Revisiting Path Contraction and Cycle Contraction

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. Path Contraction admits an algorithm running in 𝒪^*(2^k) time. This improves over the current algorithm known for the problem [Algorithmica 2014]. Cycle Contraction admits an algorithm running in 𝒪^*((2 + ϵ_ℓ)^k) time where 0 < ϵ_ℓ≤ 0.5509 is inversely proportional to ℓ = n - k. 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^*(2.5191^n)-time algorithm to solve the optimization version of Cycle Contraction. Next, we turn our attention to restricted graph classes and show the following results. Path Contraction on planar graphs admits a polynomial-time algorithm. Path Contraction on chordal graphs does not admit an algorithm running in time 𝒪(n^2-ϵ· 2^o(tw)) for any ϵ > 0, unless the Orthogonal Vectors Conjecture fails. Here, tw is the treewidth of the input graph. The second result complements the 𝒪(nm)-time, i.e., 𝒪(n^2 · tw)-time, algorithm known for the problem [Discret. Appl. Math. 2014].