Revisiting Path Contraction and Cycle Contraction
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911792666509312 |
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| author | Krithika, R. Malu, V. K. Kutty Tale, Prafullkumar |
| author_facet | Krithika, R. Malu, V. K. Kutty Tale, Prafullkumar |
| contents | 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 $\mathcal{O}^*(2^{k})$ time. This improves over the current algorithm known for the problem [Algorithmica 2014]. Cycle Contraction admits an algorithm running in $\mathcal{O}^*((2 + ε_{\ell})^k)$ time where $0 < ε_{\ell} \leq 0.5509$ is inversely proportional to $\ell = n - k$.
Central to these results is an algorithm for a general variant of Path Contraction, namely, Path Contraction With Constrained Ends. We also give an $\mathcal{O}^*(2.5191^n)$-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 $\mathcal{O}(n^{2-ε} \cdot 2^{o(tw)})$ for any $ε> 0$, unless the Orthogonal Vectors Conjecture fails. Here, $tw$ is the treewidth of the input graph. The second result complements the $\mathcal{O}(nm)$-time, i.e., $\mathcal{O}(n^2 \cdot tw)$-time, algorithm known for the problem [Discret. Appl. Math. 2014]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_06290 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Revisiting Path Contraction and Cycle Contraction Krithika, R. Malu, V. K. Kutty Tale, Prafullkumar Data Structures and Algorithms F.2.2 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 $\mathcal{O}^*(2^{k})$ time. This improves over the current algorithm known for the problem [Algorithmica 2014]. Cycle Contraction admits an algorithm running in $\mathcal{O}^*((2 + ε_{\ell})^k)$ time where $0 < ε_{\ell} \leq 0.5509$ is inversely proportional to $\ell = n - k$. Central to these results is an algorithm for a general variant of Path Contraction, namely, Path Contraction With Constrained Ends. We also give an $\mathcal{O}^*(2.5191^n)$-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 $\mathcal{O}(n^{2-ε} \cdot 2^{o(tw)})$ for any $ε> 0$, unless the Orthogonal Vectors Conjecture fails. Here, $tw$ is the treewidth of the input graph. The second result complements the $\mathcal{O}(nm)$-time, i.e., $\mathcal{O}(n^2 \cdot tw)$-time, algorithm known for the problem [Discret. Appl. Math. 2014]. |
| title | Revisiting Path Contraction and Cycle Contraction |
| topic | Data Structures and Algorithms F.2.2 |
| url | https://arxiv.org/abs/2403.06290 |