Stefan Hougardy

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For any integer k, we prove the existence of a uniquely k-colourable graph of girth at least g on at most k 12(g+1) vertices whose maximal degree is at most 5k 13. From this we deduce that, unless NP=RP, no polynomial time algorithm for k-Colourability on graphs G of girth g(G) log jGj 13 log k and maximum degree (G) 6k 13 can exist. We also study several(More)
We present a general iterative framework for improving the performance ratio of Steiner tree approximation algorithms. By applying this framework to one specific algorithm we obtain a new polynomial time approximation algorithm for the Steiner tree problem in graphs that achieves a performance ratio of 1.598 after 11 iterations. This beats the so far best(More)
Approximation algorithms have so far mainly been studied for problems that are not known to have polynomial time algorithms for solving them exactly. Here we propose an approximation algorithm for the weighted matching problem in graphs which can be solved in polynomial time. The weighted matching problem is to find a matching in an edge weighted graph that(More)
The terminal Steiner tree problem is a special version of the Steiner tree problem, where a Steiner minimum tree has to be found in which all terminals are leaves. We prove that no polynomial time approximation algorithm for the terminal Steiner tree problem can achieve an approximation ratio less than (1− o(1)) lnn unless NP has slightly superpolynomial(More)
The weighted matching problem is to find a matching in a weighted graph that has maximum weight. The fastest known algorithm for this problem has running time O(nm + n log n). Many real world problems require graphs of such large size that this running time is too costly. We present a linear time approximation algorithm for the weighted matching problem(More)
The Steiner tree problem asks for a shortest subgraph connecting a given set of terminals in a graph. It is known to be APX-complete, which means that no polynomial time approximation scheme can exist for this problem, unless P=NP. Currently, the best approximation algorithm for the Steiner tree problem has a performance ratio of + , , whereas the(More)