Martin Pál

Group strategy proof mechanisms via primal-dual algorithms

We develop a general method for turning a primal-dual algorithm into a group strategy proof cost-sharing mechanism. We use our method to design approximately budget balanced cost sharing mechanisms for two NP-complete problems: metric facility location, and single source rent-or-buy network design. Both mechanisms are competitive, group strategyproof and recover a constant fraction of the cost. For the facility location game our cost-sharing method recovers a 1/3rd of the total cost, while in the network design game the cost shares pay for a 1/15 fraction of the cost of the solution.

A recursive greedy algorithm for walks in directed graphs

Given an arc-weighted directed graph G = (V, A, /spl lscr/) and a pair of nodes s, t, we seek to find an s-t walk of length at most B that maximizes some given function f of the set of nodes visited by the walk. The simplest case is when we seek to maximize the number of nodes visited: this is called the orienteering problem. Our main result is a quasi-polynomial time algorithm that yields an O(log OPT) approximation for this problem when f is a given submodular set function. We then extend it to the case when a node v is counted as visited only if the walk reaches v in its time window [R(v), D(v)]. We apply the algorithm to obtain several new results. First, we obtain an O(log OPT) approximation for a generalization of the orienteering problem in which the profit for visiting each node may vary arbitrarily with time. This captures the time window problem considered earlier for which, even in undirected graphs, the best approximation ratio known [Bansal, N et al. (2004)] is O(log/sup 2/ OPT). The second application is an O(log/sup 2/ k) approximation for the k-TSP problem in directed graphs (satisfying asymmetric triangle inequality). This is the first non-trivial approximation algorithm for this problem. The third application is an O(log/sup 2/ k) approximation (in quasi-poly time) for the group Steiner problem in undirected graphs where k is the number of groups. This improves earlier ratios (Garg, N et al.) by a logarithmic factor and almost matches the inapproximability threshold on trees (Halperin and Krauthgamer, 2003). This connection to group Steiner trees also enables us to prove that the problem we consider is hard to approximate to a ratio better than /spl Omega/(log/sup 1-/spl epsi// OPT), even in undirected graphs. Even though our algorithm runs in quasi-poly time, we believe that the implications for the approximability of several basic optimization problems are interesting.

Boosted sampling: approximation algorithms for stochastic optimization

Several combinatorial optimization problems choose elements to minimize the total cost of constructing a feasible solution that satisfies requirements of clients. In the Steiner Tree problem, for example, edges must be chosen to connect terminals (clients); in Vertex Cover, vertices must be chosen to cover edges (clients); in Facility Location, facilities must be chosen and demand vertices (clients) connected to these chosen facilities. We consider a stochastic version of such a problem where the solution is constructed in two stages: Before the actual requirements materialize, we can choose elements in a first stage. The actual requirements are then revealed, drawn from a pre-specified probability distribution π thereupon, some more elements may be chosen to obtain a feasible solution for the actual requirements. However, in this second (recourse) stage, choosing an element is costlier by a factor of σ> 1. The goal is to minimize the first stage cost plus the expected second stage cost.We give a general yet simple technique to adapt approximation algorithms for several deterministic problems to their stochastic versions via the following method. First stage: Draw σ independent sets of clients from the distribution π and apply the approximation algorithm to construct a feasible solution for the union of these sets. Second stage: Since the actual requirements have now been revealed, augment the first-stage solution to be feasible for these requirements. We use this framework to derive constant factor approximations for stochastic versions of Vertex Cover, Steiner Tree and Uncapacitated Facility Location for arbitrary distributions π in one fell swoop. For special (product) distributions, we obtain additional and improved results. Our techniques adapt and use the notion of strict cost-shares introduced in [5].

Facility location with nonuniform hard capacities

The authors give the first constant factor approximation algorithm for the facility location problem with nonuniform, hard capacities. Facility location problems have received a great deal of attention in recent years. Approximation algorithms have been developed for many variants. Most of these algorithms are based on linear programming, but the LP techniques developed thus far have been unsuccessful in dealing with hard capacities. A local-search based approximation algorithm (M. Korupolu et al., 1998; F.A. Chudak and D.P. Williamson, 1999) is known for the special case of hard but uniform capacities. We present a local-search heuristic that yields an approximation guarantee of 9 + /spl epsi/ for the case of nonuniform hard capacities. To obtain this result, we introduce new operations that are natural in this context. Our proof is based on network flow techniques.

Universal Facility Location

In the Universal Facility Location problem we are given a set of demand points and a set of facilities. The goal is to assign the demands to facilities in such a way that the sum of service and facility costs is minimized. The service cost is proportional to the distance each unit of demand has to travel to its assigned facility, whereas the facility cost of each facility i depends on the amount of demand assigned to that facility and is given by a cost function f i (·). We present a (7.88 + e)-approximation algorithm for the Universal Facility Location problem based on local search, under the assumption that the cost functions f i are nondecreasing. The algorithm chooses local improvement steps by solving a knapsack-like subproblem using dynamic programming. This is the first constant-factor approximation algorithm for this problem. Our algorithm also slightly improves the best known approximation ratio for the capacitated facility location problem with non-uniform hard capacities.

Approximation via cost-sharing: a simple approximation algorithm for the multicommodity rent-or-buy problem

We study the multicommodity rent-or-buy problem, a type of network design problem with economies of scale. In this problem, capacity on an edge can be rented, with cost incurred on a per-unit of capacity basis, or bought, which allows unlimited use after payment of a large fixed cost. Given a graph and a set of source-sink pairs, we seek a minimum-cost way of installing sufficient capacity on edges so that a prescribed amount of flow can be sent simultaneously from each source to the corresponding sink. The first constant-factor approximation algorithm for this problem was recently given by Kumar et al.; however, this algorithm and its analysis are both quite complicated, and its performance guarantee is extremely large. In this paper, we give a conceptually simple 12-approximation algorithm for this problem. Our analysis of this algorithm makes crucial use of cost sharing, the task of allocating the cost of an object to many users of the object in a fair manner. While techniques from approximation algorithms have recently yielded new progress on cost sharing problems, our work is the first to show the converse - those ideas from cost sharing can be fruitfully applied in the design and analysis of approximation algorithms.

Sampling and Cost-Sharing: Approximation Algorithms for Stochastic Optimization Problems

We consider two- and multistage versions of stochastic combinatorial optimization problems with recourse: in this framework, the instance for the combinatorial optimization problem is drawn from a known probability distribution

Optimization, Games, and Quantified Constraint Satisfaction

pi

Maximizing a Monotone Submodular Function subject to a Matroid Constraint

and is only revealed to the algorithm over two (or multiple) stages. At each stage, on receiving some more information about the instance, the algorithm is allowed to build some partial solution. Since the costs of elements increase with each passing stage, there is a natural tension between waiting for later stages, to gain more information about the instance, and purchasing elements in earlier stages, to take advantages of lower costs. We provide approximation algorithms for stochastic combinatorial optimization problems (such as the Steiner tree problem, the Steiner network problem, and the vertex cover problem) by means of a simple sampling-based algorithm. In every stage, our algorithm samples the probability distribution of the requirements and constructs a partial solution to serve the resulting sample. We show that if one can construct cost-sharing functions associated with the algorithms used to construct these partial solutions, then this strategy results in provable approximation guarantees for the overall stochastic optimization problem. We also extend this approach to provide an approximation algorithm for the stochastic version of the uncapacitated facility location problem, a problem that does not fit into the simpler framework of our main model.

Sharing the cost more efficiently: improved approximation for multicommodity rent-or-buy

Optimization problems considered in the literature generally assume a passive environment that does not react to the actions of an agent. In this paper, we introduce and study a class of optimization problems in which the environment plays an active, adversarial role and responds dynamically to the actions of an agent; this class of problems is based on the framework of quantified constraint satisfaction. We formalize a new notion of approximation algorithm for these optimization problems, and consider certain restricted versions of the general problem obtained by restricting the types of constraints that may appear. Our main result is a dichotomy theorem classifying exactly those restricted versions having a constant factor approximation algorithm.