Greedy Algorithms for Steiner Forest

In the Steiner Forest problem, we are given terminal pairs $\{s_i, t_i\}$, and need to find the cheapest subgraph which connects each of the terminal pairs together. In 1991, Agrawal, Klein, and Ravi, and Goemans and Williamson gave primal-dual constant-factor approximation algorithms for this problem; until now, the only constant-factor approximations we know are via linear programming relaxations. We consider the following greedy algorithm: Given terminal pairs in a metric space, call a terminal "active" if its distance to its partner is non-zero. Pick the two closest active terminals (say $s_i, t_j$), set the distance between them to zero, and buy a path connecting them. Recompute the metric, and repeat. Our main result is that this algorithm is a constant-factor approximation. We also use this algorithm to give new, simpler constructions of cost-sharing schemes for Steiner forest. In particular, the first "group-strict" cost-shares for this problem implies a very simple combinatorial sampling-based algorithm for stochastic Steiner forest

Smooth Inequalities and Equilibrium Inefficiency in Scheduling Games

We study coordination mechanisms for Scheduling Games (with unrelated machines). In these games, each job represents a player, who needs to choose a machine for its execution, and intends to complete earliest possible. Our goal is to design scheduling policies that always admit a pure Nash equilibrium and guarantee a small price of anarchy for the l_k-norm social cost --- the objective balances overall quality of service and fairness. We consider policies with different amount of knowledge about jobs: non-clairvoyant, strongly-local and local. The analysis relies on the smooth argument together with adequate inequalities, called smooth inequalities. With this unified framework, we are able to prove the following results. First, we study the inefficiency in l_k-norm social costs of a strongly-local policy SPT and a non-clairvoyant policy EQUI. We show that the price of anarchy of policy SPT is O(k). We also prove a lower bound of Omega(k/log k) for all deterministic, non-preemptive, strongly-local and non-waiting policies (non-waiting policies produce schedules without idle times). These results ensure that SPT is close to optimal with respect to the class of l_k-norm social costs. Moreover, we prove that the non-clairvoyant policy EQUI has price of anarchy O(2^k). Second, we consider the makespan (l_infty-norm) social cost by making connection within the l_k-norm functions. We revisit some local policies and provide simpler, unified proofs from the framework's point of view. With the highlight of the approach, we derive a local policy Balance. This policy guarantees a price of anarchy of O(log m), which makes it the currently best known policy among the anonymous local policies that always admit a pure Nash equilibrium.Comment: 25 pages, 1 figur

LP-based Covering Games with Low Price of Anarchy

We present a new class of vertex cover and set cover games. The price of anarchy bounds match the best known constant factor approximation guarantees for the centralized optimization problems for linear and also for submodular costs -- in contrast to all previously studied covering games, where the price of anarchy cannot be bounded by a constant (e.g. [6, 7, 11, 5, 2]). In particular, we describe a vertex cover game with a price of anarchy of 2. The rules of the games capture the structure of the linear programming relaxations of the underlying optimization problems, and our bounds are established by analyzing these relaxations. Furthermore, for linear costs we exhibit linear time best response dynamics that converge to these almost optimal Nash equilibria. These dynamics mimic the classical greedy approximation algorithm of Bar-Yehuda and Even [3]

Special Section on the Forty-First Annual ACM Symposium on Theory of Computing (STOC 2009)

This issue of SICOMP contains nine specially selected papers from the Forty-first Annual ACM Symposium on the Theory of Computing, otherwise known as STOC 2009, held May 31 to June 2 in Bethesda, Maryland. The papers here were chosen to represent both the excellence and the broad range of the STOC program. The papers have been revised and extended by the authors, and subjected to the standard thorough reviewing process of SICOMP. The program committee consisted of Susanne Albers, Andris Ambainis, Nikhil Bansal, Paul Beame, Andrej Bogdanov, Ran Canetti, David Eppstein, Dmitry Gavinsky, Shafi Goldwasser, Nicole Immorlica, Anna Karlin, Jonathan Katz, Jonathan Kelner, Subhash Khot, Ravi Kumar, Leslie Ann Goldberg, Michael Mitzenmacher (Chair), Kamesh Munagala, Rasmus Pagh, Anup Rao, Rocco Servedio, Mikkel Thorup, Chris Umans, and Lisa Zhang. They accepted 77 papers out of 321 submissions

Network Cournot Competition

Cournot competition is a fundamental economic model that represents firms competing in a single market of a homogeneous good. Each firm tries to maximize its utility---a function of the production cost as well as market price of the product---by deciding on the amount of production. In today's dynamic and diverse economy, many firms often compete in more than one market simultaneously, i.e., each market might be shared among a subset of these firms. In this situation, a bipartite graph models the access restriction where firms are on one side, markets are on the other side, and edges demonstrate whether a firm has access to a market or not. We call this game \emph{Network Cournot Competition} (NCC). In this paper, we propose algorithms for finding pure Nash equilibria of NCC games in different situations. First, we carefully design a potential function for NCC, when the price functions for markets are linear functions of the production in that market. However, for nonlinear price functions, this approach is not feasible. We model the problem as a nonlinear complementarity problem in this case, and design a polynomial-time algorithm that finds an equilibrium of the game for strongly convex cost functions and strongly monotone revenue functions. We also explore the class of price functions that ensures strong monotonicity of the revenue function, and show it consists of a broad class of functions. Moreover, we discuss the uniqueness of equilibria in both of these cases which means our algorithms find the unique equilibria of the games. Last but not least, when the cost of production in one market is independent from the cost of production in other markets for all firms, the problem can be separated into several independent classical \emph{Cournot Oligopoly} problems. We give the first combinatorial algorithm for this widely studied problem

Efficiency in Multi-objective Games

In a multi-objective game, each agent individually evaluates each overall action-profile on multiple objectives. I generalize the price of anarchy to multi-objective games and provide a polynomial-time algorithm to assess it. This work asserts that policies on tobacco promote a higher economic efficiency

Designing cost-sharing methods for Bayesian games

We study the design of cost-sharing protocols for two fundamental resource allocation problems, the Set Cover and the Steiner Tree Problem, under environments of incomplete information (Bayesian model). Our objective is to design protocols where the worst-case Bayesian Nash equilibria, have low cost, i.e. the Bayesian Price of Anarchy (PoA) is minimized. Although budget balance is a very natural requirement, it puts considerable restrictions on the design space, resulting in high PoA. We propose an alternative, relaxed requirement called budget balance in the equilibrium (BBiE).We show an interesting connection between algorithms for Oblivious Stochastic optimization problems and cost-sharing design with low PoA. We exploit this connection for both problems and we enforce approximate solutions of the stochastic problem, as Bayesian Nash equilibria, with the same guarantees on the PoA. More interestingly, we show how to obtain the same bounds on the PoA, by using anonymous posted prices which are desirable because they are easy to implement and, as we show, induce dominant strategies for the players

Improving the Price of Anarchy for Selfish Routing via Coordination Mechanisms

We reconsider the well-studied Selfish Routing game with affine latency functions. The Price of Anarchy for this class of games takes maximum value 4/3; this maximum is attained already for a simple network of two parallel links, known as Pigou's network. We improve upon the value 4/3 by means of Coordination Mechanisms. We increase the latency functions of the edges in the network, i.e., if $\ell_e(x)$ is the latency function of an edge $e$, we replace it by $\hat{\ell}_e(x)$ with $\ell_e(x) \le \hat{\ell}_e(x)$ for all $x$. Then an adversary fixes a demand rate as input. The engineered Price of Anarchy of the mechanism is defined as the worst-case ratio of the Nash social cost in the modified network over the optimal social cost in the original network. Formally, if \CM(r) denotes the cost of the worst Nash flow in the modified network for rate $r$ and \Copt(r) denotes the cost of the optimal flow in the original network for the same rate then [\ePoA = \max_{r \ge 0} \frac{\CM(r)}{\Copt(r)}.] We first exhibit a simple coordination mechanism that achieves for any network of parallel links an engineered Price of Anarchy strictly less than 4/3. For the case of two parallel links our basic mechanism gives 5/4 = 1.25. Then, for the case of two parallel links, we describe an optimal mechanism; its engineered Price of Anarchy lies between 1.191 and 1.192.Comment: 17 pages, 2 figures, preliminary version appeared at ESA 201

Scheduling Games with Machine-Dependent Priority Lists

We consider a scheduling game in which jobs try to minimize their completion time by choosing a machine to be processed on. Each machine uses an individual priority list to decide on the order according to which the jobs on the machine are processed. We characterize four classes of instances in which a pure Nash equilibrium (NE) is guaranteed to exist, and show, by means of an example, that none of these characterizations can be relaxed. We then bound the performance of Nash equilibria for each of these classes with respect to the makespan of the schedule and the sum of completion times. We also analyze the computational complexity of several problems arising in this model. For instance, we prove that it is NP-hard to decide whether a NE exists, and that even for instances with identical machines, for which a NE is guaranteed to exist, it is NP-hard to approximate the best NE within a factor of $2-\frac{1}{m}-\epsilon$ for all $\epsilon>0$. In addition, we study a generalized model in which players' strategies are subsets of resources, each having its own priority list over the players. We show that in this general model, even unweighted symmetric games may not have a pure NE, and we bound the price of anarchy with respect to the total players' costs.Comment: 19 pages, 2 figure