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 − ǫ) 2 for any positive ǫ. Next we study the same problem with weighted edges. In this case we present an O(log n) approximation algorithm. Zhang [5] has given an exact O(n 2 ) complexity algorithm for trees in case of uniform degree constraint. We improve their result by giving O(n · log n) complexity exact algorithm for trees with general degree constraint.
Given two graphs G1 and G2 on n vertices each, we define a graph G on vertex set V1 × V2 and the edge set as the union of edges ofWe consider the completely-positive Lovász ϑ function, i.e., cpϑ function for G. We show that the function evaluates to n whenever G1 and G2 are isomorphic and to less than n − 1/(4n 4 ) when non-isomorphic. Hence this function provides a test for graph isomorphism. We also provide some geometric insight into the feasible region of the completely positive program.
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