Optimal partitions in additively separable hedonic games

We study the problem of allocating indivisible binary-valued items on a path among agents. The objective is to find a fair and efficient allocation in which each agent's bundle forms a contiguous block on the path. We demonstrate that deciding whether every item can be allocated to an agent who wants it is NP-complete. Consequently, we provide fixed-parameter tractable (FPT) algorithms for maximizing utilitarian social welfare, with respect to the optimum value and the number of agents. Additionally, we present a 2-approximation algorithm for the special case when the maximum utility is equal to the number of items. Furthermore, we establish that deciding whether the maximum egalitarian social welfare is at least 2 or at most 1 is an NP-complete problem. We also explore the case where the order of the blocks of items allocated to the agents is predetermined. In this case, we show that both maximum utilitarian social welfare and egalitarian social welfare can be computed in polynomial time. However, we determine that checking the existence of an EF1 allocation is NP-complete.

Multiagent resource allocation provides mechanisms to allocate bundles of resources to agents, where resources are assumed to be indivisible and nonshareable. A central goal is to maximize social welfare of such allocations, which can be measured in terms of the sum of utilities realized by the agents (utilitarian social welfare), in terms of their minimum (egalitarian social welfare), and in terms of their product (Nash product social welfare). Unfortunately, social welfare optimization is a computationally intractable task in many settings. We survey recent approximability and inapproximability results on social welfare optimization in multiagent resource allocation, focusing on the two most central representation forms for utility functions of agents, the bundle form and the k-additive form. In addition, we provide some new (in)approximability results on maximizing egalitarian social welfare and social welfare with respect to the Nash product when restricted to certain special cases.

We study the problem of fair allocation of items / goods / resources to m agents. The problem appears in two broad variants: (a) fair allocation of a divisible item wherein one item is to be divided amongst m agents; the problem is popularly known as cake-cutting problem, (b) fair allocation of indivisible items wherein n indivisible or non-shareable items are to be allocated amongst m agents. Further, the item/items can be homogeneous or heterogeneous. Both the variants have several real-time applications such as land-division amongst heirs is an example of allocation of a divisible item, whereas in divorce settlement, division of assets such as house, car, artifacts, etc. is an example of allocation of indivisible items. There are many notions of fairness and social welfare that are important and have been studied: envy-freeness, proportionality, Pareto optimality, maximin share guarantee, Borda maximin share guarantee, utilitarian social welfare, egalitarian social welfare and Nash social welfare. In this chapter, we discuss about the fair-allocation problem and their related results.

A Latin square is an n×n matrix filled with n distinct symbols, each appearing exactly once in each row and column. We introduce a problem of allocating n indivisible items among n agents over n rounds while satisfying the Latin square constraint. This constraint ensures that each agent receives no more than one item per round and receives each item at most once. Each agent has an additive valuation on the item--round pairs. Real-world applications like scheduling, resource management, and experimental design require the Latin square constraint to satisfy fairness or balancedness in allocation. Our goal is to find a partial or complete allocation that maximizes the sum of the agents' valuations (utilitarian social welfare) or the minimum of the agents' valuations (egalitarian social welfare). For maximizing utilitarian social welfare, we prove NP-hardness even when the valuations are binary additive. We then provide (1-1/e) and (1-1/e)/4-approximation algorithms for partial and complete settings, respectively. Additionally, we present fixed-parameter tractable (FPT) algorithms with respect to the order of Latin square and the optimum value for both partial and complete settings. For maximizing egalitarian social welfare, we establish that deciding whether the optimum value is at most 1 or at least 2 is NP-hard for both the partial and complete settings, even when the valuations are binary. Furthermore, we prove that checking the existence of a complete allocation satisfying each of envy-free, proportional, equitable, envy-free up to any good, proportional up to any good, or equitable up to any good is NP-hard, even when the valuations are identical.

This paper considers the multi-unit resource allocation problem with a few distinct (indivisible and non-shareable) goods (or items, objects), each has a certain amount of copies. Every agent expresses her preference over multisets of goods by mean of utility function. The goal is to distribute goods among agents in such a way that maximizes either utilitarian social welfare or egalitarian social welfare or Nash product. Mathematically, such a problem can be formulated as an integer programming with linear constraints. We explore some special structure of this problem which facilitates the development of efficient (exact or approximation) algorithms. In addition, we concern with various types of utility functions, ranging from multi-minded functions, additive functions, submodular functions, to general functions.

Optimal cuts and partitions in tree metrics in polynomial time

An important task in multiagent resource allocation, which provides mechanisms to allocate bundles of (indivisible and nonshareable) resources to agents, is to maximize social welfare. We study the computational complexity of exact social welfare optimization by the Nash product, which can be seen as a sensible compromise between the well-known notions of utilitarian and egalitarian social welfare. When utilitiy functions are represented in the bundle or the k-additive form, for k ≥ 3, we prove that the corresponding computational problems are DP-complete (where DP denotes the second level of the boolean hierarchy over NP), thus confirming two conjectures raised by Roos and Rothe [10]. We also study the approximability of social welfare optimization problems.

We investigate pure Nash equilibria in generalized graphk-coloring games where we are given an edge-weighted undirected graph together with a set of k colors. Nodes represent players and edges capture their mutual interests. The strategy set of each player consists of k colors. The utility of a player v in a given state or coloring is given by the sum of the weights of edges {v, u} incident to v such that the color chosen by v is different than the one chosen by u, plus the profit gained by using the chosen color. Such games form some of the basic payoff structures in game theory, model lots of real-world scenarios with selfish players and extend or are related to several fundamental class of games. We first show that generalized graph k-coloring games are potential games. In particular, they are convergent and thus Nash Equilibria always exist. We then evaluate their performance by means of the widely used notions of price of anarchy and price of stability and provide tight bounds for two natural and widely used social welfare, i.e., utilitarian and egalitarian social welfare.

We consider a finite society with of individuals distributed along the real line. The individuals form jurisdictions to consume public projects, equally share their costs and, in addition, bear a transportation cost to the location of the project. We examine a core and Nash notions of stable jurisdiction structures and show that in hedonic games both solution sets could be empty. We demonstrate that in a quasi-hedonic set-up there is a Nash stable partition, but, in general, there are no core stable partitions. We then examine a subclass of societies that admits the existence of both types of stable partitions.

We consider a team formation setting where agents have varying levels of expertise in a global set of required skills, and teams are ranked with respect to how well the expertise of teammates complement each other. We model this setting as a hedonic game, and we show that this class of games possesses many desirable properties, some of which are as follows: A partition that is Nash stable, core stable and Pareto optimal is always guaranteed to exist. A contractually individually stable partition (and a Nash stable partition in a restricted setting) can be found in polynomial-time. A core stable partition can be approximated within a factor of \(1 - \frac{1}{e}\) and this bound is tight. We discover a larger and relatively general class of hedonic games, where the above existence guarantee holds. For this larger class, we present simple dynamics that converge to a Nash stable partition in a relatively low number of moves.

Stability of Jurisdiction Structures Under the Equal Share and Median Rules

Given n heterogeneous traffic sources which generate multiple types of traffic among themselves, we consider the problem of finding a set of disjoint clusters to cover n traffic sources. The objective is to minimize the total communication cost for the entire system in the context that the intracluster communication is less expensive than the intercluster communication. Different from the general graph partitioning problem, our work takes into account the physical topology constraints of the linear arrangement of physical cells in highway cellular systems and the hexagonal mesh arrangement of physical cells in cellular systems. In our partitioning schemes, the optimal partitioning problem is transformed into an equivalent problem with a relative cost function, which generates the communication cost differences between the intracluster communications and the intercluster communications. For highway cellular systems, we have designed an efficient optimal partitioning algorithm of O(mn/sup 2/) by dynamic programming, where m is the number of clusters of n base stations. The algorithm also finds all the valid partitions in the same polynomial time, given the size constraint on a cluster and the total allowable communication cost for the entire system. For hexagonal cellular systems, we have developed four heuristics for the optimal partitioning based on the techniques of moving or interchanging boundary nodes between adjacent clusters. The heuristics have been evaluated and compared through experimental testing and analysis.

We prove that the problem of deciding whether a Nash stable partition exists in an Additively Separable Hedonic Game is NP-complete. We also show that the problem of deciding whether a non trivial Nash stable partition exists in an Additively Separable Hedonic Game with non-negative and symmetric preferences is NP-complete. We motivate our study of the computational complexity by linking Nash stable partitions in Additively Separable Hedonic Games to community structures in networks. Our results formally justify that computing community structures in general is hard.

Hedonic games provide a model of coalition formation in which a set of agents is partitioned into coalitions and the agents have preferences over which set they belong to. Recently, Aziz et. al. (2014) have initiated the study of hedonic games with dichotomous preferences, where each agent either approves or disapproves of a given coalition. In this work, we study the computational complexity of questions related to finding optimal and stable partitions in dichotomous hedonic games under various ways of restricting and representing the collection of approved coalitions. Encouragingly, many of these problems turn out to be polynomial-time solvable. In particular, we show that an individually stable outcome always exists and can be found in polynomial time. We also provide efficient algorithms for cases in which agents approve only few coalitions, in which they only approve intervals, and in which they only approve sets of size 2 (the roommates case). These algorithms are complemented by NP-hardness results, especially for representations that are very expressive, such as in the case when agents' goals are given by propositional formulas.

In this paper, we develop a game theoretic approach for clustering features in a learning problem. Feature clustering can serve as an important preprocessing step in many problems such as feature selection, dimensionality reduction, etc. In this approach, we view features as rational players of a coalitional game where they form coalitions (or clusters) among themselves in order to maximize their individual payoffs. We show how Nash Stable Partition (NSP), a well known concept in the coalitional game theory, provides a natural way of clustering features. Through this approach, one can obtain some desirable properties of the clusters by choosing appropriate payoff functions. For a small number of features, the NSP based clustering can be found by solving an integer linear program (ILP). However, for large number of features, the ILP based approach does not scale well and hence we propose a hierarchical approach. Interestingly, a key result that we prove on the equivalence between a k-size NSP of a coalitional game and minimum k-cut of an appropriately constructed graph comes in handy for large scale problems. In this paper, we use feature selection problem (in a classification setting) as a running example to illustrate our approach. We conduct experiments to illustrate the efficacy of our approach.