Abstract
We consider a still NP-complete partial case of the unconstrained binary quadratic optimization problem that can be described in terms of an undirected graph with red edges having negative weights and green edges having positive weights. The maximum vertex degree of the graph is three. It can be assumed w.l.o.g. that every vertex is incident to a red and a green edge. We are looking for a vertex cover with respect to the red edges which covers a subset of green edges of total weight as small as possible. We prove that for all connected such graphs except a subclass of special graphs having exactly five green edges it is possible to find a vertex cover with respect to the red edges for which the total weight of uncovered green edges is at least 1/4 fraction of the total weight of all green edges.
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Palubeckis, G. A Tight Lower Bound for a Special Case of Quadratic 0–1 Programming. Computing 77, 131–145 (2006). https://doi.org/10.1007/s00607-005-0153-z
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DOI: https://doi.org/10.1007/s00607-005-0153-z