Partial Degree Bounded Edge Packing Problem with Arbitrary Bounds

TL;DR

This paper investigates the Partial Degree Bounded Edge Packing problem with arbitrary degree constraints, proposing new approximation algorithms, improved exact algorithms for trees, and solutions for weighted edges, advancing the understanding of this NP-hard problem.

Contribution

It introduces two combinatorial approximation algorithms with improved factors, an iterative rounding method, and a faster exact algorithm for trees with general degree constraints.

Findings

Presented approximation algorithms with factors 4 and 2.

Developed an O(n log n) exact algorithm for trees with arbitrary degree constraints.

Provided an O(log n) approximation for weighted edge cases.

Abstract

We study the Partial Degree Bounded Edge Packing (PDBEP) problem introduced in [5] by Zhang. They have shown that this problem is NP-Hard even for uniform degree constraint. They also presented approximation algorithms for the case when all the vertices have degree constraint of 1 and 2 with approximation ratio of 2 and 32=11 respectively. 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 integer program based solution and present an iterative rounding algorithm with approximation factor 3/(1 - \epsilon)^2 for any positive \epsilon. Next we study the same problem with weighted edges. In this case we present an O(log n) approximation algorithm. Zhang has given an exact O(n^2) complexity algorithm for trees in case of uniform degree constraint. We improve their result by giving O(nlog n) complexity exact algorithm for trees with general degree constraint.