Tight SoS-Degree Bounds for Approximate Nash Equilibria

Nash equilibria always exist, but are widely conjectured to require time to find that is exponential in the number of strategies, even for two-player games. By contrast, a simple quasi-polynomial time algorithm, due to Lipton, Markakis and Mehta (LMM), can find approximate Nash equilibria, in which no player can improve their utility by more than ε by changing their strategy. The LMM algorithm can also be used to find an approximate Nash equilibrium with near-maximal total welfare. Matching hardness results for this optimization problem were found assuming the hardness of the planted-clique problem (by Hazan and Krauthgamer) and assuming the Exponential Time Hypothesis (by Braverman, Ko and Weinstein). In this paper we consider the application of the sum-squares (SoS) algorithm from convex optimization to the problem of optimizing over Nash equilibria. We show the first unconditional lower bounds on the number of levels of SoS needed to achieve a constant factor approximation to this problem. While it may seem that Nash equilibria do not naturally lend themselves to convex optimization, we also describe a simple LP (linear programming) hierarchy that can find an approximate Nash equilibrium in time comparable to that of the LMM algorithm, although neither algorithm is obviously a generalization of the other. This LP can be viewed as arising from the SoS algorithm at log n levels – matching our lower bounds. The lower bounds involve a modification of the Braverman-Ko-Weinstein embedding of CSPs into strategic games and techniques from sum-of-squares proof systems. The upper bound (i.e. analysis of the LP) uses information-theory techniques that have been recently applied to other linear- and semidefinite programming hierarchies

Computation of Approximate Welfare-Maximizing Correlated Equilibria and Pareto-Optima with Applications to Wireless Communication

In a wireless application with multiple communication links, the data rate of each link is subject to degradation due to transmitting interference from other links. A competitive wireless game then arises as each link acts as a player maximizing its own data rate. The game outcome can be evaluated using the solution concept of game equilibria. However, when significant interference among the links arises, uniqueness of equilibrium is not guaranteed. To select among multiple equilibria, the sum of network rate or social welfare is used as the selection criterion. This thesis aims to offer the theoretical foundation and the computational tool for determining approximate correlated equilibria with global maximum expected social welfare in polynomial games. Using sum of utilities as the global objective, we give two theoretical and two wireless-specific contributions. 1. We give a problem formulation for computing near-exact ε -correlated equilibria with highest possible expected social welfare. We then give a sequential Semidefinite Programming (SDP) algorithm that computes the solution. The solution consists of bounds information on the social welfare. 2. We give a novel reformulation to arrive at a leaner problem for computing near-exact ε -correlated equilibria using Kantorovich polynomials with sparsity. 3. Forgoing near-exactness, we consider approximate correlated equilibria. To account for the loss in precision, we introduce the notion of regret. We give theoretical bounds on the regrets at any iteration of the sequential SDP algorithm. Moreover, we give a heuristic procedure for extracting a discrete probability distribution. Subject to players’ acceptance of the regrets, the computed distributions can be used to implement central arbitrators to facilitate real-life implementation of the correlated equilibrium concept. 4. We demonstrate how to compute Pareto-optimal solutions by dropping the correlated equilibria constraints. For demonstration purpose, we focus only on Pareto-optima with equal weights among the players

The Strongish Planted Clique Hypothesis and Its Consequences

We formulate a new hardness assumption, the Strongish Planted Clique Hypothesis (SPCH), which postulates that any algorithm for planted clique must run in time n^?(log n) (so that the state-of-the-art running time of n^O(log n) is optimal up to a constant in the exponent). We provide two sets of applications of the new hypothesis. First, we show that SPCH implies (nearly) tight inapproximability results for the following well-studied problems in terms of the parameter k: Densest k-Subgraph, Smallest k-Edge Subgraph, Densest k-Subhypergraph, Steiner k-Forest, and Directed Steiner Network with k terminal pairs. For example, we show, under SPCH, that no polynomial time algorithm achieves o(k)-approximation for Densest k-Subgraph. This inapproximability ratio improves upon the previous best k^o(1) factor from (Chalermsook et al., FOCS 2017). Furthermore, our lower bounds hold even against fixed-parameter tractable algorithms with parameter k. Our second application focuses on the complexity of graph pattern detection. For both induced and non-induced graph pattern detection, we prove hardness results under SPCH, improving the running time lower bounds obtained by (Dalirrooyfard et al., STOC 2019) under the Exponential Time Hypothesis

Combinatorial Optimization

This report summarizes the meeting on Combinatorial Optimization where new and promising developments in the field were discussed. Th