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- Björklund, Andreas, et al.
(författare)
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Spotting Trees with Few Leaves
- 2015
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Ingår i: Lecture Notes in Computer Science/Automata, Languages, and Programming. - Berlin, Heidelberg : Springer Berlin Heidelberg. - 0302-9743 .- 1611-3349. - 9783662476727 ; 9134, s. 243-255
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Konferensbidrag (refereegranskat)abstract
- We show two results related to the Hamiltonicity and k -Path algorithms in undirected graphs by Björklund [FOCS’10], and Björklund et al., [arXiv’10]. First, we demonstrate that the technique used can be generalized to finding some k-vertex tree with l leaves in an n-vertex undirected graph in O∗(1.657^k2^{l/2}) time. It can be applied as a subroutine to solve the k -Internal Spanning Tree (k-IST) problem in O∗(min(3.455^k,1.946^n)) time using polynomial space, improving upon previous algorithms for this problem. In particular, for the first time, we break the natural barrier of O∗(2^n). Second, we show that the iterated random bipartition employed by the algorithm can be improved whenever the host graph admits a vertex coloring with few colors; it can be an ordinary proper vertex coloring, a fractional vertex coloring, or a vector coloring. In effect, we show improved bounds for k-Path and Hamiltonicity in any graph of maximum degree Δ=4,…,12 or with vector chromatic number at most 8.
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