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(Almost) Ruling Out SETH Lower Bounds for All-Pairs Max-Flow

Ohad Trabelsi

TL;DR

It is shown that for most problem settings, deterministic reductions based on the Strong Exponential Time Hypothesis (SETH) cannot rule out $O(n^{4-\varepsilon})$ time algorithms for some small $\varepsilon>0$, under a hypothesis called NSETH.

Abstract

The All-Pairs Max-Flow problem has gained significant popularity in the last two decades, and many results are known regarding its fine-grained complexity. Despite this, wide gaps remain in our understanding of the time complexity for several basic variants of the problem. In this paper, we aim to bridge these gaps by providing algorithms, conditional lower bounds, and non-reducibility results. Notably, we show that for most problem settings, deterministic reductions based on the Strong Exponential Time Hypothesis (SETH) cannot rule out $O(n^{4-\varepsilon})$ time algorithms for some small $\varepsilon>0$, under a hypothesis called NSETH. To obtain our result for the setting of undirected graphs with unit node-capacities, we design a new randomized Las Vegas $O(m^{2+o(1)})$ time combinatorial algorithm. This is our main technical result, improving over the recent $O(m^{11/5+o(1)})$ time Monte Carlo algorithm [Huang et al., STOC 2023] and matching their $m^{2-o(1)}$ lower bound (up to subpolynomial factors), thus essentially settling the time complexity for this setting of the problem.

(Almost) Ruling Out SETH Lower Bounds for All-Pairs Max-Flow

TL;DR

It is shown that for most problem settings, deterministic reductions based on the Strong Exponential Time Hypothesis (SETH) cannot rule out time algorithms for some small , under a hypothesis called NSETH.

Abstract

The All-Pairs Max-Flow problem has gained significant popularity in the last two decades, and many results are known regarding its fine-grained complexity. Despite this, wide gaps remain in our understanding of the time complexity for several basic variants of the problem. In this paper, we aim to bridge these gaps by providing algorithms, conditional lower bounds, and non-reducibility results. Notably, we show that for most problem settings, deterministic reductions based on the Strong Exponential Time Hypothesis (SETH) cannot rule out time algorithms for some small , under a hypothesis called NSETH. To obtain our result for the setting of undirected graphs with unit node-capacities, we design a new randomized Las Vegas time combinatorial algorithm. This is our main technical result, improving over the recent time Monte Carlo algorithm [Huang et al., STOC 2023] and matching their lower bound (up to subpolynomial factors), thus essentially settling the time complexity for this setting of the problem.
Paper Structure (35 sections, 23 theorems, 12 equations, 3 figures, 1 table)

This paper contains 35 sections, 23 theorems, 12 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. A New Algorithm for Graphs with Unit Node-Capacities
  3. Other Results
  4. Unit Node-Capacities
  5. General node-capacities.
  6. Directed edge-capacities.
  7. Technical Overview
  8. An algorithm for undirected graph with unit node-capacities
  9. Nondeterministic algorithms
  10. A related nondeterministic algorithm in AKT20_soda.
  11. Obstacles in accelerating current algorithms.
  12. Our approach.
  13. Preliminaries
  14. Non-Reducibility Results
  15. Verifying lower bounds.
  16. ...and 20 more sections

Key Result

Theorem 1.1

There is a randomized Las Vegas algorithm that given an undirected graph $G=(V,E)$ with unit node-capacities, makes $\tilde{O}(m)$ calls to Max-Flow in $G$ and spends $\tilde{O}(mn)$ time outside of these calls, and can:

Figures (3)

  • Figure 1: New and old bounds for All-Pairs Max-Flow on undirected graphs with unit node-capacities.
  • Figure 2: If $v\in V_{u}$ then both $u,v$ are queried directly as a high degree pair. Otherwise, with constant probability $v$ must have been separated from $u$ in a higher recursion node along $P$, in which case $v\notin C_{u,v}$ at that moment of separation.
  • Figure 3: The figure on the left shows an anti-directed walk of length $8$, while the figure on the right illustrates its elimination.

Theorems & Definitions (39)

  • Theorem 1.1
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 1.8: Definition $3.1$ from AKT20_soda. See also CGIMPS16
  • Definition 1.9: Definition $3.12$ from AKT20_soda. See also WILL18
  • Theorem 1.10
  • Theorem 2.1
  • ...and 29 more