Complexity results for the spanning tree congestion problem

Authors:
Yota Otachi;Hans L. Bodlaender;Erik Jan Van Leeuwen
Affiliations:
Graduate School of Information Sciences, Tohoku University, Sendai, Japan;Institute of Information and Computing Sciences, Utrecht University, Utrecht, The Netherlands;Department of Informatics, University of Bergen, Bergen, Norway
Venue:
WG'10 Proceedings of the 36th international conference on Graph-theoretic concepts in computer science
Year:
2010

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Abstract

We study the problem of determining the spanning tree congestion of a graph. We present some sharp contrasts in the complexity of this problem. First, we show that for every fixed k and d the problem to determine whether a given graph has spanning tree congestion at most k can be solved in linear time for graphs of degree at most d. In contrast, if we allow only one vertex of unbounded degree, the problem immediately becomes NP-complete for any fixed k ≥ 10. For very small values of k however, the problem becomes polynomially solvable. We also show that it is NP-hard to approximate the spanning tree congestion within a factor better than 11/10. On planar graphs, we prove the problem is NP-hard in general, but solvable in linear time for fixed k.