Elliot Anshelevich

Near-optimal network design with selfish agents

We introduce a simple network design game that models how independent selfish agents can build or maintain a large network. In our game every agent has a specific connectivity requirement, i.e. each agent has a set of terminals and wants to build a network in which his terminals are connected. Possible edges in the network have costs and each agents goal is to pay as little as possible. Determining whether or not a Nash equilibrium exists in this game is NP-complete. However, when the goal of each player is to connect a terminal to a common source, we prove that there is a Nash equilibrium as cheap as the optimal network, and give a polynomial time algorithm to find a (1+ε)-approximate Nash equilibrium that does not cost much more. For the general connection game we prove that there is a 3-approximate Nash equilibrium that is as cheap as the optimal network, and give an algorithm to find a (4.65+ε)-approximate Nash equilibrium that does not cost much more.

Deformable volumes in path planning applications

This paper addresses the problem of path planning for a class of deformable volumes under fairly general manipulation constraints. The underlying geometric model for the volume is provided by a mass-spring representation. It is augmented by a realistic mechanical model. The latter permits the computation of the shape of the considered object with respect to the grasping constraints by minimizing the energy function of the deformation of the object. Previous research in planning for deformable objects considered the case of elastic plates and proposed a randomized framework for planning paths for plates under manipulation constraints. The present paper modifies and extends the previously proposed framework to handle simple volumes. Our planner builds a roadmap in the configuration space. The nodes of the roadmap are equilibrium configurations of the considered volume under the manipulation constraints, while its edges correspond to quasi-static equilibrium paths. Paths are found by searching the roadmap. We present experimental results that illustrate our approach.

Anarchy, stability, and utopia: creating better matchings

Historically, the analysis of matching has centered on designing algorithms to produce stable matchings as well as on analyzing the incentive compatibility of matching mechanisms. Less attention has been paid to questions related to the social welfare of stable matchings in cardinal utility models. We examine the loss in social welfare that arises from requiring matchings to be stable, the natural equilibrium concept under individual rationality. While this loss can be arbitrarily bad under general preferences, when there is some structure to the underlying graph corresponding to natural conditions on preferences, we prove worst case bounds on the price of anarchy. Surprisingly, under simple distributions of utilities, the average case loss turns out to be significantly smaller than the worst-case analysis would suggest. Furthermore, we derive conditions for the existence of approximately stable matchings that are also close to socially optimal, demonstrating that adding small switching costs can make socially (near-)optimal matchings stable. Our analysis leads to several concomitant results of interest on the convergence of decentralized partner-switching algorithms, and on the impact of heterogeneity of tastes on social welfare.

Stability of load balancing algorithms in dynamic adversarial systems

In the dynamic load balancing problem, we seek to keep the job load roughly evenly distributed among the processors of a given network. The arrival and departure of jobs is modeled by an adversary restricted in its power. Muthukrishnan and Rajaraman (1998) gave a clean characterization of a restriction on the adversary that can be considered the natural analogue of a cut condition. They proved that a simple local balancing algorithm proposed by Aiello et. al. (1993) is stable against such an adversary if the insertion rate is restricted to a (1—ε) fraction of the cut size. They left as an open question whether the algorithm is stable at rate 1.In this paper, we resolve this question positively, by proving stability of the local algorithm at rate 1. Our proof techniques are very different from the ones used by Muthukrishnan and Rajaraman, and yield a simpler proof and tighter bounds on the difference in loads.In addition, we introduce a multi-commodity version of this load balancing model, and show how to extend the result to the case of balancing two different kinds of loads at once (obtaining as a corollary a new proof of the 2-commodity Max-Flow Min-Cut Theorem). We also show how to apply the proof techniques to the problem of routing packets in adversarial systems. Awerbuch et. al. (2001) showed that the same load balancing algorithm is stable against an adversary inserting packets at rate 1 with a single destination, in dynamically changing networks. Our techniques give a much simpler proof for a different model of adversarially changing networks.

Approximability of the Firefighter Problem: Computing Cuts over Time

We provide approximation algorithms for several variants of the Firefighter problem on general graphs. The Firefighter problem models the case where a diffusive process such as an infection (or an idea, a computer virus, a fire) is spreading through a network, and our goal is to contain this infection by using targeted vaccinations. Specifically, we are allowed to vaccinate at most a fixed number (called the budget) of nodes per time step, with the goal of minimizing the effect of the infection. The difficulty of this problem comes from its temporal component, since we must choose nodes to vaccinate at every time step while the infection is spreading through the network, leading to notions of “cuts over time”.We consider two versions of the Firefighter problem: a “non-spreading” model, where vaccinating a node means only that this node cannot be infected; and a “spreading” model where the vaccination itself is an infectious process, such as in the case where the infection is a harmful idea, and the vaccine to it is another infectious beneficial idea. We look at two measures: the MaxSave measure in which we want to maximize the number of nodes which are not infected given a fixed budget, and the MinBudget measure, in which we are given a set of nodes which we have to save and the goal is to minimize the budget. We study the approximability of these problems in both models.

Stability of Load Balancing Algorithms in Dynamic Adversarial Systems

In the dynamic load balancing problem, we seek to keep the job load roughly evenly distributed among the processors of a given network. The arrival and departure of jobs is modeled by an adversary restricted in its power. Muthukrishnan and Rajaraman [An adversarial model for distributed dynamic load balancing, in Proceedings of the 10th ACM Symposium on Parallel Algorithms and Architectures, ACM, New York, 1998] gave a clean characterization of a restriction on the adversary that can be considered the natural analogue of a cut condition. They proved that a simple local balancing algorithm proposed by Aiello et al. [Approximate load balancing on dynamic and asynchronous networks, in Proceedings of the 25th ACM Symposium on Theory of Computing, ACM, New York, 1993] is stable against such an adversary if the insertion rate is restricted to a

Terminal backup, 3D matching, and covering cubic graphs

(1-\varepsilon)

Strategic network formation through peering and service agreements

fraction of the cut size. They left as an open question whether the algorithm is stable at rate 1. In this paper, we resolve this question positively, by proving stability of the local algorithm at rate 1. Our proof techniques are very different from the ones used by Muthukrishnan and Rajaraman and yield a simpler proof and tighter bounds on the difference in loads. In addition, we introduce a multicommodity version of this load balancing model and show how to extend the result to the case of balancing two different kinds of loads at once (obtaining as a corollary a new proof of the 2-commodity Max-Flow Min-Cut Theorem). We also show how to apply the proof techniques to the problem of routing packets in adversarial systems. Awerbuch et al. [Simple routing strategies for adversarial systems, in Proceedings of the 42nd IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, Los Alamitos, CA, 2001] showed that the same load balancing algorithm is stable against an adversary, inserting packets at rate 1 with a single destination in dynamically changing networks. Our techniques give a much simpler proof for a different model of adversarially changing networks.

Anarchy, Stability, and Utopia: Creating Better Matchings

We define a problem called Simplex Matching, and show that it is solvable in polynomial time. While Simplex Matching is interesting in its own right as a nontrivial extension of non-bipartite min-cost matching, its main value lies in many(seemingly very different) problems that can be solved using ouralgorithm. For example, suppose that we are given a graph with terminal nodes, non-terminal nodes, and edge costs. Then, the Terminal Backup problem, which consists of finding the cheapest forest connecting every terminal to at least one other terminal, is reducible to Simplex Matching. Simplex Matching is also useful for various tasks that involve forming groups of at least two members, such as project assignment and variants of facility location. In an instance of Simplex Matching, we are given a hypergraphH with edge costs, and edge size at most 3. We show how to find the min-cost perfect matching of H efficiently, if the edge costs obey a simple and realistic inequality that we call the SimplexCondition. The algorithm we provide is relatively simple to understand and implement, but difficult to prove correct. In the process of this proof we show some powerful new results about covering cubic graphs with simple combinatorial objects.

Exact and Approximate Equilibria for Optimal Group Network Formation

We introduce a game theoretic model of network formation in an effort to understand the complex system of business relationships between various Internet entities (e.g., Autonomous Systems, enterprise networks, residential customers). In our model we are given a network topology of nodes and links where the nodes act as the players of the game, and links represent potential contracts. Nodes wish to satisfy their demands, which earn potential revenues, but may have to pay their neighbors for links incident to them. We incorporate some of the qualities of Internet business relationships, including customer-provider and peering contracts. We show that every Nash equilibrium can be represented by a circulation flow of utility with certain constraints. This allows us to prove bounds on the prices of anarchy and stability. We also focus on the quality of equilibria achievable through centralized incentives.