Committee selection with multimodal preferences

On the Rationale of Group Decision-making

Abstract: When a decision is reached by voting or is arrived at by a group all of whose members are not in complete accord, there is no part of economic theory which applies. This paper is intended to help fill this gap; to provide a type of reasoning which will contribute to the development of the theory of tradeunions, the firm, and the cartel; and to provide the basis for a theory of the equilibrium distribution of taxation or of public expenditure. Still other uses of the theory might be not less important. For reasons of space we avoid discussion of many points that demand fuller treatment and only attempt to indicate the course of the argument.

Fixed-parameter tractability and completeness II: on completeness for W [1]

Abstract: For many fixed-parameter problems that are trivially solvable in polynomial-time, such as k -DOMINATING SET, essentially no better algorithm is presently known than the one which tries all possible solutions. Other problems, such as FEEDBACK VERTEX SET, exhibit fixed-parameter tractability : for each fixed k the problem is solvable in time bounded by a polynomial of degree c , where c is a constant independent of k . In a previous paper, the W Hierarchy of parameterized problems was defined, and complete problems were identified for the classes W [ t ] for t ⩾ 2. Our main result shows that INDEPENDENT SET is complete for W [1].

On the computation of fully proportional representation

Abstract: We investigate two systems of fully proportional representation suggested by Chamberlin & Courant and Monroe. Both systems assign a representative to each voter so that the "sum of misrepresentations" is minimized. The winner determination problem for both systems is known to be NP-hard, hence this work aims at investigating whether there are variants of the proposed rules and/or specific electorates for which these problems can be solved efficiently. As a variation of these rules, instead of minimizing the sum of misrepresentations, we considered minimizing the maximal misrepresentation introducing effectively two new rules. In the general case these "minimax" versions of classical rules appeared to be still NP-hard. We investigated the parameterized complexity of winner determination of the two classical and two new rules with respect to several parameters. Here we have a mixture of positive and negative results: e.g., we proved fixed-parameter tractability for the parameter the number of candidates but fixed-parameter intractability for the number of winners. For single-peaked electorates our results are overwhelmingly positive: we provide polynomial-time algorithms for most of the considered problems. The only rule that remains NP-hard for single-peaked electorates is the classical Monroe rule.

On the complexity of achieving proportional representation

Abstract: We demonstrate that winner selection in two prominent proportional representation voting systems is a computationally intractable problem—implying that these systems are impractical when the assembly is large. On a different note, in settings where the size of the assembly is constant, we show that the problem can be solved in polynomial time.

Finding a collective set of items: from proportional multirepresentation to group recommendation

Abstract: We consider the following problem: There is a set of items (e.g., movies) and a group of agents (e.g., passengers on a plane); each agent has some intrinsic utility for each of the items. Our goal is to pick a set of K items that maximize the total derived utility of all the agents (i.e., in our example we are to pick K movies that we put on the plane's entertainment system). However, the actual utility that an agent derives from a given item is only a fraction of its intrinsic one, and this fraction depends on how the agent ranks the item among the chosen, available, ones. We provide a formal specification of the model and provide concrete examples and settings where it is applicable. We show that the problem is hard in general, but we show a number of tractability results for its natural special cases.