Blazing a Trail via Matrix Multiplications: A Faster Algorithm for Non-shortest Induced Paths

TL;DR

The paper addresses the problem of finding non-shortest induced $uv$-paths (uv-trails) in undirected graphs, building on a prior $O(n^{18})$-time algorithm. It introduces a guess-and-verify framework based on $uv$-straight graphs, wings, and winged quadruples, and then exploits poly-logarithmic matrix multiplications and a witness-matrix data structure to dramatically accelerate preprocessing. The main contribution is an $O(n^{4.75})$-time algorithm achieved through $O(n^ ext{ω})$ preprocessing to obtain a $uv$-straight instance, a $ ilde{O}(n^{2 ext{ω}})$ winged-quadruple data structure, and a degree-two trailblazer made feasible by dynamic connectivity, culminating in substantial speedups for related recognition tasks. This work broadens the practical tractability of induced-subgraph problems and raises open questions about shortest uv-trails and all-pairs variants, with implications for problems like hole-length recognition.

Abstract

For vertices $u$ and $v$ of an $n$-vertex graph $G$, a $uv$-trail of $G$ is an induced $uv$-path of $G$ that is not a shortest $uv$-path of $G$. Berger, Seymour, and Spirkl [Discrete Mathematics 2021] gave the previously only known polynomial-time algorithm, running in $O(n^{18})$ time, to either output a $uv$-trail of $G$ or ensure that $G$ admits no $uv$-trail. We reduce the complexity to the time required to perform a poly-logarithmic number of multiplications of $n^2\times n^2$ Boolean matrices, leading to a largely improved $O(n^{4.75})$-time algorithm.

Figures (6)

• Figure 1: The red $uv$-path $P$ is the only $uv$-trail of the $uv$-straight graph $G$. The twist pair of $P$ is $(c,b)$. The twist of $P$ is $6$. $P[a^*,c]$ and $P[b,d^*]$ form a pair of wings for the quadruple $(a,b,c,d)$ of $V(G)$ in $G$.
• Figure 2: The blue dotted $uc$-path is a sidetrack $S$ for the red $uv$-trail $P$ of the $uv$-straight graph $G$. Each of $P[t,v]$ and the green dotted $tv$-path can be a monotone $tv$-path $T$ satisfying Condition S\ref{['S1']}.
• Figure 3: An illustration for the proof of Lemma \ref{['lemma:lemma4']}. The red path denotes a shortest $uv$-trail $P$ of the $uv$-straight graph $G$. The blue dotted monotone path denotes a sidetrack $S$ for $P$. The green dotted path denotes a monotone path $T$ satisfying Condition S\ref{['S1']}.
• Figure 4: An illustration for the proof that $B$ is a $uv$-trailblazer of degree four. The red path denotes a shortest $uv$-trail of the $uv$-straight graph $G$. The blue and green dotted paths denote a monotone $us$-path and a monotone $tv$-path of $G$ containing a precomputed pair of wings for $(a,t,s,d)$ that need not coincide with $P$ except at $a$, $t$, $s$, and $d$.
• Figure 5: An illustration for the proof that $B_1$ is a $uv$-trailblazer of degree two. The red path denotes a shortest $uv$-trail $P$ of the $uv$-straight graph $G$. The blue and green dotted paths denote a monotone $uc$-path $S_c$ and a monotone $tv$-path $T_c$ of $G$ containing a precomputed pair of wings for $(a,t,c,d_c)$ that need not coincide with $P$ except at $a$ and $t$.

Theorems & Definitions (12)

• Theorem 1
• Lemma 1: Berger et al. BergerSS21-dm
• Lemma 2: Implicit in Berger et al. BergerSS21-dm
• Lemma 3: Berger et al. BergerSS21-dm
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6