The Complexity of Planar Counting Problems

Hunt, Harry B.; Marathe, Madhav V.; Radhakrishnan, Venkatesh Babu; Stearns, Richard E.

doi:10.1137/s0097539793304601

Cited by 80 publications

(38 citation statements)

References 16 publications

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“…, x n ∈ {0, 1} such that ∑ n i=1 x i a i = q? It is known that #SUBSETSUM is #P-parsimonious-complete (for parsimonious reductions from #3-SAT via #EXACTCOVERBY3-SETS to #SUBSETSUM see, e.g., Hunt, Marathe, Radhakrishnan, & Stearns, 1998;Papadimitriou, 1995). Hence, by Lemma 4.2, we have the following.…”

Section: 3)mentioning

confidence: 99%

False-Name Manipulation in Weighted Voting Games is Hard for Probabilistic Polynomial Time

Rey¹,

Rothe²

2014

jair

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False-name manipulation refers to the question of whether a player in a weighted voting game can increase her power by splitting into several players and distributing her weight among these false identities. Relatedly, the beneficial merging problem asks whether a coalition of players can increase their power in a weighted voting game by merging their weights. For the problems of whether merging or splitting players in weighted voting games is beneficial in terms of the Shapley-Shubik and the normalized Banzhaf index, merely NP-hardness lower bounds are known, leaving the question about their exact complexity open. For the Shapley-Shubik and the probabilistic Banzhaf index, we raise these lower bounds to hardness for PP, "probabilistic polynomial time," a class considered to be by far a larger class than NP. For both power indices, we provide matching upper bounds for beneficial merging and, whenever the new players' weights are given, also for beneficial splitting, thus resolving previous conjectures in the affirmative. Relatedly, we consider the beneficial annexation problem, asking whether a single player can increase her power by taking over other players' weights. It is known that annexation is never disadvantageous for the Shapley-Shubik index, and that beneficial annexation is NP-hard for the normalized Banzhaf index. We show that annexation is never disadvantageous for the probabilistic Banzhaf index either, and for both the Shapley-Shubik index and the probabilistic Banzhaf index we show that it is NP-complete to decide whether annexing another player is advantageous. Moreover, we propose a general framework for merging and splitting that can be applied to different classes and representations of games.

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Section: 3)mentioning

confidence: 99%

False-Name Manipulation in Weighted Voting Games is Hard for Probabilistic Polynomial Time

Rey¹,

Rothe²

2014

jair

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“…For the planar case one uses additional sets of clauses that act as crossovers and make the formula planar, as described by Lichtenstein [L82]. These can also be made to preserve the number of solutions [HMRS98]. These additional clauses can be viewed as part of the circuit encoding, and then yield a segregating reduction to the planar versions of Pl-3CNF and ⊕Pl-3CNF as needed.…”

Section: Diversitymentioning

confidence: 99%

Some Observations on Holographic Algorithms

Valiant

2010

LATIN 2010: Theoretical Informatics

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Abstract. We define the notion of diversity for families of finite functions, and express the limitations of a simple class of holographic algorithms in terms of limitations on diversity. We go on to describe polynomial time holographic algorithms for computing the parity of the following quantities for degree three planar undirected graphs: the number of 3-colorings up to permutation of colors, the number of connected vertex covers, and the number of induced forests or feedback vertex sets. In each case the parity can be computed for any slice of the problem, in particular for colorings where the first color is used a certain number of times, or where the connected vertex cover, feedback set or induced forest has a certain number of nodes. These holographic algorithms use bases of three components, rather than two.

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“…In the Counting Constraint Satisfaction Problem, #CSP(H), over a finite relational structures H the objective is, given a finite relational structure G, to compute the number of homomorphisms from G to H. Various particular cases of the #CSP arise and have been extensively studied in a wide range of areas from logic and graph theory [4,19,29,38,42,51,57,61,62], to artificial intelligence [56,58], to statistical physics [3,17,49]. In different areas this problem often appears in different equivalent forms: (1) the problem of finding the number of models of a conjunctive formula, (2) the problem of computing the size (number of tuples) of the evaluation Q(D) of a conjunctive query (without projection) Q on a database D and also (3) the problem of counting the number of assignments to a set of variables subject to specified constraints.…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, some other variants of the #H -COLORING problem for undirected graphs have been intensively studied during the last few years [23,24]. Another direction in this area is the study of problems with restricted input, that is subproblems of the #H -COLORING problem in which the input graph G must be planar [42,60], a partial k-tree [22], sparse or of low degree [38,39], etc. Finally, we should mention the approach to counting problems using approximation and randomized algorithms, see e.g.…”

Section: Introductionmentioning

confidence: 99%