Adrian Vetta

Finding odd cycle transversals

We present an O(mn) algorithm to determine whether a graph G with m edges and n vertices has an odd cycle transversal of order at most k, for any fixed k. We also obtain an algorithm that determines, in the same time, whether a graph has a half integral packing of odd cycles of weight k.

Nash equilibria in competitive societies, with applications to facility location, traffic routing and auctions

We consider the following class of problems. The value of an outcome to a society is measured via a submodular utility function (submodularity has a natural economic interpretation: decreasing marginal utility). Decisions, however, are controlled by non-cooperative agents who seek to maximise their own private utility. We present, under basic assumptions, guarantees on the social performance of Nash equilibria. For submodular utility functions, any Nash equilibrium gives an expected social utility within a factor 2 of optimal, subject to a function-dependent additive term. For non-decreasing, submodular utility functions, any Nash equilibrium gives an expected social utility within a factor 1+/spl delta/ of optimal, where 0/spl les//spl delta//spl les/1 is a number based upon discrete curvature of the function. A condition under which all sets of social and private utility functions induce pure strategy Nash equilibria is presented. The case in which agents themselves make use of approximation algorithms in decision making is discussed and performance guarantees given. Finally we present specific problems that fall into our framework. These include competitive versions of the facility location problem and k-median problem, a maximisation version of the traffic routing problem studied by Roughgarden and Tardos (2000), and multiple-item auctions.

Sink equilibria and convergence

We introduce the concept of a sink equilibrium. A sink equilibrium is a strongly connected component with no outgoing arcs in the strategy profile graph associated with a game. The strategy profile graph has a vertex set induced by the set of pure strategy profiles; its arc set corresponds to transitions between strategy profiles that occur with nonzero probability. (Here our focus will just be on the special case in which the strategy profile graph is actually a best response graph; that is, its arc set corresponds exactly to best response moves that result from myopic or greedy behaviour). We argue that there is a natural convergence process to sink equilibria in games where agents use pure strategies. This leads to an alternative measure of the social cost of a lack of coordination, the price of sinking, which measures the worst case ratio between the value of a sink equilibrium and the value of the socially optimal solution. We define the value of a sink equilibrium to be the expected social value of the steady state distribution induced by a random walk on that sink. We illustrate the value of this measure in three ways. Firstly, we show that it may more accurately reflects the inefficiency of uncoordinated solutions in competitive games when the use of pure strategies is the norm. In particular, we give an example (a valid-utility game) in which the game converges to solutions which are a factor n worse than socially optimal. The price of sinking is indeed n, but the price of anarchy is close to 1. Secondly, sink equilibria always exist. Thus, even in games in which pure strategy Nash equilibria (PSNE) do not exist, we can still calculate the price of sinking. Thirdly, we show that bounding the price of sinking can have important implications for the speed of convergence to socially good solutions in games where the agents make best response moves in a random order. We present two examples to illustrate our ideas. (i) Unsplittable selfish routing (and weighted congestion games):we prove that the price of sinking for the weighted unsplittable flow version of the selfish routing problem (for bounded-degree polynomial latency functions) is at most O(2/sup 2d/ d/sup 2d + 3/). In comparison, we give instances of these games without any PSNE. Moreover, our proof technique implies fast convergence to socially good (approximate) solutions. This is in contrast to the negative result of Fabrikant, Papadimitriou, and Talwar (2004) showing the existence of exponentially long best-response paths. (ii) Valid-utility games: we show that for valid-utility games the price of sinking is at most n+1; thus the worst case price of sinking in a valid-utility game is between it and n+1. We use our proof to show fast convergence to constant factor approximate solutions in basic-utility games. In addition, we present a hardness result which shows that, in general, there might be states that are exponentially far from any sink equilibrium in valid-utility games. We prove this by showing that the problem of finding a sink equilibrium (or a PSNE) in valid-utility games is PLS-complete.

Convergence issues in competitive games

We study the speed of convergence to approximate solutions in iterative competitive games. We also investigate the value of Nash equilibria as a measure of the cost of the lack of coordination in such games. Our basic model uses the underlying best response graph induced by the selfish behavior of the players. In this model, we study the value of the social function after multiple rounds of best response behavior by the players. This work therefore deviates from other attempts to study the outcome of selfish behavior of players in non-cooperative games in that we dispense with the insistence upon only evaluating Nash equilibria. A detailed theoretical and practical justification for this approach is presented. We consider non-cooperative games with a submodular social utility function; in particular, we focus upon the class of valid-utility games introduced in [13]. Special cases include basic-utility games and market sharing games which we examine in depth. On the positive side we show that for basic-utility games we obtain extremely quick convergence. After just one round of iterative selfish behavior we are guaranteed to obtain a solution with social value at least \(\frac13\) that of optimal. For n-player valid-utility games, in general, after one round we obtain a \(\frac{1}{2n}\)-approximate solution. For market sharing games we prove that one round of selfish response behavior of players gives \(\Omega({1\over \ln n})\)-approximate solutions and this bound is almost tight. On the negative side we present an example to show that even in games in which every Nash equilibrium has high social value (at least half of optimal), iterative selfish behavior may “converge” to a set of extremely poor solutions (each being at least a factor n from optimal). In such games Nash equilibria may severely underestimate the cost of the lack of coordination in a game, and we discuss the implications of this.

AN APPROXIMATION ALGORITHM FOR THE MINIMUM-COST k-VERTEX CONNECTED SUBGRAPH

We present an approximation algorithm for the problem of finding a minimum-cost k-vertex connected spanning subgraph, assuming that the number of vertices is at least 6k 2 . The approximation guarantee is six times the kth harmonic number (which is O(log k)), and this is also an upper bound on the integrality ratio for a standard linear programming relaxation.

Approximation algorithms for minimum-cost k-vertex connected subgraphs

(MATH) We present two new algorithms for the problem of finding a minimum-cost k-vertex connected spanning subgraph. The first algorithm works on undirected graphs with at least 6k2 vertices and achieves an approximation factor of 6 times the kth harmonic number, which is

Network Design Via Iterative Rounding Of Setpair Relaxations

O(\log k)

Factor 4/3 approximations for minimum 2-connected subgraphs

. The second algorithm works on directed and undirected graphs. It gives an

Simultaneous Clustering of Multiple Gene Expression and Physical Interaction Datasets

O(\sqrt{ n /\keps})

Approximation Algorithms for Network Design with Metric Costs

-approximation algorithm for any