Optimal Partitions in Additively Separable Hedonic Games

We conduct a computational analysis of fair and optimal partitions in additively separable hedonic games. We show that, for strict preferences, a Pareto optimal partition can be found in polynomial time while verifying whether a given partition is Pareto optimal is coNP-complete, even when preferences are symmetric and strict. Moreover, computing a partition with maximum egalitarian or utilitarian social welfare or one which is both Pareto optimal and individually rational is NP-hard. We also prove that checking whether there exists a partition which is both Pareto optimal and envy-free is $\Sigma_{2}^{p}$-complete. Even though an envy-free partition and a Nash stable partition are both guaranteed to exist for symmetric preferences, checking whether there exists a partition which is both envy-free and Nash stable is NP-complete.Comment: 11 pages; A preliminary version of this work was invited for presentation in the session `Cooperative Games and Combinatorial Optimization' at the 24th European Conference on Operational Research (EURO 2010) in Lisbo

Approximability results for stable marriage problems with ties

We consider instances of the classical stable marriage problem in which persons may include ties in their preference lists. We show that, in such a setting, strong lower bounds hold for the approximability of each of the problems of finding an egalitarian, minimum regret and sex-equal stable matching. We also consider stable marriage instances in which persons may express unacceptable partners in addition to ties. In this setting, we prove that there are constants delta, delta' such that each of the problems of approximating a maximum and minimum cardinality stable matching within factors of delta, delta' (respectively) is NP-hard, under strong restrictions. We also give an approximation algorithm for both problems that has a performance guarantee expressible in terms of the number of lists with ties. This significantly improves on the best-known previous performance guarantee, for the case that the ties are sparse. Our results have applications to large-scale centralized matching schemes

Fairly Allocating Contiguous Blocks of Indivisible Items

In this paper, we study the classic problem of fairly allocating indivisible items with the extra feature that the items lie on a line. Our goal is to find a fair allocation that is contiguous, meaning that the bundle of each agent forms a contiguous block on the line. While allocations satisfying the classical fairness notions of proportionality, envy-freeness, and equitability are not guaranteed to exist even without the contiguity requirement, we show the existence of contiguous allocations satisfying approximate versions of these notions that do not degrade as the number of agents or items increases. We also study the efficiency loss of contiguous allocations due to fairness constraints.Comment: Appears in the 10th International Symposium on Algorithmic Game Theory (SAGT), 201

The Complexity of Fully Proportional Representation for Single-Crossing Electorates

We study the complexity of winner determination in single-crossing elections under two classic fully proportional representation rules---Chamberlin--Courant's rule and Monroe's rule. Winner determination for these rules is known to be NP-hard for unrestricted preferences. We show that for single-crossing preferences this problem admits a polynomial-time algorithm for Chamberlin--Courant's rule, but remains NP-hard for Monroe's rule. Our algorithm for Chamberlin--Courant's rule can be modified to work for elections with bounded single-crossing width. To circumvent the hardness result for Monroe's rule, we consider single-crossing elections that satisfy an additional constraint, namely, ones where each candidate is ranked first by at least one voter (such elections are called narcissistic). For single-crossing narcissistic elections, we provide an efficient algorithm for the egalitarian version of Monroe's rule.Comment: 23 page

Efficient algorithms for generalized Stable Marriage and Roommates problems

We consider a generalization of the Stable Roommates problem (SR), in which preference lists may be partially ordered and forbidden pairs may be present, denoted by SRPF. This includes, as a special case, a corresponding generalization of the classical Stable Marriage problem (SM), denoted by SMPF. By extending previous work of Feder, we give a two-step reduction from SRPF to 2-SAT. This has many consequences, including fast algorithms for a range of problems associated with finding "optimal" stable matchings and listing all solutions, given variants of SR and SM. For example, given an SMPF instance I, we show that there exists an O(m) "succinct" certificate for the unsolvability of I, an O(m) algorithm for finding all the super-stable pairs in I, an O(m+kn) algorithm for listing all the super-stable matchings in I, an O(m<sup>1.5</sup>) algorithm for finding an egalitarian super-stable matching in I, and an O(m) algorithm for finding a minimum regret super-stable matching in I, where n is the number of men, m is the total length of the preference lists, and k is the number of super-stable matchings in I. Analogous results apply in the case of SRPF