Path Contraction Faster Than 2n
🤖 AI Summary
This paper addresses the Path Contraction problem: given a graph $G$, determine whether it can be transformed into a path graph $P_k$ of length $k$ via a sequence of edge contractions. A classical graph modification problem, it has long lacked exact algorithms with runtime better than $2^n$. We present the first exact algorithm with runtime $O^*(c^n)$ for some $c < 2$, thereby strictly breaking the exponential base-2 barrier. Our approach introduces the notion of *contraction-sensitive vertices*, integrates structured branching with pruning, graph induction, and dynamic state compression, and employs refined recursive analysis. The theoretical correctness is fully established. The algorithm is deterministic and achieves provable exponential speedup on $n$-vertex graphs. Moreover, our framework yields a generalizable analytical paradigm for broader graph contraction problems.
Application Category
📝 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 $X$...
Problem
Research questions and friction points this paper is trying to address.
Develop faster algorithm for Path Contraction problem
Improve exponential time complexity below 2^n
Introduce 3-Disjoint Connected Subgraphs as subroutine
Innovation
Methods, ideas, or system contributions that make the work stand out.
Developed faster exact exponential algorithm for Path Contraction
Introduced 3-Disjoint Connected Subgraphs problem
Used 1.88^n time algorithm as subroutine
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