Co-density and fractional edge cover packing

Given a multigraph G = ( V , E ) , the edge cover packing problem (ECPP) on G is to find a coloring of edges of G using the maximum number of colors such that at each vertex all colors occur. ECPP can be formulated as an integer program and is NP-hard in general. In this paper, we consider the fract...

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Bibliographic Details
Published in:Journal of combinatorial optimization Vol. 39; no. 4; pp. 955 - 987
Main Authors: Zhao, Qiulan, Chen, Zhibin, Sang, Jiajun
Format: Journal Article
Language:English
Published: New York Springer US 01.05.2020
Springer Nature B.V
Subjects:
Algorithms
Combinatorial analysis
Combinatorics
Convex and Discrete Geometry
Density
Mathematical Modeling and Industrial Mathematics
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Optimization
Polynomials
Theory of Computation
Weight
Co-density
68Q25
Edge cover
Combinatorial algorithm
Multigraph
Fractional packing
Primary 90C27
ISSN:1382-6905, 1573-2886
Online Access:Get full text
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Summary:Given a multigraph G = ( V , E ) , the edge cover packing problem (ECPP) on G is to find a coloring of edges of G using the maximum number of colors such that at each vertex all colors occur. ECPP can be formulated as an integer program and is NP-hard in general. In this paper, we consider the fractional edge cover packing problem , the LP relaxation of ECPP. We focus on the more general weighted setting, the weighted fractional edge cover packing problem (WFECPP), which can be formulated as the following linear program Maximize 1 T x subject to A x ≤ w x ≥ 0 , where A is the edge–edge cover incidence matrix of G , w = ( w ( e ) : e ∈ E ) , and w ( e ) is a positive rational weight on each edge e of G . The weighted co-density problem , closely related to WFECPP, is to find a subset S ⊆ V with | S | ≥ 3 and odd, such that 2 w ( E + ( S ) ) | S | + 1 is minimized, where E + ( S ) is the set of all edges of G with at least one end in S and w ( E + ( S ) ) is the total weight of all edges in E + ( S ) . We present polynomial combinatorial algorithms for solving these two problems exactly.
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ISSN:1382-6905
1573-2886
DOI:10.1007/s10878-020-00535-x