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A Primal-Dual Extension of the Goemans--Williamson Algorithm for the Weighted Fractional Cut-Covering Problem

Nathan Benedetto Proença, Marcel K. de Carli Silva, Cristiane M. Sato, Levent Tunçel

TL;DR

The paper studies the weighted fractional cut-covering problem through gauge duality, linking it to the maximum cut problem via antiblocker concepts. It introduces SDP relaxations η(G, w) and η^{∘}(G, z) and shows how Goemans–Williamson random hyperplane rounding extends to produce (1/α_{GW})-approximately optimal fractional cut covers, with certificates of approximate optimality tying the primal and dual problems together. A primal-dual extension of GW yields β-certificates that certify near-optimality for both maximum cut and fractional cut-covering, and it connects these results to geometric graph representations via hypersphere encodings. The work demonstrates near-optimality in several aspects and discusses tightness through sparsification limits, SDP perturbations, and complexity barriers, offering a cohesive framework for simultaneous approximation and certification in related graph problems.

Abstract

We study a weighted generalization of the fractional cut-covering problem, which we relate to the maximum cut problem via antiblocker and gauge duality. This relationship allows us to introduce a semidefinite programming (SDP) relaxation whose solutions may be rounded into fractional cut covers by sampling via the random hyperplane technique. We then provide a $1/α_{\scriptscriptstyle \mathrm{GW}}$-approximation algorithm for the weighted fractional cut-covering problem, where $α_{\scriptscriptstyle \mathrm{GW}} \approx 0.878$ is the approximation factor of the celebrated Goemans--Williamson algorithm for the maximum cut problem. Nearly optimal solutions of the SDPs in our duality framework allow one to consider instances of the maximum cut and the fractional cut-covering problems as primal-dual pairs, where cuts and fractional cut covers simultaneously certify each other's approximation quality. We exploit this relationship to introduce new combinatorial certificates for both problems, as well as a randomized polynomial-time algorithm for producing such certificates. In~particular, we~show how the Goemans--Williamson algorithm implicitly approximates a weighted instance of the fractional cut-covering problem, and how our new algorithm explicitly approximates a weighted instance of the maximum cut problem. We conclude by discussing the role played by geometric representations of graphs in our results, and by proving our algorithms and analyses to be optimal in several aspects.

A Primal-Dual Extension of the Goemans--Williamson Algorithm for the Weighted Fractional Cut-Covering Problem

TL;DR

The paper studies the weighted fractional cut-covering problem through gauge duality, linking it to the maximum cut problem via antiblocker concepts. It introduces SDP relaxations η(G, w) and η^{∘}(G, z) and shows how Goemans–Williamson random hyperplane rounding extends to produce (1/α_{GW})-approximately optimal fractional cut covers, with certificates of approximate optimality tying the primal and dual problems together. A primal-dual extension of GW yields β-certificates that certify near-optimality for both maximum cut and fractional cut-covering, and it connects these results to geometric graph representations via hypersphere encodings. The work demonstrates near-optimality in several aspects and discusses tightness through sparsification limits, SDP perturbations, and complexity barriers, offering a cohesive framework for simultaneous approximation and certification in related graph problems.

Abstract

We study a weighted generalization of the fractional cut-covering problem, which we relate to the maximum cut problem via antiblocker and gauge duality. This relationship allows us to introduce a semidefinite programming (SDP) relaxation whose solutions may be rounded into fractional cut covers by sampling via the random hyperplane technique. We then provide a -approximation algorithm for the weighted fractional cut-covering problem, where is the approximation factor of the celebrated Goemans--Williamson algorithm for the maximum cut problem. Nearly optimal solutions of the SDPs in our duality framework allow one to consider instances of the maximum cut and the fractional cut-covering problems as primal-dual pairs, where cuts and fractional cut covers simultaneously certify each other's approximation quality. We exploit this relationship to introduce new combinatorial certificates for both problems, as well as a randomized polynomial-time algorithm for producing such certificates. In~particular, we~show how the Goemans--Williamson algorithm implicitly approximates a weighted instance of the fractional cut-covering problem, and how our new algorithm explicitly approximates a weighted instance of the maximum cut problem. We conclude by discussing the role played by geometric representations of graphs in our results, and by proving our algorithms and analyses to be optimal in several aspects.
Paper Structure (20 sections, 30 theorems, 200 equations, 2 figures, 4 algorithms)

This paper contains 20 sections, 30 theorems, 200 equations, 2 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Organization of the Text
  3. Notation
  4. A Randomized Rounding Algorithm for Weighted Fractional Cut Covering
  5. A Primal-Dual Extension of the Goemans--Williamson Algorithm With Certificates of Approximate Optimality
  6. Gauge Duality
  7. β-pairings
  8. Existence of α-Certificates
  9. Algorithmically Obtaining β-Certificates
  10. Geometric Representation of Graphs
  11. Tightness of our Results
  12. Sparsification of Rounded Solution
  13. Edge Thickening or SDP Perturbation
  14. Asymptotic Support Size
  15. Computational Complexity of Fractional Cut Covering
  16. ...and 5 more sections

Key Result

Proposition 1

Let $G = (V, E)$ be a graph and let $z \in \mathbb{R}_+^E$. Let $(\mu,Y)$ be feasible for eq:GW-polar-def-intro. Set $y \in \mathbb{R}^{\mathcal{P}(V)}$ by Then $y$ is a fractional cut cover for $(G, z)$ with objective value $\mathbbm{1}^{\mathsf{T}}y = \tfrac{1}{\alpha_{ \mathrm{GW}}}\mu$. In particular,

Figures (2)

  • Figure 1: \ref{['sec:certificates']} produces solutions accompanied by certificates of their approximate optimality. These developments parallel \ref{['sec:rounding']}: we exploit the properties of optimal solutions to SDP relaxations in a rounding procedure. As an auxiliary step, we study simultaneous approximate solutions to both problems.
  • Figure 2: Geometric equivalence between hypersphere representations illustrating \ref{['thm:geometric-representation-equivalence']}.

Theorems & Definitions (71)

  • Proposition 1
  • proof
  • Corollary 2
  • proof
  • Remark 3
  • Proposition 4
  • proof
  • Remark 5
  • Theorem 6
  • proof
  • ...and 61 more