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The Maximum Clique Problem under Adversarial Uncertainty: a min-max approach

Immanuel Bomze, Chiara Faccio, Francesco Rinaldi, Giovanni Spisso

TL;DR

The paper addresses the problem of finding large cliques in graphs subject to adversarial edge perturbations, formalized as a two-player zero-sum game between a clique seeker and an adversary. It advances a penalized, continuous min–max reformulation that ensures the global minimizers correspond to maximum common cliques across all perturbations, by introducing an epigraphical QCQP and a penalty parameter threshold. A novel projection-free algorithm based on a Goldstein–Clarke subdifferential framework is developed to solve the nonsmooth reformulation, with global sublinear convergence to Goldstein and Clarke stationary points. Empirical results on DIMACS-like graphs show the method reliably detects large cliques common to all perturbations, highlighting the approach’s potential for robust clique detection in uncertain networks.

Abstract

We analyze the problem of identifying large cliques in graphs that are affected by adversarial uncertainty. More specifically, we consider a new formulation, namely the adversarial maximum clique problem, which extends the classical maximum-clique problem to graphs with edges strategically perturbed by an adversary. The proposed mathematical model is thus formulated as a two-player zero-sum game between a clique seeker and an opposing agent. Inspired by regularized continuous reformulations of the maximum-clique problem, we derive a penalized continuous formulation leading to a nonconvex and nonsmooth optimization problem. We further introduce the notion of stable global solutions, namely points remaining optimal under small perturbations of the penalty parameters, and prove an equivalence between stable global solutions of the continuous reformulation and largest cliques that are common to all the adversarially perturbed graphs. In order to solve the given nonsmooth problem, we develop a first-order and projection-free algorithm based on generalized subdifferential calculus in the sense of Clarke and Goldstein, and establish global sublinear convergence rates for it. Finally, we report numerical experiments on benchmark instances showing that the proposed method efficiently detects large common cliques.

The Maximum Clique Problem under Adversarial Uncertainty: a min-max approach

TL;DR

The paper addresses the problem of finding large cliques in graphs subject to adversarial edge perturbations, formalized as a two-player zero-sum game between a clique seeker and an adversary. It advances a penalized, continuous min–max reformulation that ensures the global minimizers correspond to maximum common cliques across all perturbations, by introducing an epigraphical QCQP and a penalty parameter threshold. A novel projection-free algorithm based on a Goldstein–Clarke subdifferential framework is developed to solve the nonsmooth reformulation, with global sublinear convergence to Goldstein and Clarke stationary points. Empirical results on DIMACS-like graphs show the method reliably detects large cliques common to all perturbations, highlighting the approach’s potential for robust clique detection in uncertain networks.

Abstract

We analyze the problem of identifying large cliques in graphs that are affected by adversarial uncertainty. More specifically, we consider a new formulation, namely the adversarial maximum clique problem, which extends the classical maximum-clique problem to graphs with edges strategically perturbed by an adversary. The proposed mathematical model is thus formulated as a two-player zero-sum game between a clique seeker and an opposing agent. Inspired by regularized continuous reformulations of the maximum-clique problem, we derive a penalized continuous formulation leading to a nonconvex and nonsmooth optimization problem. We further introduce the notion of stable global solutions, namely points remaining optimal under small perturbations of the penalty parameters, and prove an equivalence between stable global solutions of the continuous reformulation and largest cliques that are common to all the adversarially perturbed graphs. In order to solve the given nonsmooth problem, we develop a first-order and projection-free algorithm based on generalized subdifferential calculus in the sense of Clarke and Goldstein, and establish global sublinear convergence rates for it. Finally, we report numerical experiments on benchmark instances showing that the proposed method efficiently detects large common cliques.
Paper Structure (13 sections, 11 theorems, 118 equations, 1 figure, 3 tables, 1 algorithm)

This paper contains 13 sections, 11 theorems, 118 equations, 1 figure, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminary formulation
  3. Contributions
  4. Exact penalization leads to the maximum common clique
  5. Adding two penalty terms; epigraphic QCQP formulation
  6. The role of common cliques and their characteristic vectors
  7. Going global: exact penalty parameter values
  8. A projection-free algorithm for the continuous min-max reformulation
  9. Preliminaries
  10. Algorithmic scheme
  11. Theoretical convergence guarantees
  12. Experimental results
  13. Conclusion

Key Result

lemma thmcounterlemma

The Karush/Kuhn/Tucker (KKT) conditions for problem kkt-formulation are Therefore, under Equations opt to compl we have for the $\bm{\lambda}$-weighted average ${\mathbf h}^x(\mathbf x|\bm{\lambda})$ of the gradients $\nabla_{\mathbf x} g_{\mathsf U}(\mathbf x, \mathbf y)$ it holds ${\mathbf h}^x(\mathbf x|\bm{\lambda}) - \sigma \mathbf e + \delta =\bm{\mu} + \bm{\phi}\in \mathbb R^n_+$ and for

Figures (1)

  • Figure 1: original graph $\mathcal{G}$, and edge configurations $\mathcal{G}({\mathsf U}_1)$, $\mathcal{G}({\mathsf U}_2)$, $\mathcal{G}({\mathsf U}_3)$, $\mathcal{G}({\mathsf U}_4)$ that may occur due to adversarial intervention.

Theorems & Definitions (31)

  • lemma thmcounterlemma
  • proof
  • proposition thmcounterproposition
  • proof
  • lemma thmcounterlemma
  • proof
  • proposition thmcounterproposition
  • proof
  • proposition thmcounterproposition
  • proof
  • ...and 21 more