DIMACS TR: 95-36

Nonoverlapping Local Alignments, Weighted Independent Sets of Axis Parallel Rectangles

Authors: Vineet Bafna, Babu Narayanan, R. Ravi


We consider the following problem motivated by an application in computational molecular biology. We are given a set of weighted axis-parallel rectangles such that for any pair of rectangles and either axis, the projection of one rectangle does not enclose that of the other. Define a pair to be independent if their projections in both axes are disjoint. The problem is to find a maximum-weight independent subset of rectangles.

We show that the problem is NP-hard even in the uniform case when all the weights are the same. We analyze the performance of a natural local- improvement heuristic for the general problem and prove a performance ration of 3.25. We extend the heuristic to the problem of finding a maximum-weight independent set in (d+1)-claw free graphs, and show a tight performance ration of (d - 1 + (1/d)). A performance ratio of d/2 was known for the heuristic when applied to the uniform case. Our contributions are proving the hardness of the problem and providing a tight analysis of the local-improvement algorithm for the general weighted case.

Paper available at: ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/1995/95-36.ps.gz

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