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Title: On NP-hardness of the clique partition : independence number gap recognition and related problems
Authors: Busygin, Stanislav.
Pasechnik, Dmitrii V.
Keywords: DRNTU::Science::Mathematics::Discrete mathematics::Theory of computation
Issue Date: 2006
Source: Busygina, S., & Pasechnikb, D. V. (2006). On NP-hardness of the clique partition: independence number gap recognition and related problems. Discrete Mathematics, 306(4), 460–463.
Series/Report no.: Discrete mathematics
Abstract: We show that for a graph G it is NP-hard to decide whether its independence number α(G) equals its clique partition number ¯χ(G) even when some minimum clique partition of G is given. This implies that any α(G)-upper bound provably better than ¯χ (G) is NP-hard to compute. To establish this result we use a reduction of the quasigroup completion problem (QCP, known to be NP-complete) to the maximum independent set problem. A QCP instance is satisfiable if and only if the independence number α(G) of the graph obtained within the reduction is equal to the number of holes h in the QCP instance. At the same time, the inequality ¯χ (G)≤ h always holds. Thus, QCP is satisfiable if and only if α(G) = ¯χ (G) = h. Computing the Lovász number ϑ(G) we can detect QCP unsatisfiability at least when ¯χ (G) <h. In the other cases QCP reduces to ¯χ (G) − α(G) >0 gap recognition, with one minimum clique partition of G known.
DOI: 10.1016/j.disc.2006.01.004
Schools: School of Physical and Mathematical Sciences 
Rights: © 2006 Elsevier. This is the author created version of a work that has been peer reviewed and accepted for publication by Discrete Mathematics, Elsevier. It incorporates referee’s comments but changes resulting from the publishing process, such as copyediting, structural formatting, may not be reflected in this document. The published version is available at: DOI [].
Fulltext Permission: open
Fulltext Availability: With Fulltext
Appears in Collections:SPMS Journal Articles

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