Complexity of the exact domatic number problem and of the exact conveyor flow shop problem

Riege, Tobias; Rothe, Jörg

doi:10.1109/ictta.2004.1307935

“…To conclude this paper, we mention in passing that Riege and this author [17] recently obtained similar BH 2k (NP)-completeness results for the exact versions of the domatic number problem and the conveyor flow shop problem.…”

Section: Corollarymentioning

confidence: 95%

“…Their papers initiated an intensive work and many papers on the Boolean hierarchy; e.g., [20,15,13,21,2,5,1,6,12,17] to name just a few.…”

Section: Exact Colorability and The Boolean Hierarchy Over Npmentioning

confidence: 99%

Exact complexity of Exact-Four-Colorability

Rothe¹

2003

Information Processing Letters

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Let M k be a given set of k integers. Define Exact-M k -Colorability to be the problem of determining whether or not χ(G), the chromatic number of a given graph G, equals one of the k elements of the set M k exactly. In 1987, Wagner [Theoret. Comput. Sci. 51 (1987 proved that Exact-M k -Colorability is BH 2k (NP)-complete, where M k = {6k + 1, 6k + 3, . . . , 8k − 1} and BH 2k (NP) is the 2kth level of the Boolean hierarchy over NP. In particular, for k = 1, it is DP-complete to determine whether or not χ(G) = 7, where DP = BH 2 (NP). Wagner raised the question of how small the numbers in a k-element set M k can be chosen such that Exact-M k -Colorability still is BH 2k (NP)-complete. In particular, for k = 1, he asked if it is DP-complete to determine whether or not χ(G) = 4.In this note, we solve Wagner's question and prove the optimal result: For each k 1, Exact-M k -Colorability is BH 2k (NP)-complete for M k = {3k + 1, 3k + 3, . . . , 5k − 1}. In particular, for k = 1, we determine the precise threshold of the parameter t ∈ {4, 5, 6, 7} for which the problem Exact-{t}-Colorability jumps from NP to DP-completeness: It is DP-complete to determine whether or not χ(G) = 4, yet Exact-{3}-Colorability is in NP. troduced DP, the class of differences of two NP problems. They showed that DP contains various interesting types of problems, including uniqueness problems, critical graph problems, and exact optimization problems. For example, Cai and Meyer [7] proved the DPcompleteness of Minimal-3-Uncolorability, a critical graph problem that asks whether a given graph is not 3-colorable, but deleting any of its vertices makes it 3-colorable. A graph is said to be k-colorable if its vertices can be colored using no more than k colors such that no two adjacent vertices receive the same color. The chromatic number of a graph G, denoted χ(G), is defined to be the smallest k such that G is k-colorable.

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“…This number, denoted by δ(G), is called the domatic number of G. The domatic number problem arises in the area of computer networks and is related to the tasks of distributing resources and of locating facilities in the network. This problem has been intensely studied, see, e.g., [CH77,Far84,Bon85,KS94,HT98,FHK00,FHKS02,RR04].…”

Section: Introductionmentioning

confidence: 99%

An improved exact algorithm for the domatic number problem

Riege¹,

Rothe²,

Spakowski³

et al. 2007

Information Processing Letters

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The 3-domatic number problem asks whether a given graph can be partitioned into three dominating sets. We prove that this problem can be solved by a deterministic algorithm in time 2.695 n (up to polynomial factors). This result improves the previous bound of 2.8805 n , which is due to Fomin, Grandoni, Pyatkin, and Stepanov. To prove our result, we combine an algorithm by Fomin et al. with Yamamoto's algorithm for the satisfiability problem. In addition, we show that the 3-domatic number problem can be solved for graphs G with bounded maximum degree ∆(G) by a randomized algorithm, whose running time is better than the previous bound due to Riege and Rothe [RR05] whenever ∆(G) ≥ 5. Our new randomized algorithm employs Schöning's approach to constraint satisfaction problems.

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An Improved Exact Algorithm for the Domatic Number Problem

Riege

¹

,

Rothe

²

,

Spakowski

³

et al.

2006 2nd International Conference on Information &Amp; Communication Technologies

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The 3-domatic number problem asks whether a given graph can be partitioned into three dominating sets. We prove that this problem can be solved by a deterministic algorithm in time 2.695 n (up to polynomial factors). This result improves the previous bound of 2.8805 n , which is due to Fomin, Grandoni, Pyatkin, and Stepanov. To prove our result, we combine an algorithm by Fomin et al. with Yamamoto's algorithm for the satisfiability problem. In addition, we show that the 3-domatic number problem can be solved for graphs G with bounded maximum degree ∆(G) by a randomized algorithm, whose running time is better than the previous bound due to Riege and Rothe [RR05] whenever ∆(G) ≥ 5. Our new randomized algorithm employs Schöning's approach to constraint satisfaction problems.

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Complexity of the exact domatic number problem and of the exact conveyor flow shop problem

Cited by 4 publications

References 65 publications

Exact complexity of Exact-Four-Colorability

Exact complexity of Exact-Four-Colorability

An improved exact algorithm for the domatic number problem

An Improved Exact Algorithm for the Domatic Number Problem

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