Consider the problem of discovering (or verifying) the edges and nonedges of a network, modelled as a connected undirected graph, using a minimum number of queries. A query at a vertex v discovers (or verifies) all edges and nonedges whose endpoints have different distance from v. In the network discovery problem, the edges and non-edges are initially unknown, and the algorithm must select the next query based only on the results of previous queries. We study the problem using competitive analysis and give a randomized on-line algorithm with competitive ratio O( √ n log n) for graphs with n vertices. We also show that no deterministic algorithm can have competitive ratio better than 3. In the network verification problem, the graph is known in advance and the goal is to compute a minimum number of queries that verify all edges and non-edges. This problem has previously been studied as the problem of placing landmarks in graphs or determining the metric dimension of a graph. We show that there is no approximation algorithm for this problem with ratio o(log n) unless P = N P. Furthermore, we prove that the optimal number of queries for d-dimensional hypercubes is Θ(d/ log d).

For a given graph with weighted vertices, the goal of the minimum-weight dominating set problem is to compute a vertex subset of smallest weight such that each vertex of the graph is contained in the subset or has a neighbor in the subset. A unit disk graph is a graph in which each vertex corresponds to a unit disk in the plane and two vertices are adjacent if and only if their disks have a non-empty intersection. We present the first constant-factor approximation algorithm for the minimum-weight dominating set problem in unit disk graphs, a problem motivated by applications in wireless ad-hoc networks. The algorithm is obtained in two steps: First, the problem is reduced to the problem of covering a set of points located in a small square using a minimumweight set of unit disks. Then, a constant-factor approximation algorithm for the latter problem is obtained using enumeration and dynamic programming techniques exploiting the geometry of unit disks. Furthermore, we show how to obtain a constant-factor approximation algorithm for the minimum-weight connected dominating set problem in unit disk graphs.Our techniques also yield a constant-factor approximation algorithm for the weighted disk cover problem (covering a set of points in the plane with unit disks of minimum total weight) and a 3-approximation algorithm for the weighted forwarding set problem (covering a set of points in the plane with weighted unit disks whose centers are all contained in a given unit disk).

We study the price of anarchy and the structure of equilibria in network creation games. A network creation game is played by n players {1, 2, . . . , n}, each identified with a vertex of a graph (network), where the strategy of player i, i = 1, . . . , n, is to build some edges adjacent to i. The cost of building an edge is α > 0, a fixed parameter of the game. The goal of every player is to minimize its creation cost plus its usage cost. The creation cost of player i is α times the number of built edges. In the SUMGAME variant, the usage cost of player i is the sum of distances from i to every node of the resulting graph. In the MAXGAME variant, the usage cost is the eccentricity of i in the resulting graph of the game. In this paper we improve previously known bounds on the price of anarchy of the game (of both variants) for various ranges of α, and give new insights into the structure of equilibria for various values of α. The two main results of the paper show that for α > 273 · n all equilibria in SUMGAME are trees and thus the price of anarchy is constant, and that for α > 129 all equilibria in MAXGAME are trees and the price of anarchy is constant. For SUMGAME this answers (almost completely) one of the fundamental open problems in the field-is price of anarchy of the network creation game constant for all values of α?-in an affirmative way, up to a tiny range of α.

Abstract. We study the price of anarchy and the structure of equilibria in network creation games.

No abstract

International audienceData Delivery by Energy-Constrained Mobile Agent

Orthogonal Variable Spreading Factor (OVSF) codes are used in UMTS to share the radio spectrum among several connections of possibly different bandwidth requirements. The combinatorial core of the OVSF code assignment problem is to assign some nodes of a complete binary tree of height h (the code tree) to n simultaneous connections, such that no two assigned nodes (codes) are on the same root-to-leaf path. A connection that uses a 2 −d fraction of the total bandwidth requires some code at depth d in the tree, but this code assignment is allowed to change over time. Requests for connections that would exceed the total available bandwidth are rejected. We consider the one-step code assignment problem: Given an assignment, move the minimum number of codes to serve a new request. Minn and Siu propose the so-called DCA algorithm to solve the problem optimally. In contrast, we show that DCA does not always return an optimal solution, and that the problem is NP-hard. We give an exact n O(h) -time algorithm, and a polynomial-time greedy algorithm that achieves approximation ratio (h). A more practically relevant version is the online code assignment problem, where future requests are not known in advance. Our objective is to minimize the overall number of code reassignments. We present a (h)-competitive online algorithm, and show that no deterministic online algorithm can achieve a competitive ratio better than 1.5. We show that the greedy strategy (minimizing the number of reassignments in every step) is not better than (h) competitive. We give a 2-resource augmented online algorithm that achieves an amortized constant number of (re-)assignments. Finally, we show that the problem is fixed-parameter tractable.

T his paper considers mathematical optimization for the multistage train formation problem, which at the core is the allocation of classification yard formation tracks to outbound freight trains, subject to realistic constraints on train scheduling, arrival and departure timeliness, and track capacity. The problem formulation allows the temporary storage of freight cars on a dedicated mixed-usage track. This real-world practice increases the capacity of the yard, measured in the number of simultaneous trains that can be successfully handled. Two optimization models are proposed and evaluated for the multistage train formation problem. The first one is a column-based integer programming model, which is solved using branch and price. The second model is a simplified reformulation of the first model as an arc-indexed integer linear program, which has the same linear programming relaxation as the first model. Both models are adapted for rolling horizon planning and evaluated on a five-month historical data set from the largest freight yard in Scandinavia. From this data set, 784 instances of different types and lengths, spanning from two to five days, were created. In contrast to earlier approaches, all instances could be solved to optimality using the two models. In the experiments, the arc-indexed model proved optimality on average twice as fast as the column-based model for the independent instances, and three times faster for the rolling horizon instances. For the arc-indexed model, the average solution time for a reasonably sized planning horizon of three days was 16 seconds. Regardless of size, no instance took longer than eight minutes to be solved. The results indicate that optimization approaches are suitable alternatives for scheduling and track allocation at classification yards.