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Continuous iterative algorithms for anti-Cheeger cut

Sihong Shao, Chuan Yang

TL;DR

This paper tackles the anti-Cheeger cut problem by developing three continuous iterative algorithms (CIA0, CIA1, CIA2) grounded in a Dinkelbach-type reformulation to avoid rounding. CIA0 uses a basic subgradient, while CIA1 refines subgradient selection to ensure monotone improvement, and CIA2 couples with maxcut iterations to escape local optima, all with inner subproblems that admit closed-form solutions and provable finite-step local convergence. Numerical results on the G-set show that the CIAs achieve solution quality competitive with MOH and generally faster than CirCut, with CIA2 achieving the best ratios (up to about 0.994) when enhanced with population updates or perturbations. The study highlights the strong connection between anti-Cheeger cut and maxcut within a unified continuous-optimization framework and suggests avenues for further efficiency improvements and theoretical insights.

Abstract

As a judicious correspondence to the classical maxcut, the anti-Cheeger cut has more balanced structure, but few numerical results on it have been reported so far. In this paper, we propose a continuous iterative algorithm (CIA) for the anti-Cheeger cut 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 objective function values are monotonically updated and the iteration points converge to a local optimum in finite steps via an appropriate subgradient selection. It can also be easily combined with the maxcut iterations for breaking out of local optima and improving the solution quality thanks to the similarity between the anti-Cheeger cut problem and the maxcut problem. The performance of CIAs is fully demonstrated through numerical experiments on G-set from two aspects: one is on the solution quality where we find that the approximate solutions obtained by CIAs are of comparable quality to those by the multiple search operator heuristic method; the other is on the computational cost where we show that CIAs always run faster than the often-used continuous iterative algorithm based on the rank-two relaxation.

Continuous iterative algorithms for anti-Cheeger cut

TL;DR

This paper tackles the anti-Cheeger cut problem by developing three continuous iterative algorithms (CIA0, CIA1, CIA2) grounded in a Dinkelbach-type reformulation to avoid rounding. CIA0 uses a basic subgradient, while CIA1 refines subgradient selection to ensure monotone improvement, and CIA2 couples with maxcut iterations to escape local optima, all with inner subproblems that admit closed-form solutions and provable finite-step local convergence. Numerical results on the G-set show that the CIAs achieve solution quality competitive with MOH and generally faster than CirCut, with CIA2 achieving the best ratios (up to about 0.994) when enhanced with population updates or perturbations. The study highlights the strong connection between anti-Cheeger cut and maxcut within a unified continuous-optimization framework and suggests avenues for further efficiency improvements and theoretical insights.

Abstract

As a judicious correspondence to the classical maxcut, the anti-Cheeger cut has more balanced structure, but few numerical results on it have been reported so far. In this paper, we propose a continuous iterative algorithm (CIA) for the anti-Cheeger cut 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 objective function values are monotonically updated and the iteration points converge to a local optimum in finite steps via an appropriate subgradient selection. It can also be easily combined with the maxcut iterations for breaking out of local optima and improving the solution quality thanks to the similarity between the anti-Cheeger cut problem and the maxcut problem. The performance of CIAs is fully demonstrated through numerical experiments on G-set from two aspects: one is on the solution quality where we find that the approximate solutions obtained by CIAs are of comparable quality to those by the multiple search operator heuristic method; the other is on the computational cost where we show that CIAs always run faster than the often-used continuous iterative algorithm based on the rank-two relaxation.

Paper Structure

This paper contains 9 sections, 4 theorems, 53 equations, 6 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Iterative algorithms
  3. Local convergence
  4. Numerical experiments
  5. Comparison with MOH in solution quality
  6. Comparison with CirCut in computational cost
  7. Conclusion and outlook
  8. Multiple search operator heuristic for the anti-Cheeger cut
  9. Rank-two relaxation for the anti-Cheeger cut

Key Result

proposition 1

Suppose $\hbox{\boldmath$x$}^k$, $r^k$ and $\hbox{\boldmath$s$}^k$ are generated by the simple iteration iter1. Then for $k\geq 1$, we always have $0\le r^{k-1} \le r^k\leq \Vert \hbox{\boldmath$s$}^k\Vert_1$. In particular, (1) $r^k= \Vert \hbox{\boldmath$s$}^k\Vert_1$ if and only if $\hbox{\boldma denotes the minimal value of eq:twostep_x2 and $\mathrm{Sgn}(\hbox{\boldmath$x$}) =\{(s_1, s_2,\ldo

Figures (6)

  • Figure 1: Anti-Cheeger cut (left) v.s. maxcut (right) on the Petersen graph: A bipartition $(S, S^\prime)$ of the vertex set $V$ is displayed in red bullets ($S$) and blue circles ($S^\prime$). For the anti-Cheeger cut, we have $\mathrm{cut}(S) = 11$, $\mathrm{vol}(S) = \mathrm{vol}(S^\prime)=15$, $h_{\mathrm{anti}}(G) = \frac{11}{15}$, and the remaining four uncut edges are divided equally in $S$ and $S^\prime$ ( judicious!). For the maxcut, we have $\mathrm{cut}(S) = 12$, $\mathrm{vol}(V) = 30$, $h_{\max}(G) = \frac{12}{15}$, but the remaining three uncut edges are all contained in $S$ ( biased!).
  • Figure 2: The flowchart of CIA2 --- a continuous iterative algorithm for the anti-Cheeger cut equipped with breaking out of local optima by the maxcut. $F_{\text{anti}}(\hbox{\boldmath$x$})$ denotes the objective function of the anti-Cheeger cut problem \ref{['eq:antiCheeger-I(x)']} and $F_{\max}(\hbox{\boldmath$x$})$ the objective function of the maxcut problem \ref{['eq:maxcut-continuous']}. $T_{\text{tot}}$ gives the total iterative steps and $T_{=}$ counts the number of successive iterative steps in which the objective function values $F_{\text{anti}}(\hbox{\boldmath$x$})$ or $F_{\max}(\hbox{\boldmath$x$})$ keep unchanged.
  • Figure 3: The maximum objective function value chosen among 100 runs of CIA1 from the same initial data v.s. iterative steps. We can clearly observe that CIA1 converges fast to a local optimum after a few iteration steps. It stops increasing the maximum objective function values (see the fourth column of Table \ref{['tab:1']}) after 53 steps for G1, 30 for G14, 25 for G22, 30 for G35, 17 for G43 and 39 for G51.
  • Figure 4: The minimum, mean, and maximum anti-Cheeger cut values produced by CIA0, CIA1 and CIA0+CIA1. CIA0+CIA1 means the output of CIA0 serves as the input to CIA1 for possible solution quality improvements. We find that, CIA1 improves the results generated by CIA0 in most cases, and its approximate solutions are of comparable quality to CIA0+CIA1.
  • Figure 5: The minimum, mean, and maximum anti-Cheeger cut values obtained by CirCut0 and CIA2 are presented in (a), and others obtained by CirCut and CIA2-P are provided in (b). The population updating manner is not used here.
  • ...and 1 more figures

Theorems & Definitions (7)

  • proposition 1: Lemma 3.1 and Theorem 3.5 in SZZ2018
  • proposition 2: Theorem 3.3 and Corollary 3.4 in SZZ2018
  • theorem 1
  • proof
  • theorem 2: finite-step local convergence
  • proof
  • remark 1