Search results for "Hamming graph"
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On the Low-Dimensional Steiner Minimum Tree Problem in Hamming Metric
2011
It is known that the d-dimensional Steiner Minimum Tree Problem in Hamming metric is NP-complete if d is considered to be a part of the input. On the other hand, it was an open question whether the problem is also NP-complete in fixed dimensions. In this paper we answer this question by showing that the problem is NP-complete for any dimension strictly greater than 2. We also show that the Steiner ratio is 2 - 2/d for d ≥ 2. Using this result, we tailor the analysis of the so-called k-LCA approximation algorithm and show improved approximation guarantees for the special cases d = 3 and d = 4.
On the low-dimensional Steiner minimum tree problem in Hamming metric
2013
While it is known that the d-dimensional Steiner minimum tree problem in Hamming metric is NP-complete if d is part of the input, it is an open question whether this also holds for fixed dimensions. In this paper, this question is answered by showing that the Steiner minimum tree problem in Hamming metric is already NP-complete in 3 dimensions. Furthermore, we show that, the minimum spanning tree gives a 2-2d approximation on the Steiner minimum tree for d>=2. Using this result, we analyse the so-called k-LCA and A"k approximation algorithms and show improved approximation guarantees for low dimensions.