Revisiting Path Contraction and Cycle Contraction

TL;DR

The paper revisits Path Contraction and Cycle Contraction, two NP-complete graph editing problems, to achieve faster algorithms and finer classifications on restricted graph classes. It introduces Path Contraction With Constrained Ends and builds a dynamic programming framework based on potential $k$-witness and potential $k$-prefix sets, yielding an overall running time of ${\mathcal{O}}^*(2^{k})$ for Path Contraction. Cycle Contraction is then tackled with an ${\mathcal{O}}^*((2 + \epsilon_{\ell})^{k})$-time FPT algorithm (where $\epsilon_{\ell} \leq 0.5509$ and $\ell = n - k$) and a faster exponential-time exact method, culminating in ${\mathcal{O}}^*(2.5191^{n})$ for the optimization problem. Additionally, the authors establish polynomial-time solvability for Path Contraction on planar graphs, show a tight fine-grained lower bound for Path Contraction on chordal graphs under the Orthogonal Vectors Conjecture, and discuss the linear-time solvability of Cycle Contraction on chordal graphs, highlighting a nuanced landscape of tractability across graph classes.

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. 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].

Figures (2)

• Figure 1: (Top) $W_q$ is one of the potential $k$-witness sets and $S_X$ and $S_Y$ are the corresponding potential $k$-prefix sets. The objective is to determine whether $G[S_X]$ and $G[S_Y]$ admit path witness structures $(W_1, W_2, \dots, W_{q})$ and $(W'_1, W'_2, \dots, W'_p)$ such that $X \subseteq W_1$, $Y \subseteq W'_1$, $W_q = W'_p$ and total number of edges contracted is at most $k$ (Bottom) $W_1$, $W_2$, $W_\ell$, $W_{\ell-1}$ (shaded sets) are identified during the initialization phase. For a potential $k$-witness set $W$ and the corresponding potential $k$-prefix set $S_X$, the algorithm enumerates all the possible sets $A$ that can be appended to a path witness structure of $G[S_X]$. The witness set $W_q$, mentioned in the top figure, lies in the unshaded area. The idea is to expand $S_X$ until it contains $W_{\ell - 2}$. The sets with sizes $\beta_1, \alpha, a$ and $b$ corresponds to four internal witness sets (except in some corner cases) justifying the search for sets $A$ satisfying $\beta_1 + \alpha + a + b \le k + 6 - |X|-|Y|$.
• Figure 2: The (chordal) graph $G$ in the reduction from Orthogonal Vectors to Path Contraction where $X \cup Y$ is an independent set and $Z$ is a clique.

Theorems & Definitions (36)

• Theorem 1
• corollary thmcountercorollary
• Theorem 2
• Theorem 3
• Theorem 4
• definition thmcounterdefinition: $(Q,\alpha,\beta)$-connected Set
• definition thmcounterdefinition: $(\alpha,\beta)$-connected Set
• proposition thmcounterproposition
• lemma thmcounterlemma
• proof