Partial Degree Bounded Edge Packing Problem

Abstract

In [1], whether a target binary string s can be represented from a boolean formula with operands chosen from a set of binary strings W was studied. In this paper, we first examine selecting a maximum subset X from W, so that for any string t in X, t is not representable by X ∖ {t}. We rephrase this problem as graph, and surprisingly find it give rise to a broad model of edge packing problem, which itself falls into the model of forbidden subgraph problem. Specifically, given a graph G(V,E) and a constant c, the problem asks to choose as many as edges to form a subgraph G′. So that in G′, for each edge, at least one of its endpoints has degree no more than c. We call such G′ partial c degree bounded. This edge packing problem model also has a direct interpretation in resource allocation. There are n types of resources and m jobs. Each job needs two types of resources. A job can be accomplished if either one of its necessary resources is shared by no more than c other jobs. The problem then asks to finish as many jobs as possible. For edge packing problem, when c = 1, it turns out to be the complement of dominating set and able to be 2-approximated. When c = 2, it can be 32/11-approximated. We also prove it is NP-complete for any constant c on graphs and is O(|V|²) solvable on trees. We believe this partial bounded graph problem is intrinsic and merits more attention.