Partial Degree Bounded Edge Packing Problem

TL;DR

This paper introduces a new edge packing problem called partial c degree bounded subgraph, explores its computational complexity, and provides algorithms for trees and approximation methods for general graphs.

Contribution

It formulates the partial degree bounded edge packing problem, analyzes its hardness, and offers exact algorithms for trees and approximation strategies for general graphs.

Findings

Problem is NP-hard in general graphs.

Efficient exact algorithms exist for trees.

Approximation algorithms are proposed for general graphs.

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. When c = 1, it turns out to be the complement of dominating set. We present several results about hardness, approximation for the general graph and efficient exact algorithm on trees. 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 nish as many jobs as possible. We believe this partial degree bounded graph problem merits more attention.