Complexity and Approximation

This chapter is dedicated to Prof. Richard Karp whose Turing Award Lecture \Combinatorics, Complexity and Randomness" (Communications of the ACM, Feb 1986) inspired this author to start working in the eld of algorithms.

Approximation algorithms for finding highly connected subgraphs

Improved Approximation Algorithms for Uniform Connectivity Problems

The problem of finding minimum weight spanning subgraphs with a given connectivity requirement is considered. The problem is NP-hard when the connectivity requirement is greater than one. Polynomial time approximation algorithms for various weighted and unweighed connectivity problems are given. The following results are presented:

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Improved approximation algorithms for uniform connectivity problems

An efficient heuristic is presented for the problem of finding a minimum-size k-connected spanning subgraph of an (undirected or directed) simple graph G=(V,E). There are four versions of the problem, and the approximation guarantees are as follows:
minimum-size k-node connected spanning subgraph of an undirected graph 1 + [1/k], minimum-size k-node connected spanning subgraph of a directed graph 1 + [1/k], minimum-size k-edge connected spanning subgraph of an undirected graph 1+[2/(k+1)], minimum-size k-edge connected spanning subgraph of a directed graph 1 + [4/\sqrt{k}]. 
The heuristic is based on a subroutine for the degree-constrained subgraph (b-matching) problem. It is simple and deterministic and runs in time O(k|E|2).
The following result on simple undirected graphs is used in the analysis: The number of edges required for augmenting a graph of minimum degree k to be k-edge connected is at most k,|V|/(k+1).
For undirected graphs and k=2, a (deterministic) parallel NC version of the heuristic finds a 2-node connected (or 2-edge connected) spanning subgraph whose size is within a factor of ($1.5+\epsilon$) of minimum, where $\epsilon>0$ is a constant.

/pdf/approximating-minimum-size-k-connected-spanning-subgraphs-43qvqkeuny.pdf

Approximating Minimum-Size k -Connected Spanning Subgraphs via Matching

We consider the problems of finding minimum P-edge connected and P-vertex connected subgraphs in a given graph. These problems are NP-hard. We provide better techniques to lower bound the size of the minimum subgraphs. This allows us to achieve approximation factors of a and $ respectively, thereby improving on existing algorithms that achieve

Aman Singla

Papers

Improved approximation algorithms for biconnected subgraphs via better lower bounding techniques