On the computational complexity of defining sets

Hatami, Hamed; Maserrat, Hossein

doi:10.1016/j.dam.2005.03.004

Cited by 5 publications

(4 citation statements)

References 9 publications

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“…In a recent paper [25], Hatami and Maserrat obtain a result that, in a sence, is complementary to our work and by this reason is also shown in Figure 1. Let Force χ ( * ) = { (x, k) : F χ (x) ≤ k}.…”

Section: Introductionsupporting

confidence: 86%

“…This inclusion follows from the facts that US ⊆ C = P and that C = P is closed under dtt-reductions (see [27,Theorem 9.9]). In a recent paper [25], Hatami and Maserrat obtain a result that, in a sence, is complementary to our work and by this reason is also shown in Figure 1…”

Section: Introductionsupporting

confidence: 85%

“…Let Force χ ( * ) = { (x, k) : F χ (x) ≤ k}. The authors of [25] prove that the recognition of membership in Force χ ( * ) is a Σ P 2 -complete problem. Note that [25] and our paper use different techniques and, moreover, the two approaches apparently cannot be used in place of one another.…”

Section: Introductionmentioning

confidence: 99%

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On the Computational Complexity of the Forcing Chromatic Number

2005

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We consider vertex colorings of graphs in which adjacent vertices have distinct colors. A graph is s-chromatic if it is colorable in s colors and any coloring of it uses at least s colors. The forcing chromatic number F χ (G) of an s-chromatic graph G is the smallest number of vertices which must be colored so that, with the restriction that s colors are used, every remaining vertex has its color determined uniquely. We estimate the computational complexity of F χ (G) relating it to the complexity class US introduced by Blass and Gurevich. We prove that recognizing if F χ (G) ≤ 2 is US-hard with respect to polynomial-time many-one reductions. Moreover, this problem is coNP-hard even under the promises that F χ (G) ≤ 3 and G is 3-chromatic. On the other hand, recognizing if F χ (G) ≤ k, for each constant k, is reducible to a problem in US via a disjunctive truth-table reduction.Similar results are obtained also for forcing variants of the clique and the domination numbers of a graph.

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Section: Introductionsupporting

confidence: 86%

Section: Introductionsupporting

confidence: 85%

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On the Computational Complexity of the Forcing Chromatic Number

2005

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“…Latin squares (and, of course, the special subclass of them that comprise Sudoku boards), matching theory, design theory, and the study of dominating vertex sets all feature variants of this idea. (See [3] for a more comprehensive list of related topics and references.) Therefore, we define the following parameters.…”

mentioning

confidence: 99%

Critical sets for Sudoku and general graph colorings

Cooper

Kirkpatrick

2014

Discrete Mathematics

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We discuss the problem of finding critical sets in graphs, a concept which has appeared in a number of guises in the combinatorics and graph theory literature. The case of the Sudoku graph receives particular attention, because critical sets correspond to minimal fair puzzles. We define four parameters associated with the sizes of extremal critical sets and (a) prove several general results about these parameters' properties, including their computational intractability, (b) compute their values exactly for some classes of graphs, (c) obtain bounds for generalized Sudoku graphs, and (d) offer a number of open questions regarding critical sets and the aforementioned parameters.

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The Minimum Information Dominating Set for Opinion Sampling in Social Networks

Gao

Zhao

Swami

2016

IEEE Trans. Netw. Sci. Eng.

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We consider the problem of inferring the opinions of a social network through strategically sampling a minimum subset of nodes by exploiting correlations in node opinions. We first introduce the concept of information dominating set (IDS). A subset of nodes in a given network is an IDS if knowing the opinions of nodes in this subset is sufficient to infer the opinion of the entire network. We focus on two fundamental algorithmic problems: (i) given a subset of the network, how to determine whether it is an IDS; (ii) how to construct a minimum IDS. Assuming binary opinions and the local majority rule for opinion correlation, we show that the first problem is co-NP-complete and the second problem is NP-hard in general networks. We then focus on networks with special structures, in particular, acyclic networks. We show that in acyclic networks, both problems admit linear-complexity solutions by establishing a connection between the IDS problems and the vertex cover problem. Our technique for establishing the hardness of the IDS problems is based on a novel graph transformation that transforms the IDS problems in a general network to that in an odd-degree network. This graph transformation technique not only gives an approximation algorithm to the IDS problems, but also provides a useful tool for general studies related to the local majority rule. Besides opinion sampling for applications such as political polling and market survey, the concept of IDS and the results obtained in this paper also find applications in data compression and identifying critical nodes in information networks.

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On the computational complexity of defining sets

Cited by 5 publications

References 9 publications

On the Computational Complexity of the Forcing Chromatic Number

On the Computational Complexity of the Forcing Chromatic Number

Critical sets for Sudoku and general graph colorings

The Minimum Information Dominating Set for Opinion Sampling in Social Networks

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