2024/03/10 by R. Krithika, Krithika, R., V. K. Kutty Malu +3
Economics, Econometrics and Finance · #Data Structures and Algorithms (cs.DS) #F.2.2 #FOS: Computer and information sciences #Merger and Competition Analysis
paper · pdf · doi:10.48550/arxiv.2403.06290
openalex publication_date 2024/03/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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^*(2k) time. This improves over the current algorithm known for the problem [Algorithmica 2014]. Cycle Contraction admits an algorithm running in O^*((2 + εℓ)k) time where 0 < εℓ ≤ 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^*(2.5191n)-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(n2-ε ⋅ 2o(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(n2 ⋅ tw)-time, algorithm known for the problem [Discret. Appl. Math. 2014].