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A simple iterative algorithm for maxcut

Sihong Shao, Dong Zhang, Weixi Zhang

TL;DR

The paper tackles the maxcut problem by formulating it as an equivalent continuous fractional program and developing a rounding-free, simple iterative (SI) algorithm. The inner subproblem is solved analytically, enabling efficient, monotone updates of the cut value $r^k$, with convergence to a local optimum via a carefully designed subgradient rule. A three-step scheme plus a rigorous analysis yields finite-step local convergence and actionable analytic solutions, while a perturbation strategy (SI-P) further improves solution quality. Empirical results on G-set show SI achieving high-quality cuts close to or surpassing state-of-the-art continuous methods, and SI-P attains ratios near 0.997 relative to combinatorial references. The approach bridges discrete and continuous optimization for maxcut and suggests potential extensions to other fractional programming problems.

Abstract

We propose a simple iterative (SI) algorithm for the maxcut problem through fully using an equivalent continuous formulation. It does not need rounding at all and has advantages that all subproblems have explicit analytic solutions, the cut values are monotonically updated and the iteration points converge to a local optima in finite steps via an appropriate subgradient selection. Numerical experiments on G-set demonstrate the performance. In particular, the ratios between the best cut values achieved by SI and those by some advanced combinatorial algorithms in [Ann. Oper. Res. 248 (2017) 365] are at least $0.986$ and can be further improved to at least $0.997$ by a preliminary attempt to break out of local optima.

A simple iterative algorithm for maxcut

TL;DR

The paper tackles the maxcut problem by formulating it as an equivalent continuous fractional program and developing a rounding-free, simple iterative (SI) algorithm. The inner subproblem is solved analytically, enabling efficient, monotone updates of the cut value , with convergence to a local optimum via a carefully designed subgradient rule. A three-step scheme plus a rigorous analysis yields finite-step local convergence and actionable analytic solutions, while a perturbation strategy (SI-P) further improves solution quality. Empirical results on G-set show SI achieving high-quality cuts close to or surpassing state-of-the-art continuous methods, and SI-P attains ratios near 0.997 relative to combinatorial references. The approach bridges discrete and continuous optimization for maxcut and suggests potential extensions to other fractional programming problems.

Abstract

We propose a simple iterative (SI) algorithm for the maxcut problem through fully using an equivalent continuous formulation. It does not need rounding at all and has advantages that all subproblems have explicit analytic solutions, the cut values are monotonically updated and the iteration points converge to a local optima in finite steps via an appropriate subgradient selection. Numerical experiments on G-set demonstrate the performance. In particular, the ratios between the best cut values achieved by SI and those by some advanced combinatorial algorithms in [Ann. Oper. Res. 248 (2017) 365] are at least and can be further improved to at least by a preliminary attempt to break out of local optima.

Paper Structure

This paper contains 11 sections, 9 theorems, 119 equations, 2 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Equivalent continuous problems
  3. A simple iterative algorithm
  4. Exact solution to the inner subproblem
  5. Subgradient selection and convergence analysis
  6. Numerical experiments
  7. Implementation and cost analysis
  8. Quality check
  9. Breaking out of local optima
  10. Comparison with CirCut
  11. Conclusion and outlook

Key Result

Theorem 2.1

The maxcut problem eq:cut can be rewritten into and any vector $\hbox{\boldmath$x$}^*$ reaching the maximum of $F(\hbox{\boldmath$x$})$ produces a maxcut $(S,S^\prime)$ where the subset $S$ satisfies $S^+(\hbox{\boldmath$x$}^*)\subset S\subset (S^-(\hbox{\boldmath$x$}^*))^c$.

Figures (2)

  • Figure 1: Quality check: Histogram of the ratios for all $2700\times 3$ runs depicted in Tab. \ref{['tab1']} (More explanations are referred to Tab. \ref{['tab1']}). The percent of runs obtained a ratio larger than $0.980$, which lie on the right of the black vertical line, exceeds $95\%$.
  • Figure 2: Comparison with CirCut: The minimum, mean, and maximum cut values (normalized by the number of edges $|E|$) produced by CirCut0, SI, and CirCut0+SI from multiple starting points are displayed in (a), while those obtained by CirCut, SI$\_$PERTURB, and CirCut+SI are presented in (b). CirCut0 refers to the pure rank-two relaxation which consists of the line-search and Procedure-CUT, only the first two steps of Alg. 1 in BurerMonteiroZhang2001. CirCut0+SI means the output of CirCut0 serves as the input to SI for possible solution quality improvements and so does CirCut+SI.

Theorems & Definitions (18)

  • Theorem 2.1
  • proof
  • Theorem 2.2: global convergence
  • proof
  • Remark 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3: exact solution
  • ...and 8 more