2 research outputs found
Maximum Bipartite Subgraph of Geometric Intersection Graphs
We study the Maximum Bipartite Subgraph (MBS) problem, which is defined as
follows. Given a set of geometric objects in the plane, we want to
compute a maximum-size subset such that the intersection graph
of the objects in is bipartite. We first give a simple -time
algorithm that solves the MBS problem on a set of intervals. We also give
an -time algorithm that computes a near-optimal solution for the
problem on circular-arc graphs. We show that the MBS problem is NP-hard on
geometric graphs for which the maximum independent set is NP-hard (hence, it is
NP-hard even on unit squares and unit disks). On the other hand, we give a PTAS
for the problem on unit squares and unit disks. Moreover, we show fast
approximation algorithms with small-constant factors for the problem on unit
squares, unit disks and unit-height rectangles. Finally, we study a closely
related geometric problem, called Maximum Triangle-free Subgraph (TFS), where
the objective is the same as that of MBS except the intersection graph induced
by the set needs to be triangle-free only (instead of being bipartite).Comment: 32 pages, 7 figure
The stable set problem in graphs with bounded genus and bounded odd cycle packing number
Consider the family of graphs without node-disjoint odd cycles, where is a constant. Determining the complexity of the stable set problem for
such graphs is a long-standing problem. We give a polynomial-time
algorithm for the case that can be further embedded in a (possibly
non-orientable) surface of bounded genus. Moreover, we obtain polynomial-size
extended formulations for the respective stable set polytopes.
To this end, we show that -sided odd cycles satisfy the Erd\H{o}s-P\'osa
property in graphs embedded in a fixed surface. This extends the fact that odd
cycles satisfy the Erd\H{o}s-P\'osa property in graphs embedded in a fixed
orientable surface (Kawarabayashi & Nakamoto, 2007).
Eventually, our findings allow us to reduce the original problem to the
problem of finding a minimum-cost non-negative integer circulation of a certain
homology class, which turns out to be efficiently solvable in our case