[Paper Review] Quasi-polynomial-time algorithm for Independent Set in P t -free and C >t -free graphs via shrinking the space of connecting subgraphs.

Marcin Pilipczuk, Michał Pilipczuk+1 more|arXiv (Cornell University)|Sep 28, 2020

Complexity and Algorithms in Graphs5 citations

TL;DR

This paper presents a simplified quasi-polynomial-time algorithm for Maximum Weight Independent Set in $P_t$-free and $C_{>t}$-free graphs, achieving a running time of $n^{O(/log^2 n)}$ for $P_t$-free graphs and $n^{O(/log^4 n)}$ for $C_{>t}$-free graphs by exploiting structural properties of connecting subgraphs. The key insight is that in $P_t$-free graphs, a vertex with high connectivity to induced paths enables efficient branching, reducing the search space quasi-polynomially.

ABSTRACT

In a recent work, Gartland and Lokshtanov [FOCS 2020] gave a quasi-polynomial-time algorithm for Maximum Weight Independent Set in $P_t$-free graphs, that is, graphs excluding a path on $t$ vertices as an induced subgraph. Their algorithm runs in time $n^{O(\log^3 n)}$, where $t$ is assumed to be a~constant. Inspired by their ideas, we present an arguably simpler algorithm with an improved running time bound of $n^{O(\log^2 n)}$. Our main insight is that a connected $P_t$-free graph always contains a vertex $w$ whose neighborhood intersects, for a constant fraction of pairs $\{u,v\} \in \binom{V(G)}{2}$, a constant fraction of induced $u-v$ paths. Since a $P_t$-free graph contains $O(n^{t-1})$ induced paths in total, branching on such a vertex and recursing independently on the connected components leads to a quasi-polynomial running time bound. In a subsequent and very recent work, Gartland and Lokshtanov [arXiv:2007.11402] extended their ideas to $C_{> t}$-free graphs: graphs that do not contain a cycle on more than $t$ vertices as an induced subgraph. They obtained an algorithm for Maximum Weight Independent Set in this graph class with running time $n^{O(\log^5 n)}$. We show that it is possible to combine their ideas with our new understanding, and thus obtain an algorithm that runs in time $n^{O(\log^4 n)}$. We also show how to use the same approach to obtain quasi-polynomial-time algorithms for related problems, including Maximum Weight Induced Matching and 3-Coloring, in $P_t$-free and $C_{>t}$-free~graphs.

Motivation & Objective

• To improve the running time of quasi-polynomial algorithms for Maximum Weight Independent Set in $P_t$-free graphs beyond the $n^{O(/log^3 n)}$ bound of Gartland and Lokshtanov (2020).
• To extend the improved approach to $C_{>t}$-free graphs, where no prior quasi-polynomial algorithm was known for the problem.
• To unify and simplify the structural insights from prior work using a new vertex-based branching strategy based on induced path connectivity.
• To adapt the method to related problems such as Maximum Weight Induced Matching and 3-Coloring in the same graph classes.
• To achieve tighter quasi-polynomial time bounds by reducing the exponent in the logarithmic term through a refined analysis of connecting subgraphs.

Proposed method

• Leverage the structural property that in $P_t$-free graphs, there exists a vertex $w$ whose neighborhood intersects a constant fraction of induced $u$-$v$ paths for many vertex pairs $\{u,v\}$.
• Use this vertex $w$ as a branching point: recursively solve the problem on connected components after removing $w$, exploiting the fact that $P_t$-free graphs have $O(n^{t-1})$ induced paths.
• Apply a recursive decomposition strategy that reduces the search space by a constant factor per level, leading to a quasi-polynomial time bound.
• Extend the method to $C_{>t}$-free graphs by combining it with the structural insights from Gartland and Lokshtanov’s recent work on cycle-free induced subgraphs.
• Adapt the branching rule to maintain quasi-polynomial time complexity in both $P_t$-free and $C_{>t}$-free settings by bounding the number of critical connecting subgraphs.
• Generalize the approach to other problems by identifying similar structural constraints on connecting subgraphs in the same graph classes.

Experimental results

Research questions

• RQ1 Can the running time of the Maximum Weight Independent Set algorithm in $P_t$-free graphs be improved beyond $n^{O(\log^3 n)}$?
• RQ2 Is it possible to extend the improved algorithmic framework to $C_{>t}$-free graphs, which exclude long induced cycles?
• RQ3 Can the structural insight about high-connectivity vertices be formalized and used to reduce the search space in a way that yields a better quasi-polynomial exponent?
• RQ4 To what extent can this method be generalized to other NP-hard problems on $P_t$-free and $C_{>t}$-free graphs?
• RQ5 What is the minimal structural property of connecting subgraphs that enables quasi-polynomial-time algorithms in these graph classes?

Key findings

• The paper achieves a running time of $n^{O(\log^2 n)}$ for Maximum Weight Independent Set in $P_t$-free graphs, improving upon the prior $n^{O(\log^3 n)}$ bound.
• For $C_{>t}$-free graphs, the algorithm runs in time $n^{O(\log^4 n)}$, improving the previous $n^{O(\log^5 n)}$ bound from Gartland and Lokshtanov (2020).
• The key innovation is identifying a vertex $w$ whose neighborhood intersects a constant fraction of induced $u$-$v$ paths in $P_t$-free graphs, enabling efficient branching.
• The method simplifies and generalizes prior approaches by focusing on the connectivity of neighborhoods to induced paths, reducing algorithmic complexity.
• The framework is extended to solve Maximum Weight Induced Matching and 3-Coloring in $P_t$-free and $C_{>t}$-free graphs in quasi-polynomial time.
• The approach demonstrates that structural constraints on connecting subgraphs can be exploited to achieve improved quasi-polynomial time bounds in sparse graph classes.