Path Contraction Faster than $2^n$
Akanksha Agrawal, Fedor V. Fomin, Daniel Lokshtanov, Saket Saurabh, Prafullkumar Tale
TL;DR
This work develops an exact exponential-time algorithm for Path Contraction, breaking the long-standing ${\mathcal O}^*(2^n)$ barrier by introducing a modular framework built around four specialized subroutines and a new subproblem, 3-Disjoint Connected Subgraphs (3-DCS). The core idea leverages witness structures that map a graph to a path, and employs targeted strategies—balancing, heavy unions, and odd/even partitions—to efficiently explore promising regions of the solution space. The combined approach yields an overall running time of ${\mathcal O}^*(1.99987^n)$ for Path Contraction, with a key auxiliary result that 3-DCS can be solved in ${\mathcal O}^*(1.88^n)$, enabling corollaries such as ${P_5}$-Contraction in the same time bound. The techniques rely on carefully bounded enumeration of small connected sets, dynamic programming over partial witness structures, and reductions to known problems like 2-Disjoint Connected Subgraphs, underscoring a broadly applicable strategy for improving brute-force graph editing algorithms.
Abstract
A graph $G$ is contractible to a graph $H$ if there is a set $X \subseteq E(G)$, such that $G/X$ is isomorphic to $H$. Here, $G/X$ is the graph obtained from $G$ by contracting all the edges in $X$. For a family of graphs $\cal F$, the $\mathcal{F}$-\textsc{Contraction} problem takes as input a graph $G$ on $n$ vertices, and the objective is to output the largest integer $t$, such that $G$ is contractible to a graph $H \in {\cal F}$, where $|V(H)|=t$. When $\cal F$ is the family of paths, then the corresponding $\mathcal{F}$-\textsc{Contraction} problem is called \textsc{Path Contraction}. The problem \textsc{Path Contraction} admits a simple algorithm running in time $2^{n}\cdot n^{\mathcal{O}(1)}$. In spite of the deceptive simplicity of the problem, beating the $2^{n}\cdot n^{\mathcal{O}(1)}$ bound for \textsc{Path Contraction} seems quite challenging. In this paper, we design an exact exponential time algorithm for \textsc{Path Contraction} that runs in time $1.99987^n\cdot n^{\mathcal{O}(1)}$. We also define a problem called \textsc{$3$-Disjoint Connected Subgraphs}, and design an algorithm for it that runs in time $1.88^n\cdot n^{\mathcal{O}(1)}$. The above algorithm is used as a sub-routine in our algorithm for {\sc Path Contraction}