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Deterministic Minimum Steiner Cut in Maximum Flow Time

Matthew Ding, Jason Li

TL;DR

The paper addresses the deterministic computation of the minimum Steiner cut in undirected weighted graphs with terminal set $T$. It introduces a novel $(s, extdelta, extgamma, T)$-terminal-strong decomposition, obtained via a terminal-aware cut-matching game, and uses it to drive a sparsification-based reduction that halves the terminal set while preserving the minimum Steiner cut. The overall algorithm achieves polylogarithmic numbers of max-flow calls and near-linear time outside max-flows, improving upon prior deterministic approaches and matching the efficiency of randomized schemes up to polylog factors. A key takeaway is that a near-linear deterministic $s-t$ max-flow algorithm would directly yield near-linear time deterministic minimum Steiner cut, highlighting the broad potential impact of terminal-strong decompositions on graph algorithms and their weighted variants.

Abstract

We devise a deterministic algorithm for minimum Steiner cut, which uses $(\log n)^{O(1)}$ maximum flow calls and additional near-linear time. This algorithm improves on Li and Panigrahi's (FOCS 2020) algorithm, which uses $(\log n)^{O(1/ε^4)}$ maximum flow calls and additional $O(m^{1+ε})$ time, for $ε> 0$. Our algorithm thus shows that deterministic minimum Steiner cut can be solved in maximum flow time up to polylogarithmic factors, given any black-box deterministic maximum flow algorithm. Our main technical contribution is a novel deterministic graph decomposition method for terminal vertices that generalizes all existing $s$-strong partitioning methods, which we believe may have future applications.

Deterministic Minimum Steiner Cut in Maximum Flow Time

TL;DR

The paper addresses the deterministic computation of the minimum Steiner cut in undirected weighted graphs with terminal set . It introduces a novel -terminal-strong decomposition, obtained via a terminal-aware cut-matching game, and uses it to drive a sparsification-based reduction that halves the terminal set while preserving the minimum Steiner cut. The overall algorithm achieves polylogarithmic numbers of max-flow calls and near-linear time outside max-flows, improving upon prior deterministic approaches and matching the efficiency of randomized schemes up to polylog factors. A key takeaway is that a near-linear deterministic max-flow algorithm would directly yield near-linear time deterministic minimum Steiner cut, highlighting the broad potential impact of terminal-strong decompositions on graph algorithms and their weighted variants.

Abstract

We devise a deterministic algorithm for minimum Steiner cut, which uses maximum flow calls and additional near-linear time. This algorithm improves on Li and Panigrahi's (FOCS 2020) algorithm, which uses maximum flow calls and additional time, for . Our algorithm thus shows that deterministic minimum Steiner cut can be solved in maximum flow time up to polylogarithmic factors, given any black-box deterministic maximum flow algorithm. Our main technical contribution is a novel deterministic graph decomposition method for terminal vertices that generalizes all existing -strong partitioning methods, which we believe may have future applications.
Paper Structure (23 sections, 19 theorems, 13 equations, 1 figure, 4 algorithms)

This paper contains 23 sections, 19 theorems, 13 equations, 1 figure, 4 algorithms.

Table of Contents

  1. Introduction
  2. Minimum Steiner Cut Background
  3. Our Contributions
  4. Preliminaries
  5. Sparsity and Strength
  6. Cut-Matching Game
  7. Minimum Steiner Cut Algorithm Overview
  8. Terminal Decomposition Using Cut-Matching Game
  9. Cut Player
  10. Matching Player
  11. Trimming
  12. Final Parameters
  13. Termination
  14. Terminal Decomposition
  15. Minimum Steiner Cut Using Sparsification
  16. ...and 8 more sections

Key Result

Theorem 1.1

Given an undirected, weighted graph $G=(V,E)$ with $n$ vertices and $m$ edges, polynomially bounded edge weights, and a set of terminal vertices $T \subseteq V$, there is a deterministic minimum Steiner cut algorithm that makes $\text{polylog}(n)$ maximum flow calls on undirected, weighted graphs wi

Figures (1)

  • Figure 1: Construction of \ref{['alg:trimming']}. Blue vertices and edges are added to the original graph $G$ and red vertices mark terminals.

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.2: Informal
  • Definition 2.1: Sparsity
  • Definition 2.2: Terminal-sparsity
  • Definition 2.3: $(s, \delta, \gamma)$-strength
  • Definition 2.4: $(s, \delta, \gamma, T)$-terminal-strength
  • Definition 2.5: $(s, \delta, \gamma, T)$-terminal-strong decomposition
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3: near_linear_mincut
  • ...and 32 more