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Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds

Samuel Hwang, Noah G. Singer, Santhoshini Velusamy

TL;DR

It is shown that there exists an oblivious algorithm achieving an approximation ratio of at least $0.4853$, while every oblivious algorithm obeying a natural symmetry property achieves an approximation ratio of at most $0.4889".

Abstract

In the maximum directed cut problem, the input is a directed graph $G=(V,E)$, and the goal is to pick a partition $V = S \cup (V \setminus S)$ of the vertices such that as many edges as possible go from $S$ to $V\setminus S$. Oblivious algorithms, introduced by Feige and Jozeph (Algorithmica'17), are a simple class of algorithms for this problem. These algorithms independently and randomly assign each vertex $v$ to either $S$ or $V \setminus S$, and the distribution of $v$'s assignment is determined using only extremely local information about $v$: its bias, i.e., the relative difference between its out- and in-degrees. These algorithms have natural implementations in certain graph streaming models, where they have important implications (Saxena, Singer, Sudan, and Velusamy, SODA'23, FOCS'23, Kallaugher, Parekh, and Voronova, STOC'24). In this work, we narrow the gap between upper and lower bounds on the best approximation ratio achievable by oblivious algorithms for Max-Directed-Cut. We show that there exists an oblivious algorithm achieving an approximation ratio of at least $0.4853$, while every oblivious algorithm obeying a natural symmetry property achieves an approximation ratio of at most $0.4889$. The previous known bounds were $0.4844$ and $0.4899$, due to Singer (APPROX'23) and Feige and Jozeph, respectively. Our techniques involve designing principled parameterizations of the spaces of algorithms and lower bounds and then executing computer searches through these spaces.

Oblivious Algorithms for Maximum Directed Cut: New Upper and Lower Bounds

TL;DR

It is shown that there exists an oblivious algorithm achieving an approximation ratio of at least , while every oblivious algorithm obeying a natural symmetry property achieves an approximation ratio of at most $0.4889".

Abstract

In the maximum directed cut problem, the input is a directed graph , and the goal is to pick a partition of the vertices such that as many edges as possible go from to . Oblivious algorithms, introduced by Feige and Jozeph (Algorithmica'17), are a simple class of algorithms for this problem. These algorithms independently and randomly assign each vertex to either or , and the distribution of 's assignment is determined using only extremely local information about : its bias, i.e., the relative difference between its out- and in-degrees. These algorithms have natural implementations in certain graph streaming models, where they have important implications (Saxena, Singer, Sudan, and Velusamy, SODA'23, FOCS'23, Kallaugher, Parekh, and Voronova, STOC'24). In this work, we narrow the gap between upper and lower bounds on the best approximation ratio achievable by oblivious algorithms for Max-Directed-Cut. We show that there exists an oblivious algorithm achieving an approximation ratio of at least , while every oblivious algorithm obeying a natural symmetry property achieves an approximation ratio of at most . The previous known bounds were and , due to Singer (APPROX'23) and Feige and Jozeph, respectively. Our techniques involve designing principled parameterizations of the spaces of algorithms and lower bounds and then executing computer searches through these spaces.

Paper Structure

This paper contains 20 sections, 16 theorems, 24 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background: Directed cuts, bias, and oblivious algorithms
  3. Results
  4. Motivations
  5. Downstream applications.
  6. Oblivious algorithms for other problems.
  7. Structure of rest of the paper
  8. Code
  9. Preliminaries
  10. Directed graphs
  11. $\textsc{Max-DiCut}$ and oblivious algorithms
  12. Linear program
  13. Improved oblivious algorithms (\ref{['thm:pl-ub']})
  14. Lower bounds
  15. Methodology for finding hard instances
  16. ...and 5 more sections

Key Result

Theorem 1.1

There exists an oblivious algorithm $\mathcal{O}_{\mathsf{S}}$ achieving an approximation ratio $\alpha(\mathcal{O}_{\mathsf{S}}) \geq 0.483$.

Figures (7)

  • Figure 1: The step function $\mathsf{PLSigmoid}_{1/2}$ and its discretization into $\ell=5$ positive bias classes. The discretization is the function in red. The jump discontinuities are notated using standard marks: Open circles are open interval ends and closed circles are closed interval ends. The continuous (non-discretized) function $\mathsf{PLSigmoid}_{1/2}$ disagrees with its discretization only within the interval $[-1/2,+1/2]$ (marked by the vertical dashed line segments). The continuous function is represented by the black line segment within this interval.
  • Figure 2: A plot depicting how the fineness of discretization affects the approximation ratio calculated by the linear program of \ref{['thm:antisymmetric-lp']}, for two continuous selection functions: $\mathsf{PLSigmoid}_{1/2}$ and $\mathsf{PLSigmoid}_{149/309}$. Each point represents the approximation ratio of some oblivious algorithm, as calculated by the linear program in \ref{['thm:antisymmetric-lp']}. The horizontal axis records the number of bias classes (up to sign, i.e., as in \ref{['thm:antisymmetric-lp']}), and the vertical axis records the calculated approximation ratio. This plot was produced by https://github.com/singerng/oblivious-csps/blob/main/figures/discretization.py in the source code.
  • Figure 3: Parameter: $c > 1$. The LIGHT BLUE vertex ($1$) has bias $+\frac{c-1}{c+1}$. The PINK vertex ($2$) has bias $-\frac{c-1}{c+1}$. The assignment $1\to1,2\to0$ satisfies weight $c$. An oblivious assignment $1\to p,2 \to q$ satisfies weight $p (1-q) c + q (1-p)$. (Every two-vertex graph (without self-loops and with two nontrivial edges) is isomorphic to this graph up to rescaling.)
  • Figure 4: The (unweighted version) of the graph used in Case 2 of the proof of \ref{['thm:pl-lb']}.
  • Figure 5: Parameter: $c>1$. The LIGHT BLUE vertices ($\{1,2\}$) have bias $+\frac{c-1}{c+1}$ (note that $\frac{(c^2-1)+1-c}{(c^2-1)+1+c} = \frac{c(c-1)}{c(c+1)} = \frac{c-1}{c+1}$). The PINK vertices ($\{3,4\}$) have bias $-\frac{c-1}{c+1}$. The assignment $\{1,3\} \to 1, \{2,4\}\to0$ satisfies weight $c^2-1 + 2 \cdot 1= c^2+1$. An oblivious assignment $\{1,2\} \to p,\{3,4\}\to q$ satisfies weight $p(1-p)(c+1) + q(1-q) (c+1) + p(1-q) (c^2-1)$.
  • ...and 2 more figures

Theorems & Definitions (32)

  • Theorem 1.1: Prior upper bound, FJ15
  • Theorem 1.2: Prior lower bound for antisymmetric selection, FJ15
  • Theorem 1.3: Prior lower bound for general selection, FJ15
  • Remark
  • Theorem 1.4: Improved prior upper bound, Sin23-kand
  • Theorem 1.5: New upper bound
  • Theorem 1.6: Lower bound for PL sigmoid selection with $b=1/2$ intercept
  • Theorem 1.7: Lower bound for PL sigmoid selection with arbitrary intercept
  • Theorem 1.8: Lower bound for symmetric selection
  • Theorem 1.9: Lower bound for general selection
  • ...and 22 more