theory | Kirtan Padh

Exact matching

Fri, 01 Dec 2017 00:00:00 +0000

The matching problem is one of the most widely studied problems in complexity theory and combinatorial optimization. A blue-red graph is a graph with each of its edges coloured either red or blue. Given a parameter k, the exact matching problem asks for a perfect matching with exactly k red edges in a blue-red graph. We survey the exact matching problem, one of the lesser understood problems in the class of matching problems. We also propose an additive approximation of the problem which gives a matching with n − 1 edges with exactly k red edges for a graph with 2n vertices, or correctly asserts that no perfect matching with exactly k red edges exists in the graph.

Combinatorial Geometry

Thu, 01 Jun 2017 00:00:00 +0000

We give a survey of results on disjointness and intersection graphs of geometric objects. We start by proving Ramsey-type results for the intersection graphs of convex sets in the plane. We first show that every intersection graph of n convex sets in the plane has a clique or independent set of size n^{1/5} . After that we show that if G is the intersection graph of n convex sets in the plane, then G or Ḡ has a bi-clique of size cn for a universal constant c. We then show an upper bound for the number of edges in a geometric graph on n vertices with no k + 1 disjoint edges. Finally, we look at the colouring properties of disjointness graphs and prove that the family of disjointness graphs of segments in the plane is χ-bounded. We also show a generalization of this result to higher dimensions and discuss the χ-boundedness of related families of disjointness graphs.

Facility location on graphs

Mon, 01 Jun 2015 00:00:00 +0000

hek-center problem is that of choosing k vertices as centers in a weighted undirected graph in which the edge weights obey the triangle inequality so that the maximum distance of any vertex to its nearest center is minimized. The problem is NP-hard, but there is a simple greedy 2-approximation algorithm which has been shown to be optimal. We consider here the capacitated k-center problem, where additionally each vertex has a capacity, which is a bound on the number of ‘clients’ it can serve if it is opened as a center. Unlike the uncapacitated k-center problem, our understanding of the capacitated version is far from complete. We mainly concern ourselves with the case when all capacities are equal, which is called the uniform capacity k-center problem. We give here an L-approximation for the uniform k-center problem where each vertex has capacity L.