Revisiting path contraction and cycle contraction

Abstract

The Path Contraction and Cycle Contraction problems take as input an undirected graph G with n vertices, m edges and an integer k and determine whether one can obtain a path or a cycle, respectively, by performing at most k edge contractions in G. We revisit these NP-complete problems and prove the following results.

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  Path Contraction admits an $O^{⁎}\left(2^k\right)$-time algorithm. This improves over the current fastest parameterized algorithm known for the problem (Heggernes et al. (2014) [15]).
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  Cycle Contraction admits an $O^{⁎}\left({(2+{ϵ}_{\ell })}^k\right)$-time algorithm where $0<{ϵ}_{\ell }\le 0.5509$ and ${ϵ}_{\ell }$ decreases as $\ell =n-k$ increases.

Central to these results is an algorithm for a general variant of Path Contraction, namely Path Contraction With Constrained Ends. We also give an $O^{⁎}\left({2.5191}^n\right)$-time algorithm to solve the optimization version of Cycle Contraction. Next, we turn our attention to restricted graph classes and show the following results.

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  Path Contraction on planar graphs admits a polynomial-time algorithm.
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  Path Contraction on chordal graphs does not admit an $O(n^{2-ϵ}\cdot 2^{o\left(tw\right)})$-time algorithm for any $ϵ>0$, under the Orthogonal Vectors Conjecture. Here, tw is the treewidth of the input graph.

The second result complements the $O\left(nm\right)$-time, i.e., $O(n^2\cdot tw)$-time, algorithm known for the problem (Heggernes et al. (2014) [16]).