The Feedback Arc Set Problem with Triangle Inequality is a Vertex Cover Problem

We consider the (precedence constrained) Minimum Feedback Arc Set problem with triangle inequalities on the weights, which finds important applications in problems of ranking with inconsistent information. We present a surprising structural insight showing that the problem is a special case of the minimum vertex cover in hypergraphs with edges of size at most 3. This result leads to combinatorial approximation algorithms for the problem and opens the road to studying the problem as a vertex cover problem

Dagstuhl Reports : Volume 1, Issue 2, February 2011

Online Privacy: Towards Informational Self-Determination on the Internet (Dagstuhl Perspectives Workshop 11061) : Simone Fischer-Hübner, Chris Hoofnagle, Kai Rannenberg, Michael Waidner, Ioannis Krontiris and Michael Marhöfer Self-Repairing Programs (Dagstuhl Seminar 11062) : Mauro Pezzé, Martin C. Rinard, Westley Weimer and Andreas Zeller Theory and Applications of Graph Searching Problems (Dagstuhl Seminar 11071) : Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer and Dimitrios M. Thilikos Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081) : Maxime Crochemore, Lila Kari, Mehryar Mohri and Dirk Nowotka Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091) Klaus Jansen, Claire Mathieu, Hadas Shachnai and Neal E. Youn

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

Scheduling with precedence constraints of low fractional dimension

Abstract. We consider the single machine scheduling problem to minimize the average weighted completion time under precedence constrains. Improving on the various 2-approximation algorithms is considered one of the ten most prominent open problems in scheduling theory. Recently, research has focused on special cases of the problem, mostly by restricting the set of precedence constraints to special classes such as convex bipartite, two-dimensional, and interval orders. In this paper we extend our previous results by presenting a framework for obtaining (2 − 2/d)-approximation algorithms provided that the set of precedence constraints has fractional dimension d. Our generalized approach yields the best known approximation ratios for all previously considered classes of precedence constraints, and it provides the first results for bounded degree and interval dimension 2 orders. As a negative result we show that the addressed problem remains NPhard even when restricted to the special case of interval orders.