This paper deals with multi-unit combinatorial auctions where there are n types of goods for sale, and for each good there is some fixed number of units. We focus on the case where each bidder desires a relatively small number of units of each good. In particular, this includes the case where each good has exactly k units, and each bidder desires no more than a single unit of each good. We provide incentive compatible mechanisms for combinatorial auctions for the general case where bidders are not limited to single minded valuations. The mechanisms we give have approximation ratios close to the best possible for both on-line and off-line scenarios. This is the first result where non-VCG mechanisms are derived for non-single minded bidders for a natural model of combinatorial auctions.

This paper deals with the design of efficiently computable incentive compatible, or truthful, mechanisms for combinatorial optimization problems with multi-parameter agents. We focus on approximation algorithms for NP-hard mechanism design problems. These algorithms need to satisfy certain monotonicity properties to ensure truthfulness. Since most of the known approximation techniques do not fulfill these properties, we study alternative techniques. Our first contribution is a quite general method to transform a pseudopolynomial algorithm into a monotone FPTAS. This can be applied to various problems like, e.g., knapsack, constrained shortest path, or job scheduling with deadlines. For example, the monotone FPTAS for the knapsack problem gives a very efficient, truthful mechanism for single-minded multi-unit auctions. The best previous result for such auctions was a 2-approximation. In addition,

In this paper we consider the unsplittable ow problem (UFP): given a directed or undirected network G = (V, E) with edge capacities and a set of terminal pairs (or requests) with associated demands, find a subset of the pairs of maximum total demand for which a single flow path can be chosen for each pair so that for every edge, the sum of the demands of the paths crossing the edge does not exceed its capacity.

The approximability of the maximum edge disjoint paths problem (EDP) in directed graphs was seemingly settled by the )-hardness result of Guruswami et al. [10] and the O( m) approximation achievable via both the natural LP relaxation [19] and the greedy algorithm [11, 12]. Here m is the number of edges in the graph. However, we observe that the hardness of approximation shown in [10] applies to sparse graphs and hence when expressed as a function of n, the number of vertices, only an \Omega\Gamma n )-hardness follows. On the other hand, the O( m)-approximation algorithms do not guarantee a sub-linear (in terms of n) approximation algorithm for dense graphs. We note that a similar gap exists in the known results on the integrality gap of the natural LP relaxation: an \Omega\Gamma n) lower bound and an O( m) upper bound. Motivated by this discrepancy in the upper and lower bounds we study algorithms for the EDP in directed and undirected graphs obtaining improved approximation ratios. We show that the greedy algorithm has an approximation ratio of O(min(n m)) in undirected graphs and a ratio of O(min(n m)) in directed graphs. For ayclic graphs we give an O( n log n) approximation via LP rounding. These are the first sub-linear approximation ratios for EDP. Our results also extend to EDP with weights and to the unsplittable flow problem with uniform edge capacities.

We present approximation algorithms for the unsplittable flow problem (UFP) on undirected graphs. As is standard in this line of research, we assume that the maximum demand is at most the minimum capacity. We focus on the non-uniform capacity case in which the edge capacities can vary arbitrarily over the graph. Our results are:- For undirected graphs we obtain a O(\Delta ff \Gamma 1 log2 n) approximation ratio, where n is the number of vertices, \Delta the maximum degree, and ff the expansion of the graph. Our ratio is capacity independent and improves upon the earlier O(\Delta ff \Gamma 1(c max=cmin) log n) bound [15] for large values of cmax=cmin. Furthermore, if we specialize to the case where all edges have the same capacity, our algorithm gives an O(\Delta ff \Gamma 1 log n) approximation, which matches the performance of the best-known algorithm [15] for this special case.- For certain strong constant-degree expanders considered by Frieze [10] we obtain an O(plog n) approximation for the uniform capacity case, improving upon the current O(log n) approximation.- For UFP on the line and the ring, we give the first constant-factor approximation algorithms. Previous results addressed only the uniform capacity case.- All of the above results improve if the maximum demand is bounded

In traditional multi-commodity flow theory, the task is to send a certain amount of each commodity from its start to its target node, subject to capacity constraints on the edges. However,

In this paper we study the classical problem of finding disjoint paths in graphs. This problem has been studied by a number of authors both for specific graphs and general classes of graphs. Whereas for specific graphs many (almost) matching upper and lower bounds are known for the competitiveness of on-line algorithms, not much is known about how well on-line algorithms can perform in the general setting. In several papers the expansion has been used to measure the performance of off-line and on-line algorithms in this field. We study a class of simple deterministic on-line algorithms, called bounded greedy algorithms, and show that they achieve a competitive ratio that is asymptotically equal to the best possible competitive ratio that can be achieved by any deterministic on-line algorithm. For this we use a parameter called routing number that allows more precise results than the expansion. Interestingly, our upper bound on the competitive ratio is even better than the best approximation ratio known for off-line algorithms. Furthermore, we introduce a refined variant of the routing number and show that this variant allows to construct online algorithms with a competitive ratio that can be significantly below the best possible upper bound for deterministic on-line algorithms if only the routing number or expansion of a network is known. We also show that our on-line algorithms can be transformed into efficient algorithms for the related unsplittable flow problem.

Amitabha Bagchi, Amitabh Chaudhary, and Christian Scheideler Dept. of Computer Science Johns Hopkins University 3400 N. Charles Street Baltimore, MD 21218, USA {bagchi,amic,scheideler}@cs.jhu.edu Petr Kolman Inst. for Theoretical Computer Science Charles University Malostransk e n am. 25 118 00 Prague, Czech Republic kolman@kam.mff.cuni.cz ABSTRACT In this paper we consider the k edge-disjoint paths problem (k-EDP), a generalization of the well-known edge-disjoint paths problem. Given a graph G = (V, E) and a set of terminal pairs (or requests) T , the problem is to find a maximum subset of the pairs in T for which it is possible to select paths such that each pair is connected by k edgedisjoint paths and the paths for di#erent pairs are mutually disjoint. To the best of our knowledge, no nontrivial result is known for this problem for k > 1. To measure the performance of our algorithms we will use the recently introduced flow number F of a graph. This parameter is known to fulfill F = O(## -1 log n), where # is the maximum degree and # is the edge expansion of G. We show that a simple, greedy online algorithm achieves a competitive ratio of F ), which naturally extends the best known bound of O(F ) for k = 1 to higher k. To achieve this competitive ratio, we introduce a new method of converting a system of k disjoint paths into a system of k length-bounded disjoint paths. We also show that any deterministic online algorithm has a competitive ratio of ## k F ).

The classical network flow theory allows decomposition of flow into several chunks of arbitrary sizes traveling through the network on different paths. In the first part of this article we consider the unsplittable flow problem where all flow traveling from a source to a destination must be sent on only one path. We prove a lower bound of �(log m / log log m) on the performance of a general class of algorithms for minimizing congestion where m is the number of edges in a graph. These algorithms start with a solution for the classical multicommodity flow problem, compute a path decomposition, and select one of its paths for each commodity in order to obtain an unsplittable flow. Our result matches the best known upper bound for randomized rounding—an algorithm of this type introduced by Raghavan and Thompson. The k-splittable flow problem is a generalization of the unsplittable flow problem where the number of paths is bounded for each commodity. We study a new variant of this problem with additional constraints on the amount of flow being sent along each path. We present approximation results for two versions of this problem with the objective to minimize the congestion of the network. The key idea is to reduce the problem under consideration to an unsplittable flow problem while only losing a constant factor in the performance ratio. © 2006 Wiley Periodicals, Inc. NETWORKS, Vol. 48(2), 68–76 2006

In a recent paper Chekuri and Khanna improved the analysis of the Greedy algorithm for the Edge Disjoint Paths problem and proved the same bounds also for the related Uniform Capacity Unsplittable Flow Problem. Here we show that their ideas can be used to get the same approximation ratio even for the more general Unsplittable Flow Problem with nonuniform edge capacities.