if each arc e is used at most w(e) times by members of C. We call T cycle Mengerian (CM) if, for every nonnegative integral function w defined on A, the minimum total weight of a feedback arc set is equal to the maximum size of a cycle packing. The purpose of these two papers is to show that a tournament is CM iff it contains none of four Möbius ladders as a subgraph; such a tournament is referred to as Möbius-free. In this first paper we present a structural description of all Möbius-free tournaments, which relies heavily on a chain theorem concerning internally 2-strong tournaments

Bounding residence times for atomic dynamic routings

In this paper, we are concerned with bounding agents’ residence times in the network for a broad class of atomic dynamic routings. We explore novel token techniques to circumvent direct analysis on complicated chain effects of dynamic routing choices. Even though agents may enter the network over time for an infinite number of periods, we prove that under a mild condition, the residence time of every agent is upper bounded (by a network-dependent constant plus the total number of agents inside the network at the entry time of the agent). Applying this result to three game models of atomic dynamic routing in the recent literature, we establish that the residence times of selfish agents in a series-parallel network with a single origin-destination pair are upper bounded at equilibrium, provided the number of incoming agents at each time point does not exceed the network capacity (i.e., the smallest total capacity of edges in the network whose removal separates the origin from the destination)

Approximation algorithms for soft-capacitated facility location in capacitated network design

Answering an open question published in Operations Research (54, 73-91, 2006) in the area of network design and logistic optimization, we present the first constant-factor approximation algorithms for the problem combining facility location and cable installation in which capacity constraints are imposed on both facilities and cables. We study the problem of designing a minimum cost network to serve client demands by opening facilities for service provision and installing cables for service shipment. Both facilities and cables have capacity constraints and incur buy-at-bulk costs. This Max SNP-hard problem arises in diverse applications and is shown in this paper to admit a combinatorial 19.84-approximation algorithm of cubic running time. This is achieved by an integration of primal-dual schema, Lagrangian relaxation, demand clustering and bi-factor approximation. Our techniques extend to several variants of this problem, which include those with unsplitable demands or requiring network connectivity, and provide constant-factor approximate algorithms in strongly polynomial time

The price of atomic selfish ring routing

We study selfish routing in ring networks with respect to minimizing the maximum latency. Our main result is an establishment of constant bounds on the price of stability (PoS) for routing unsplittable flows with linear latency. We show that the PoS is at most 6.83, which reduces to 4.57 when the linear latency functions are homogeneous. We also show the existence of a (54,1)-approximate Nash equilibrium. Additionally we address some algorithmic issues for computing an approximate Nash equilibrium