Minimum Stable Cut and Treewidth

A stable or locally-optimal cut of a graph is a cut whose weight cannot be increased by changing the side of a single vertex. Equivalently, a cut is stable if all vertices have the (weighted) majority of their neighbors on the other side. Finding a stable cut is a prototypical PLS-complete problem that has been studied in the context of local search and of algorithmic game theory. In this paper we study Min Stable Cut, the problem of finding a stable cut of minimum weight, which is closely related to the Price of Anarchy of the Max Cut game. Since this problem is NP-hard, we study its complexity on graphs of low treewidth, low degree, or both. We begin by showing that the problem remains weakly NP-hard on severely restricted trees, so bounding treewidth alone cannot make it tractable. We match this hardness with a pseudo-polynomial DP algorithm solving the problem in time (?? W)^{O(tw)}n^{O(1)}, where tw is the treewidth, ? the maximum degree, and W the maximum weight. On the other hand, bounding ? is also not enough, as the problem is NP-hard for unweighted graphs of bounded degree. We therefore parameterize Min Stable Cut by both tw and ? and obtain an FPT algorithm running in time 2^{O(?tw)}(n+log W)^{O(1)}. Our main result for the weighted problem is to provide a reduction showing that both aforementioned algorithms are essentially optimal, even if we replace treewidth by pathwidth: if there exists an algorithm running in (nW)^{o(pw)} or 2^{o(?pw)}(n+log W)^{O(1)}, then the ETH is false. Complementing this, we show that we can, however, obtain an FPT approximation scheme parameterized by treewidth, if we consider almost-stable solutions, that is, solutions where no single vertex can unilaterally increase the weight of its incident cut edges by more than a factor of (1+?). Motivated by these mostly negative results, we consider Unweighted Min Stable Cut. Here our results already imply a much faster exact algorithm running in time ?^{O(tw)}n^{O(1)}. We show that this is also probably essentially optimal: an algorithm running in n^{o(pw)} would contradict the ETH

Node-Max-Cut and the Complexity of Equilibrium in Linear Weighted Congestion Games

In this work, we seek a more refined understanding of the complexity of local optimum computation for Max-Cut and pure Nash equilibrium (PNE) computation for congestion games with weighted players and linear latency functions. We show that computing a PNE of linear weighted congestion games is PLS-complete either for very restricted strategy spaces, namely when player strategies are paths on a series-parallel network with a single origin and destination, or for very restricted latency functions, namely when the latency on each resource is equal to the congestion. Our results reveal a remarkable gap regarding the complexity of PNE in congestion games with weighted and unweighted players, since in case of unweighted players, a PNE can be easily computed by either a simple greedy algorithm (for series-parallel networks) or any better response dynamics (when the latency is equal to the congestion). For the latter of the results above, we need to show first that computing a local optimum of a natural restriction of Max-Cut, which we call Node-Max-Cut, is PLS-complete. In Node-Max-Cut, the input graph is vertex-weighted and the weight of each edge is equal to the product of the weights of its endpoints. Due to the very restricted nature of Node-Max-Cut, the reduction requires a careful combination of new gadgets with ideas and techniques from previous work. We also show how to compute efficiently a (1+?)-approximate equilibrium for Node-Max-Cut, if the number of different vertex weights is constant

Search and optimization with randomness in computational economics: equilibria, pricing, and decisions

In this thesis we study search and optimization problems from computational economics with primarily stochastic inputs. The results are grouped into two categories: First, we address the smoothed analysis of Nash equilibrium computation. Second, we address two pricing problems in mechanism design, and solve two economically motivated stochastic optimization problems. Computing Nash equilibria is a central question in the game-theoretic study of economic systems of agent interactions. The worst-case analysis of this problem has been studied in depth, but little was known beyond the worst case. We study this problem in the framework of smoothed analysis, where adversarial inputs are randomly perturbed. We show that computing Nash equilibria is hard for 2-player games even when input perturbations are large. This is despite the existence of approximation algorithms in a similar regime. In doing so, our result disproves a conjecture relating approximation schemes to smoothed analysis. Despite the hardness results in general, we also present a special case of co-operative games, where we show that the natural greedy algorithm for finding equilibria has polynomial smoothed complexity. We also develop reductions which preserve smoothed analysis. In the second part of the thesis, we consider optimization problems which are motivated by economic applications. We address two stochastic optimization problems. We begin by developing optimal methods to determine the best among binary classifiers, when the objective function is known only through pairwise comparisons, e.g. when the objective function is the subjective opinion of a client. Finally, we extend known algorithms in the Pandora's box problem --- a classic optimal search problem --- to an order-constrained setting which allows for richer modelling. The remaining chapters address two pricing problems from mechanism design. First, we provide an approximately revenue-optimal pricing scheme for the problem of selling time on a server to jobs whose parameters are sampled i.i.d. from an unknown distribution. We then tackle the problem of fairly dividing chores among a collection of economic agents via a competitive equilibrium, which balances assigned tasks with payouts. We give efficient algorithms to compute such an equilibrium