Greedy is good: An experimental study on minimum clique cover and maximum independent set problems for randomly generated rectangles

TL;DR

This paper evaluates greedy algorithms for minimum clique cover and maximum independent set problems on random rectangles, showing they perform close to optimal and proposing refined algorithms for better approximation guarantees.

Contribution

It provides experimental evidence that greedy algorithms are effective for these problems on random instances and introduces refined algorithms based on simplicial rectangles.

Findings

Greedy algorithms produce solutions close to optimal for MCC and MIS.

The size of MIS is at least 2√n, and clique cover at most 3√n, in experiments.

Approximation ratios are at most 3/2 for MIS and at least 2/3 for MCC.

Abstract

Given a set ${\cal R}=\{R_1,R_2,..., R_n\}$ of $n$ randomly positioned axis parallel rectangles in 2D, the problem of computing the minimum clique cover (MCC) and maximum independent set (MIS) for the intersection graph $G({\cal R})$ of the members in $\cal R$ are both computationally hard \cite{CC05}. For the MCC problem, it is proved that polynomial time constant factor approximation is impossible to obtain \cite{PT11}. Though such a result is not proved yet for the MIS problem, no polynomial time constant factor approximation algorithm exists in the literature. We study the performance of greedy algorithms for computing these two parameters of $G({\cal R})$. Experimental results shows that for each of the MCC and MIS problems, the corresponding greedy algorithm produces a solution that is very close to its optimum solution. Scheinerman \cite{Scheinerman80} showed that the size of MIS is tightly bounded by $\sqrt{n}$ for a random instance of the 1D version of the problem, (i.e., for the interval graph). Our experiment shows that the size of independent set and the clique cover produced by the greedy algorithm is at least $2\sqrt{n}$ and at most $3\sqrt{n}$, respectively. Thus the experimentally obtained approximation ratio of the greedy algorithm for MIS problem is at most 3/2 and the same for the MCC problem is at least 2/3. Finally we will provide refined greedy algorithms based on a concept of {\it simplicial rectangle}. The characteristics of this algorithm may be of interest in getting a provably constant factor approximation algorithm for random instance of both the problems. We believe that the result also holds true for any finite dimension.