Improved Approximability Result for Test Set with Small Redundancy

TL;DR

This paper improves the theoretical approximation ratio for the set cover greedy algorithm in bioinformatics test set redundancy scenarios, especially when redundancy is small, using potential function analysis.

Contribution

It establishes a tighter approximation ratio for SGA in test sets with small redundancy, extending previous results for plain test sets.

Findings

Approximation ratio of SGA is $(2-rac{1}{2r}) ln n + rac{3}{2} ln r + O( ln ln n)$.

Result surpasses the $2 ewline ln n$ bound when $r=o(rac{ ewline ln n}{ ewline ln ewline ln n})$.

Extends approximability results from plain test sets to those with redundancy.

Abstract

Test set with redundancy is one of the focuses in recent bioinformatics research. Set cover greedy algorithm (SGA for short) is a commonly used algorithm for test set with redundancy. This paper proves that the approximation ratio of SGA can be $(2-\frac{1}{2r})\ln n+{3/2}\ln r+O(\ln\ln n)$ by using the potential function technique. This result is better than the approximation ratio $2\ln n$ which directly derives from set multicover, when $r=o(\frac{\ln n}{\ln\ln n})$, and is an extension of the approximability results for plain test set.