Showing papers by "Piotr Faliszewski published in 2013"

The Complexity of Fully Proportional Representation for Single-Crossing Electorates

Abstract: 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.

Fully proportional representation as resource allocation: approximability results

Abstract: We study the complexity of (approximate) winner determination under Monroe's and Chamberlin-Courant's multiwinner voting rules, where we focus on the total (dis)satisfaction of the voters (the utilitarian case) or the (dis)satisfaction of the worst-off voter (the egalitarian case) We show good approximation algorithms for the satisfaction-based utilitarian cases, and inapproximability results for the remaining settings

Achieving fully proportional representation is easy in practice

Abstract: We provide experimental evaluation of a number of known and new algorithms for approximate computation of Monroe's and Chamberlin-Courant's rules. Our experiments, conducted both on real-life preference-aggregation data and on synthetic data, show that even very simple and fast algorithms can in many cases find near-perfect solutions. Our results confirm and complement very recent theoretical analysis of Skowron et al., who have shown good lower bounds on the quality of (some of) the algorithms that we study.

Weighted Electoral Control

Abstract: Although manipulation and bribery have been extensively studied under weighted voting, there has been almost no work done on election control under weighted voting. This is unfortunate, since weighted voting appears in many important natural settings. In this paper, we study the complexity of controlling the outcome of weighted elections through adding and deleting voters. We obtain polynomial-time algorithms, NP-completeness results, and for many NP-complete cases, approximation algorithms. In particular, for scoring rules we completely characterize the complexity of weighted voter control. Our work shows that for quite a few important cases, either polynomial-time exact algorithms or polynomial-time approximation algorithms exist.

The Complexity of Fully Proportional Representation for Single-Crossing Electorates

Abstract: 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.

Approximating the MaxCover Problem with Bounded Frequencies in FPT Time

Abstract: We study approximation algorithms for several variants of the MaxCover problem, with the focus on algorithms that run in FPT time. In the MaxCover problem we are given a set N of elements, a family S of subsets of N, and an integer K. The goal is to find up to K sets from S that jointly cover (i.e., include) as many elements as possible. This problem is well-known to be NP-hard and, under standard complexitytheoretic assumptions, the best possible polynomial-time approximation algorithm has approximation ratio (1 − 1 ). We first consider a variant of MaxCover with bounded element frequencies, i.e., a variant where there is a constant p such that each element belongs to at most p sets in S. For this case we show that there is an FPT approximation scheme (i.e., for each β there is a β-approximation algorithm running in FPT time) for the problem of maximizing the number of covered elements, and a randomized FPT approximation scheme for the problem of minimizing the number of elements left uncovered (we take K to be the parameter). Then, for the case where there is a constant p such that each element belongs to at least p sets from S, we show that the standard greedy approximation algorithm achieves approximation ratio exactly 1 − e −max(pK/kSk,1) . We conclude by considering an unrestricted variant of MaxCover, and show approximation algorithms that run in exponential time and combine an exact algorithm with a greedy approximation. Some of our results improve currently known results for MaxVertexCover.

Achieving Fully Proportional Representation: Approximability Results

Abstract: We study the complexity of (approximate) winner determination under the Monroe and Chamberlin--Courant multiwinner voting rules, which determine the set of representatives by optimizing the total (dis)satisfaction of the voters with their representatives. The total (dis)satisfaction is calculated either as the sum of individual (dis)satisfactions (the utilitarian case) or as the (dis)satisfaction of the worst off voter (the egalitarian case). We provide good approximation algorithms for the satisfaction-based utilitarian versions of the Monroe and Chamberlin--Courant rules, and inapproximability results for the dissatisfaction-based utilitarian versions of them and also for all egalitarian cases. Our algorithms are applicable and particularly appealing when voters submit truncated ballots. We provide experimental evaluation of the algorithms both on real-life preference-aggregation data and on synthetic data. These experiments show that our simple and fast algorithms can in many cases find near-perfect solutions.

On swap-distance geometry of voting rules

Abstract: Axioms that govern our choice of voting rules are usually defined by imposing constraints on the rule's behavior under various transformations of the preference profile. In this paper we adopt a different approach, and view a voting rule as a (multi-)coloring of the election graph - the graph whose vertices are elections over a given set of candidates, and two vertices are adjacent if they can be obtained from each other by swapping adjacent candidates in one of the votes. Given this perspective, a voting rule F is characterized by the shapes of its "monochromatic components", i.e., sets of elections that have the same winner under F. In particular, it would be natural to expect each monochromatic component to be convex, or, at the very least, connected. We formalize the notions of connectivity and (weak) convexity for monochromatic components, and say that a voting rule is connected/(weakly) convex if each of its monochromatic components is connected/(weakly) convex. We then investigate which of the classic voting rules have these properties. It turns out that while all voting rules that we consider are connected, convexity and even weak convexity are much more demanding properties. Our study of connectivity suggests a new notion of monotonicity, which may be of independent interest.

Achieving Fully Proportional Representation is Easy in Practice

Abstract: We provide experimental evaluation of a number of known and new algorithms for approximate computation of Monroe's and Chamberlin-Courant's rules. Our experiments, conducted both on real-life preference-aggregation data and on synthetic data, show that even very simple and fast algorithms can in many cases find near-perfect solutions. Our results confirm and complement very recent theoretical analysis of Skowron et al., who have shown good lower bounds on the quality of (some of) the algorithms that we study.

The complexity of losing voters

Abstract: We consider the scenario of a parliament that is going to vote on a specific important issue. The voters are grouped in parties, and all voters of a party vote in the same way. The expected winner decision is known, because parties declare their intentions to vote, but before the actual vote takes place some voters may leave the leading party to join other parties. We investigate the computational complexity of the problem of determining how many voters need to leave the leading party before the winner changes. We consider both positional scoring rules (plurality, veto, k-approval, k-veto, Borda) and Condorcet-consistent methods (maximin, Copeland), and we study two versions of the problem: a pessimistic one, where we want to determine the maximal number of voters that can leave the leading party without changing the winner; and an optimistic one, where we want the minimal number of voters that must leave the leading party to be sure the winner will change. These two numbers provide a measure of the threat to the expected winner, and thus to the leading party, given by losing some voters. We show that for many positional scoring rules these problems are easy (except for the optimistic version with k-approval,for k at least 3, and Borda). Instead, for Condorcet-consistent rules, they are both computationally difficult, with both Maximin and Copeland.

Weighted electoral control

Abstract: Although manipulation and bribery have been extensively studied under weighted voting, there has been almost no work done on election control under weighted voting. This is unfortunate, since weighted voting appears in many important natural settings. In this paper, we study the complexity of controlling the outcome of weighted elections through adding and deleting voters. We obtain polynomial-time algorithms, NP-completeness results, and for many NP-complete cases, approximation algorithms. Our work shows that for quite a few important cases, either polynomial-time exact algorithms or polynomial-time approximation algorithms exist.

Complexity of Manipulation, Bribery, and Campaign Management in Bucklin and Fallback Voting

Abstract: A central theme in computational social choice is to study the extent to which voting systems computationally resist manipulative attacks seeking to influence the outcome of elections, such as manipulation (i.e., strategic voting), control, and bribery. Bucklin and fallback voting are among the voting systems with the broadest resistance (i.e., NP-hardness) to control attacks. However, only little is known about their behavior regarding manipulation and bribery attacks. We comprehensively investigate the computational resistance of Bucklin and fallback voting for many of the common manipulation and bribery scenarios; we also complement our discussion by considering several campaign management problems for Bucklin and fallback.