An Improved Approximation Algorithm for the Minimum Common Integer Partition Problem

Abstract

Given a collection of multisets $\{X_1, X_2, \ldots , X_k\}$ ($k \ge 2$) of positive integers, a multiset $S$ is a common integer partition for them if $S$ is an integer partition of every multiset $X_i, 1 \le i \le k$. The minimum common integer partition ($k$-MCIP) problem is defined as to find a CIP for $\{X_1, X_2, \ldots , X_k\}$ with the minimum cardinality. We present a $\frac{6}{5}$-approximation algorithm for the $2$-MCIP problem, improving the previous best algorithm of ratio $\frac{5}{4}$ designed in 2006. We then extend it to obtain an absolute $0.6k$-approximation algorithm for $k$-MCIP when $k$ is even (when $k$ is odd, the approximation ratio is $0.6k+0.4$).