Polynomial-time algorithms for regular set-covering and threshold synthesis

Peled, Uri N.; Simeone, Bruno

doi:10.1016/0166-218x(85)90040-x

Cited by 75 publications

(67 citation statements)

References 13 publications

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“…7 The numbers of weighted voting games and complete simple games coincide for n ≤ 5 voters but their ratio converges to zero with increasing n, see also Table 2. An asymptotic upper bound for weighted voting games is given in de Keijzer et al (2010) and an asymptotic lower bound for complete simple games, there called regular Boolean functions, is given in Peled and Simeone (1985). In order to accelerate the check whether a complete simple game is weighted we utilize the following linear program, instead of the one presented in the previous section.…”

Section: Partial Complete Simple Games and Similar Linear Programsmentioning

confidence: 99%

On minimum sum representations for weighted voting games

Kurz

2012

Ann Oper Res

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A proposal in a weighted voting game is accepted if the sum of the (non-negative) weights of the "yea" voters is at least as large as a given quota. Several authors have considered representations of weighted voting games with minimum sum, where the weights and the quota are restricted to be integers. In Freixas and Molinero (Ann. Oper. Res. 166:243-260, 2009) the authors have classified all weighted voting games without a unique minimum sum representation for up to 8 voters. Here we exhaustively classify all weighted voting games consisting of 9 voters which do not admit a unique minimum sum integer weight representation.

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Section: Partial Complete Simple Games and Similar Linear Programsmentioning

confidence: 99%

On minimum sum representations for weighted voting games

Kurz

2012

Ann Oper Res

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“…Theorem 2 (Peled & Simeone, 1985). There exists a polynomial time algorithm for testing whether a game given as a list of minimal winning coalitions is a weighted voting game.…”

Section: The Algorithmmentioning

confidence: 99%

“…, β n ), where 5. The terminology "roof" and "ceiling" is taken from Peled and Simeone (1985), while Taylor and Zwicker (1999) call these coalitions shift-minimal winning coalitions and shift-maximal losing coalitions.…”

Section: Power Indicesmentioning

confidence: 99%

“…The algorithm proposed by Peled and Simeone (1985) that has the above-named characteristics is called the Hop-Skip-and-Jump algorithm. The algorithm was originally designed for applications related to solving problems in the fields of threshold logic and set covering, but it is only a matter of changing terminology in a straightforward way to see that the Hop-Skip-and-Jump algorithm fulfills our purposes as well.…”

Section: The Algorithmmentioning

confidence: 99%

“…The Hop-Skip-and-Jump algorithm of Peled and Simeone (1985) works by first generating the list of maximal losing coalitions of a game, given the list of minimal winning coalitions of a game, and subsequently solving a linear program to check whether the game is a weighted voting game. Peled and Simeone give a first improvement by showing several ways to improve and compactify the linear program in question.…”

Section: Improvements and Optimizationsmentioning

confidence: 99%

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Finding Optimal Solutions for Voting Game Design Problems

Keijzer¹,

Klos²,

Zhang³

2014

jair

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In many circumstances where multiple agents need to make a joint decision, voting is used to aggregate the agents' preferences. Each agent's vote carries a weight, and if the sum of the weights of the agents in favor of some outcome is larger than or equal to a given quota, then this outcome is decided upon. The distribution of weights leads to a certain distribution of power. Several 'power indices' have been proposed to measure such power. In the so-called inverse problem, we are given a target distribution of power, and are asked to come up with a game-in the form of a quota, plus an assignment of weights to the players-whose power distribution is as close as possible to the target distribution (according to some specified distance measure).Here we study solution approaches for the larger class of voting game design (VGD) problems, one of which is the inverse problem. In the general VGD problem, the goal is to find a voting game (with a given number of players) that optimizes some function over these games. In the inverse problem, for example, we look for a weighted voting game that minimizes the distance between the distribution of power among the players and a given target distribution of power (according to a given distance measure).Our goal is to find algorithms that solve voting game design problems exactly, and we approach this goal by enumerating all games in the class of games of interest. We first present a doubly exponential algorithm for enumerating the set of simple games. We then improve on this algorithm for the class of weighted voting games and obtain a quadratic exponential (i.e., 2 O(n 2 ) ) algorithm for enumerating them. We show that this improved algorithm runs in output-polynomial time, making it the fastest possible enumeration algorithm up to a polynomial factor. Finally, we propose an exact anytime-algorithm that runs in exponential time for the power index weighted voting game design problem (the 'inverse problem').We implement this algorithm to find a weighted voting game with a normalized Banzhaf power distribution closest to a target power index, and perform experiments to obtain some insights about the set of weighted voting games. We remark that our algorithm is applicable to optimizing any exponential-time computable function, the distance of the normalized Banzhaf index to a target power index is merely taken as an example.

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Characterizing and decomposing classes of threshold, split, and bipartite graphs via 1‐Sperner hypergraphs

Boros

Gurvich

Milanič

2019

Journal of Graph Theory

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A hypergraph is said to be 1-Sperner if for every two hyperedges the smallest of their two set differences is of size one. We present several applications of 1-Sperner hypergraphs and their structure to graphs. In particular, we consider the classical characterizations of threshold and domishold graphs and use them to obtain further characterizations of these classes in terms of 1-Spernerness, thresholdness, and 2-asummability of their vertex cover, clique, dominating set, and closed neighborhood hypergraphs. Furthermore, we apply a decomposition property of 1-Sperner hypergraphs to derive decomposition theorems for two classes of split graphs, a class of bipartite graphs, and a class of cobipartite graphs. These decomposition theorems are based on certain matrix partitions of the corresponding graphs, giving rise to new classes of graphs of bounded clique-width and new polynomially solvable cases of several domination problems. IntroductionHypergraphs are one of the most fundamental and general combinatorial objects, encompassing various important structures such as graphs, matroids, and combinatorial designs. Results for more specific discrete structures (e.g., graphs) can often be proved more generally, in the context of suitable classes of hypergraphs (see, e.g., Schrijver [57]). Of course, applications of hypergraph theory to structural and optimization aspects of other discrete structures are not restricted to the above phenomenon. Since it is impossible to survey here all kinds of applications of hypergraphs, let us mention only a few more. First, recent work of Hujdurović et al. [37] made use of connections between hypergraphs and binary matrices, acyclic digraphs, and partially ordered sets to study two combinatorial optimization problems motivated by computational biology. Second, a common approach of applying hypergraph theory to graphs is to define and study a hypergraph derived in an appropriate way from a given graph, depending on what type of property of a graph or its vertex or edge subsets one is interested in. This includes matching hypergraphs [1,63], various clique [32,41,47,51,63], independent set [32,63], neighborhood [12,27], separator [13,60], and dominating set hypergraphs [12,13], etc. Close interrelations between hypergraphs and monotone Boolean functions can be useful in such studies, 1 arXiv:1805.03405v3 [math.CO] 29 May 2018 allowing for the transfer and applications of results from the theory of Boolean functions (see, e.g., [21]).In this work, we present several new applications of hypergraphs to graphs. Our starting point is [9] where the class of 1-Sperner hypergraphs was studied. It was shown that such hypergraphs can be decomposed in a particular way and that they form a subclass of the class of threshold hypergraphs studied by Golumbic [31], by Reiterman et al. [56] and, in the equivalent context of threshold monotone Boolean functions, also by Muroga [48] and by Peled and Simeone [52]. (Precise definitions of all relevant concepts will be given in Section 2.)While threshold h...

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Polynomial-time algorithms for regular set-covering and threshold synthesis

Cited by 75 publications

References 13 publications

On minimum sum representations for weighted voting games

On minimum sum representations for weighted voting games

Finding Optimal Solutions for Voting Game Design Problems

Characterizing and decomposing classes of threshold, split, and bipartite graphs via 1‐Sperner hypergraphs

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