Eva-Marta Lundell

Efficient approximation algorithms for shortest cycles in undirected graphs

We describe a simple combinatorial approximation algorithm for finding a shortest (simple) cycle in an undirected graph. Given an adjacency-list representation of an undirected graph G with n vertices and unknown girth k, our algorithm returns with high probability a cycle of length at most 2k for even k and 2k+2 for odd k, in time O(n^3^2logn). Thus, in general, it yields a 223 approximation. For a weighted, undirected graph, with non-negative edge weights in the range {1,2,...,M}, we present a simple combinatorial 2-approximation algorithm for a minimum weight (simple) cycle that runs in time O(n^2logn(logn+logM)).

On the approximability of maximum and minimum edge clique partition problems

We consider the following clustering problems: given a general undirected graph, partition its vertices into disjoint clusters such that each cluster forms a clique and the number of edges within the clusters is maximized (Max-ECP), or the number of edges between clusters is minimized (Min-ECP). These problems arise naturally in the DNA clone classification. We investigate the hardness of finding such partitions and provide approximation algorithms. Further, we show that greedy strategies yield constant factor approximations for graph classes for which maximum cliques can be found efficiently.

Counting and Detecting Small Subgraphs via Equations

We present a general technique for detecting and counting small subgraphs. It consists of forming special linear combinations of the numbers of occurrences of different induced subgraphs of fixed size in a graph. These combinations can be efficiently computed by rectangular matrix multiplication. Our two main results utilizing the technique are as follows. Let

Induced Subgraph Isomorphism: Are Some Patterns Substantially Easier Than Others?

H

Induced subgraph isomorphism

be a fixed graph with

Detecting and Counting Small Pattern Graphs

k

Polynomial-Time Algorithms for the Ordered Maximum Agreement Subtree Problem

vertices and an independent set of size

The Approximability of Maximum Rooted Triplets Consistency with Fan Triplets and Forbidden Triplets

s.

Unique subgraphs are not easier to find

1. Detecting if an

Unique small subgraphs are not easier to find

n