Partial degree bounded edge packing problem for graphs and $k$-uniform hypergraphs

Abstract

Given a graph $G=(V,E)$ and a non-negative integer $c_u$ for each $u\in V$, partial degree bounded edge packing problem is to find a subgraph $G^{\prime }=(V,E^{\prime })$ with maximum $|E^{\prime }|$ such that for each edge $(u,v)\in E^{\prime }$, either $deg_{G^{\prime }}(u)\le c_u$ or $deg_{G^{\prime }}(v)\le c_v$. The problem has been shown to be NP-hard even for uniform degree constraint (i.e., all $c_u$ being equal). In this work we study the general degree constraint case (arbitrary degree constraint for each vertex) and present two combinatorial approximation algorithms with approximation factors $4$ and $2$. Then we give a $\log _2 n$ approximation algorithm for edge-weighted version of the problem and an efficient exact algorithm for edge-weighted trees with time complexity $O(n\log n)$. We also consider a generalization of this problem to $k$-uniform hypergraphs and present a constant factor approximation algorithm based on linear programming using Lagrangian relaxation.