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

Incentive and normative analysis on sequencing problem

We identify the complete class of transfer rules that guarantee strategyproofness of any non-increasing in completion time allocation rule for the sequencing problem. We then characterize the class of mechanisms satisfying efficiency of decision (or aggregate cost minimization), egalitarian equivalence and strategyproofness. There is no mechanism in this class that satisfies either feasibility or weak group strategyproofness. Finally we identify the restrictions under which egalitarian equivalence, efficiency of decision, identical preference lower bound and strategyproofness are compatible

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