Abstract
In this work we consider two twocriteria optimization problems: given an input graph, the goal is to find its interval (or chordal) supergraph that minimizes the number of edges and its clique number simultaneously. For the interval supergraph, the problem can be restated as simultaneous minimization of the path width pw(G) and the profile p(G) of the input graph G. We prove that for an arbitrary graph G and an integer t ∈ {1, … , pw(G) + 1}, there exists an interval supergraph G′ of G such that for its clique number it holds ω(G′)≤(1+2t)(pw(G)+ 1) and the number of its edges is bounded by E(G′) ≤ (t + 2)p(G). In other words, the pathwidth and the profile of a graph can be simultaneously minimized within the factors of 1+2t (plus a small constant) and t + 2, respectively. Note that for a fixed t, both upper bounds provide constant factor approximations. On the negative side, we show an example that proves that, for some graphs, there is no solution in which both parameters are optimal. In case of finding a chordal supergraph, the two corresponding graph parameters that reflect its clique size and number of edges are the treewidth and fillin. We obtain that the treewidth and the fillin problems are also ‘orthogonal’ in the sense that for some graphs, a solution that minimizes one of those parameters cannot minimize the other. As a motivating example, we recall graph searching games which illustrates a need of simultaneous minimization of these pairs of graph parameters.
Citations

0
CrossRef

0
Web of Science

0
Scopus
Authors (2)
Details
 Category:
 Articles
 Type:
 artykuł w czasopiśmie wyróżnionym w JCR
 Published in:

THEORY OF COMPUTING SYSTEMS
no. 63,
pages 450  465,
ISSN: 14324350  Language:
 English
 Publication year:
 2019
 Bibliographic description:
 Dereniowski D., Stański A.: On Tradeoffs Between Width and Filllike Graph Parameters// THEORY OF COMPUTING SYSTEMS. Vol. 63, iss. 3 (2019), s.450465
 DOI:
 Digital Object Identifier (open in new tab) 10.1007/s0022401898821
 Verified by:
 Gdańsk University of Technology
seen 12 times