Partial Degree Bounded Edge Packing Problem with Arbitrary Bounds

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.