Partial Degree Bounded Edge Packing Problem with Arbitrary Bounds

Abstract

Given a graph G = (V,E) and a non-negative integer c _u for each u ∈ V. Partial Degree Bounded Edge Packing (PDBEP) problem is to find a subgraph G′ = (V,E′) with maximum |E′| such that for each edge (u,v) ∈ E′, either deg _G′(u) ≤ c _u or deg _G′(v) ≤ c _v . The problem has been shown to be NP-hard even for uniform degree constraint (i.e., all c _u being equal). Approximation algorithms for uniform degree cases with c _u equal to 1 and 2 with approximation ratio of 2 and 32/11 respectively are known. In this work we study general degree constraint case (arbitrary degree constraint for each vertex) and present two combinatorial approximation algorithms with approximation factors 4 and 2. We also study a related integer program for which we present an iterative rounding algorithm with approximation factor 1.5/(1 − ε) for any positive ε. This also leads to a 3/(1 − ε)² factor approximation algorithm for the general PDBEP problem. For special cases (large values of c _v /deg _G (v)’s) the factor improves up to 1.5/(1 − ε). Next we study the same problem with weighted edges. In this case we present a 2 + log₂ n approximation algorithm. In the literature exact O(n ²) complexity algorithm for trees is known in case of uniform degree constraint. We improve this result by giving an O(n·logn) complexity exact algorithm for trees with general degree constraint.