Partial degree bounded edge packing problem

• Authors: Peng Zhang
• Venue: FAW-AAIM'12 Proceedings of the 6th international Frontiers in Algorithmics, and Proceedings of the 8th international conference on Algorithmic Aspects in Information and Management
• Year: 2012

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.