65 research outputs found
Approximating the least core value and least core of cooperative games with supermodular costs
We study the approximation of the least core value and the least core of supermodular cost cooperative games. We provide a framework for approximation based on oracles that approximately determine maximally violated constraints. This framework yields a 3-approximation algorithm for computing the least core value of supermodular cost cooperative games, and a polynomial-time algorithm for computing a cost allocation in the 2-approximate least core of these games. This approximation framework extends naturally to submodular profit cooperative games. For scheduling games, a special class of supermodular cost cooperative games, we give a fully polynomial-time approximation scheme for computing the least core value. For matroid profit games, a special class of submodular profit cooperative games, we give exact polynomial-time algorithms for computing the least core value as well as a least core cost allocation.National Science Foundation (U.S.) (DMI-0426686
Sharing Supermodular Costs
We study cooperative games with supermodular costs. We show that supermodular costs arise in a variety of situations; in particular, we show that the problem of minimizing a linear function over a supermodular polyhedron—a problem that often arises in combinatorial optimization—has supermodular optimal costs. In addition, we examine the computational complexity of the least core and least core value of supermodular cost cooperative games. We show that the problem of computing the least core value of these games is strongly NP-hard and, in fact, is inapproximable within a factor strictly less than 17/16 unless P = NP. For a particular class of supermodular cost cooperative games that arises from a scheduling problem, we show that the Shapley value—which, in this case, is computable in polynomial time—is in the least core, while computing the least core value is NP-hard.National Science Foundation (U.S.) (DMI-0426686
Coreness of Cooperative Games with Truncated Submodular Profit Functions
Coreness represents solution concepts related to core in cooperative games,
which captures the stability of players. Motivated by the scale effect in
social networks, economics and other scenario, we study the coreness of
cooperative game with truncated submodular profit functions. Specifically, the
profit function is defined by a truncation of a submodular function
: if and
otherwise, where is a given threshold. In this paper, we
study the core and three core-related concepts of truncated submodular profit
cooperative game. We first prove that whether core is empty can be decided in
polynomial time and an allocation in core also can be found in polynomial time
when core is not empty. When core is empty, we show hardness results and
approximation algorithms for computing other core-related concepts including
relative least-core value, absolute least-core value and least average
dissatisfaction value
Algorithmic and game-theoretic perspectives on scheduling
This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2008.Includes bibliographical references (p. 103-110).(cont.) Second, for almost all 0-1 bipartite instances, we give a lower bound on the integrality gap of various linear programming relaxations of this problem. Finally, we show that for almost all 0-1 bipartite instances, all feasible schedules are arbitrarily close to optimal. Finally, we consider the problem of minimizing the sum of weighted completion times in a concurrent open shop environment. We present some interesting properties of various linear programming relaxations for this problem, and give a combinatorial primal-dual 2-approximation algorithm.In this thesis, we study three problems related to various algorithmic and game-theoretic aspects of scheduling. First, we apply ideas from cooperative game theory to study situations in which a set of agents faces super modular costs. These situations appear in a variety of scheduling contexts, as well as in some settings related to facility location and network design. Although cooperation is unlikely when costs are super modular, in some situations, the failure to cooperate may give rise to negative externalities. We study the least core value of a cooperative game -- the minimum penalty we need to charge a coalition for acting independently that ensures the existence of an efficient and stable cost allocation -- as a means of encouraging cooperation. We show that computing the least core value of supermodular cost cooperative games is strongly NP-hard, and design an approximation framework for this problem that in the end, yields a (3 + [epsilon])-approximation algorithm. We also apply our approximation framework to obtain better results for two special cases of supermodular cost cooperative games that arise from scheduling and matroid optimization. Second, we focus on the classic precedence- constrained single-machine scheduling problem with the weighted sum of completion times objective. We focus on so-called 0-1 bipartite instances of this problem, a deceptively simple class of instances that has virtually the same approximability behavior as arbitrary instances. In the hope of improving our understanding of these instances, we use models from random graph theory to look at these instances with a probabilistic lens. First, we show that for almost all 0-1 bipartite instances, the decomposition technique of Sidney (1975) does not yield a non-trivial decomposition.by Nelson A. Uhan.Ph.D
Shapley Meets Shapley
This paper concerns the analysis of the Shapley value in matching games.
Matching games constitute a fundamental class of cooperative games which help
understand and model auctions and assignments. In a matching game, the value of
a coalition of vertices is the weight of the maximum size matching in the
subgraph induced by the coalition. The Shapley value is one of the most
important solution concepts in cooperative game theory.
After establishing some general insights, we show that the Shapley value of
matching games can be computed in polynomial time for some special cases:
graphs with maximum degree two, and graphs that have a small modular
decomposition into cliques or cocliques (complete k-partite graphs are a
notable special case of this). The latter result extends to various other
well-known classes of graph-based cooperative games.
We continue by showing that computing the Shapley value of unweighted
matching games is #P-complete in general. Finally, a fully polynomial-time
randomized approximation scheme (FPRAS) is presented. This FPRAS can be
considered the best positive result conceivable, in view of the #P-completeness
result.Comment: 17 page
Computing Stable Coalitions: Approximation Algorithms for Reward Sharing
Consider a setting where selfish agents are to be assigned to coalitions or
projects from a fixed set P. Each project k is characterized by a valuation
function; v_k(S) is the value generated by a set S of agents working on project
k. We study the following classic problem in this setting: "how should the
agents divide the value that they collectively create?". One traditional
approach in cooperative game theory is to study core stability with the
implicit assumption that there are infinite copies of one project, and agents
can partition themselves into any number of coalitions. In contrast, we
consider a model with a finite number of non-identical projects; this makes
computing both high-welfare solutions and core payments highly non-trivial.
The main contribution of this paper is a black-box mechanism that reduces the
problem of computing a near-optimal core stable solution to the purely
algorithmic problem of welfare maximization; we apply this to compute an
approximately core stable solution that extracts one-fourth of the optimal
social welfare for the class of subadditive valuations. We also show much
stronger results for several popular sub-classes: anonymous, fractionally
subadditive, and submodular valuations, as well as provide new approximation
algorithms for welfare maximization with anonymous functions. Finally, we
establish a connection between our setting and the well-studied simultaneous
auctions with item bidding; we adapt our results to compute approximate pure
Nash equilibria for these auctions.Comment: Under Revie
Methodologies for Analyzing Equilibria in Wireless Games
Under certain assumptions in terms of information and models, equilibria
correspond to possible stable outcomes in conflicting or cooperative scenarios
where rational entities interact. For wireless engineers, it is of paramount
importance to be able to predict and even ensure such states at which the
network will effectively operate. In this article, we provide non-exhaustive
methodologies for characterizing equilibria in wireless games in terms of
existence, uniqueness, selection, and efficiency.Comment: To appear in IEEE Signal Processing Magazine, Sep. 200
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