An Enumerative Perspective on Connectivity

TL;DR

The paper reexamines APC and k-APC in directed graphs through a generating-function lens, replacing the prior flow-vector proofs with combinatorial interpretations of determinants of edge-adjacency-derived matrices. By encoding edge-disjoint walks as power-series weights and using random evaluations over a large finite field, the authors show that submatrix ranks of inverse-generated matrices recover exact connectivities and their bounded counterparts with high probability. A key contribution is a degree-reduction and low-rank substitution strategy that reduces problem size, enabling $ ilde{O}(m^ ext{ω})$ time for APC and $ ilde{O}((kn)^ ext{ω})$ time for k-APC. This perspective tightens the conceptual link between algebraic graph algorithms and classical combinatorial tools while yielding simpler correctness proofs and practical randomized algorithms. The work also outlines open problems, including faster APC, deterministic verifiers, and extensions to vertex-connectivity variants.

Abstract

Connectivity (or equivalently, unweighted maximum flow) is an important measure in graph theory and combinatorial optimization. Given a graph $G$ with vertices $s$ and $t$, the connectivity $λ(s,t)$ from $s$ to $t$ is defined to be the maximum number of edge-disjoint paths from $s$ to $t$ in $G$. Much research has gone into designing fast algorithms for computing connectivities in graphs. Previous work showed that it is possible to compute connectivities for all pairs of vertices in directed graphs with $m$ edges in $\tilde{O}(m^ω)$ time [Chueng, Lau, and Leung, FOCS 2011], where $ω\in [2,2.3716)$ is the exponent of matrix multiplication. For the related problem of computing "small connectivities," it was recently shown that for any positive integer $k$, we can compute $\min(k,λ(s,t))$ for all pairs of vertices $(s,t)$ in a directed graph with $n$ nodes in $\tilde{O}((kn)^ω)$ time [Akmal and Jin, ICALP 2023]. In this paper, we present an alternate exposition of these $\tilde{O}(m^ω)$ and $\tilde{O}((kn)^ω)$ time algorithms, with simpler proofs of correctness. Earlier proofs were somewhat indirect, introducing an elegant but ad hoc "flow vector framework" for showing correctness of these algorithms. In contrast, we observe that these algorithms for computing exact and small connectivity values can be interpreted as testing whether certain generating functions enumerating families of edge-disjoint paths are nonzero. This new perspective yields more transparent proofs, and ties the approach for these problems more closely to the literature surrounding algebraic graph algorithms.

Figures (2)

• Figure 5: When we substitute $x_{ef} = y_{e1}z_{1f} + \dots + y_{ek}z_{kf}$ (pictured here for $k=3$) into $X$, we get the "simpler" matrix $YZ$. While powers of $X$ enumerate walks in $G$, powers of $YZ$ intuitively enumerate walks in a modified graph where after traversing an edge $e=(u,v)$, we have $k$ different versions of $v$ we can choose to go to. The $y_{ej}$ and $z_{jf}$ variables in this enumeration only keep track of the individual edges traversed and versions of vertices we pick, instead of recording all pairs of consecutive edges traversed like the $x_{ef}$ variables. This simpler enumeration suffices to solve $k$-APC.
• Figure 6: Given edge-disjoint paths $P_1 = \langle e_1,f_1\rangle$ and $P_2=\langle e_2,f_2\rangle$ in $G$, the determinant of the matrix $(I-X)^{-1}[\left\{e_1,e_2\right\},\left\{f_1,f_2\right\}]$ enumerates this pair via the monomial $\xi(P_1,P_2) = x_{e_1f_1}\cdot x_{e_2f_2}$. The variables in this monomial provide enough information to uniquely recover $P_1$ and $P_2$. In contrast, for $k=2$, the determinant of $(I-YZ)^{-1}[\left\{e_1,e_2\right\},\left\{f_1,f_2\right\}]$ assigns this pair weight $\tilde{\xi}(P_1,P_2) = (y_{e_11}z_{1f_1} + y_{e_12}z_{2f_1})(y_{e_21}z_{1f_2} + y_{e_22}z_{2f_2}).$One of the terms in the expansion of the above product is $y_{e_11}z_{1f_1}\cdot y_{e_22}z_{2f_2}$. We can read this term as saying "the first path $P_1$ traverses $e_1$ and $f_1$, and the second path $P_2$ traverses $e_2$ and $f_2$." So this monomial provides enough information to recover the pair of paths $\langle P_1,P_2\rangle$ as well.

Theorems & Definitions (26)

• Theorem 1.1
• Theorem 1.2
• proposition 2.1: Matrix Inversion
• proposition 2.2: Matrix Rank
• proposition 2.3
• corollary 2.4: Rational Identity Testing
• proposition 3.1: Power Series Inversion
• proposition 3.2: Geometric Series Formula
• proposition 4.1
• corollary 4.2: Enumerating Walks