Approximation of MWIS on geometric intersection graphs

Abstract

We present a generic formulation of an algorithmic paradigm for approximating maximum weighted independent sets (MWIS) in arbitrary vertex weighted graphs. A special case of this paradigm has been proposed earlier for geometric intersection graphs. Here, we propose and analyse a much more general formulation. As part of this formulation, we introduce a new graph parameter which plays a role in bounding the approximation factor of the algorithms. By applying this paradigm to intersection graph classes of specific types of geometric objects, we obtain efficient algorithms which approximate a MWIS within ${(\log n)}^{O\left(1\right)}$ multiplicative factors. It is also shown that the same approach can be generalised to obtain efficient approximation algorithms for computing an optimal weight $P$-subgraphs where $P$ is a suitable hereditary property.

Applying our paradigm, we establish, for every $k\ge 2$ and $p\in [1,\infty ]$, that MWIS of the intersection graph of a given collection of weighted k-dimensional $L_p$ spheres (having a common radius) can be efficiently approximated within a multiplicative factor of ${(\log n)}^{k-1}$. The running time can be brought down to $O\left(n\right(\log n\left)\right)$ at the cost of increasing the approximation guarantee to $c_{k,p}{(\log n)}^{k-1}$, for some constant $c_{p,k}$ depending only on p and k. It is also shown that the above MWIS-approximation results can be extended to MWIS-approximation over the more general intersection graphs of finite collections of connected, full-dimensional and centrally-symmetric bodies in k-dimensional, $L_p$-spaces, for every $k\ge 2$ and $p\in (0,\infty ]$.

In a related development, we also establish the following graph theoretic result which will be of independent interest: For every $p\in [2,\infty ]$ and for every G, there is a $k\ge 1$ such that G is isomorphic to the IG of a collection of k-dimensional $L_p$-spheres of a common radius. The minimum value of such a k is referred to as the p-sphericity of G.

Also, applying our paradigm, one obtains for every $k\ge 2$, an efficient algorithm which, given a collection $B$ of weighted k-dimensional axis-parallel boxes, finds a ${(\log n)}^{k-1}$-approximation to MWIS. For the unweighted case, the running time can be improved to $O\left(n{(\log n)}^2\right)$.