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An approximation algorithm for Maximum DiCut vs. Cut

Tamio-Vesa Nakajima, Stanislav Živný

TL;DR

An SDP-based algorithm which, if given a directed graph G that has a directed cut of value rho, finds an undirected cut in G (ignoring edge directions) with value at least \rho is designed.

Abstract

Goemans and Williamson designed a 0.878-approximation algorithm for Max-Cut in undirected graphs [JACM'95]. Khot, Kindler, Mosel, and O'Donnel showed that the approximation ratio of the Goemans-Williamson algorithm is optimal assuming Khot's Unique Games Conjecture [SICOMP'07]. In the problem of maximum cuts in directed graphs (Max-DiCut), in which we seek as many edges going from one particular side of the cut to the other, the situation is more complicated but the recent work of Brakensiek, Huang, Potechin, and Zwick showed that their 0.874-approximation algorithm is tight under the Unique Games Conjecture (up to a small delta)[FOCS'23]. We consider a promise version of the problem and design an SDP-based algorithm which, if given a directed graph G that has a directed cut of value rho, finds an undirected cut in G (ignoring edge directions) with value at least ρ.

An approximation algorithm for Maximum DiCut vs. Cut

TL;DR

An SDP-based algorithm which, if given a directed graph G that has a directed cut of value rho, finds an undirected cut in G (ignoring edge directions) with value at least \rho is designed.

Abstract

Goemans and Williamson designed a 0.878-approximation algorithm for Max-Cut in undirected graphs [JACM'95]. Khot, Kindler, Mosel, and O'Donnel showed that the approximation ratio of the Goemans-Williamson algorithm is optimal assuming Khot's Unique Games Conjecture [SICOMP'07]. In the problem of maximum cuts in directed graphs (Max-DiCut), in which we seek as many edges going from one particular side of the cut to the other, the situation is more complicated but the recent work of Brakensiek, Huang, Potechin, and Zwick showed that their 0.874-approximation algorithm is tight under the Unique Games Conjecture (up to a small delta)[FOCS'23]. We consider a promise version of the problem and design an SDP-based algorithm which, if given a directed graph G that has a directed cut of value rho, finds an undirected cut in G (ignoring edge directions) with value at least ρ.
Paper Structure (4 sections, 11 theorems, 24 equations, 2 figures)

This paper contains 4 sections, 11 theorems, 24 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Results
  4. Analysis

Key Result

Theorem 1

There is a randomised algorithm which, if given a directed graph $G$ and $\varepsilon \in (0, 1)$, finds a cut $c$ in $G$ with expected value at least $\mathop{\mathrm{\textsc{DiCut}}}\nolimits(G) - \varepsilon$ in polynomial time in $|G|$ and $\log(1/\varepsilon)$.

Figures (2)

  • Figure 1: Rotation function $f_a$
  • Figure 2: Plot of $F(0.14, 0.36, z)$

Theorems & Definitions (20)

  • Theorem 1
  • Theorem 2
  • Definition 3
  • Theorem 3
  • Theorem 3
  • Lemma 3: Configuration lemma
  • proof : Proof of \ref{['thm:mainrand']}
  • proof : Proof of \ref{['thm:mainunrand']}
  • Lemma 3: Configuration lemma
  • Lemma 4
  • ...and 10 more