Faster Algorithms for All-Pairs Bounded Min-Cuts

TL;DR

This paper introduces faster algorithms and establishes lower bounds for the all-pairs bounded min-cut problem in directed graphs, significantly advancing the understanding of its computational complexity for various cases.

Contribution

It presents new randomized and deterministic algorithms for the problem, along with conditional lower bounds, improving the state-of-the-art for larger values of k.

Findings

New randomized algorithm for vertex capacities with time O((nk)^ω)

Deterministic algorithms for edge capacities in DAGs with different time complexities

First super-cubic lower bound under the 4-Clique conjecture for DAGs with unit vertex capacities

Abstract

The All-Pairs Min-Cut problem (aka All-Pairs Max-Flow) asks to compute a minimum $s$-$t$ cut (or just its value) for all pairs of vertices $s,t$. We study this problem in directed graphs with unit edge/vertex capacities (corresponding to edge/vertex connectivity). Our focus is on the $k$-bounded case, where the algorithm has to find all pairs with min-cut value less than $k$, and report only those. The most basic case $k=1$ is the Transitive Closure (TC) problem, which can be solved in graphs with $n$ vertices and $m$ edges in time $O(mn)$ combinatorially, and in time $O(n^{\omega})$ where $\omega<2.38$ is the matrix-multiplication exponent. These time bounds are conjectured to be optimal. We present new algorithms and conditional lower bounds that advance the frontier for larger $k$, as follows: (i) A randomized algorithm for vertex capacities that runs in time $O((nk)^{\omega})$. (ii) Two deterministic algorithms for edge capacities (which is more general) that work in DAGs and further reports a minimum cut for each pair. The first algorithm is combinatorial (does not involve matrix multiplication) and runs in time $O(2^{O(k^2)}\cdot mn)$. The second algorithm can be faster on dense DAGs and runs in time $O((k\log n)^{4^k+o(k)} n^{\omega})$. (iii) The first super-cubic lower bound of $n^{\omega-1-o(1)} k^2$ time under the $4$-Clique conjecture, which holds even in the simplest case of DAGs with unit vertex capacities. It improves on the previous (SETH-based) lower bounds even in the unbounded setting $k=n$. For combinatorial algorithms, our reduction implies an $n^{2-o(1)} k^2$ conditional lower bound. Thus, we identify new settings where the complexity of the problem is (conditionally) higher than that of TC.