Adversarial Maximum Clique Problem

• The adversarial maximum clique problem is defined as finding the largest clique in a graph whose edges are strategically modified by an adversary.
• Key algorithmic approaches leverage semidefinite programming relaxations and recursive combinatorial searches to recover planted cliques under worst-case perturbations.
• Robust min–max formulations and projection-free Frank–Wolfe schemes have shown strong empirical performance and theoretical guarantees on benchmark graph ensembles.

The adversarial maximum clique problem concerns identifying the largest clique in a graph where edge structure is subject to strategic modifications by an adversarial agent. Two principal adversarial settings dominate the literature: adversarially planted clique/independent set models (where an adversary "hides" a large clique or independent set within a random or semi-random background) and robust min–max frameworks (where every input graph arises as an adversarial perturbation within a prescribed uncertainty set). These models interrogate the algorithmic and complexity-theoretic boundaries of maximum clique detection under worst-case or semi-random corruptions, presenting substantial contrast with canonical random planted models. The adversarial maximum clique problem unites central concepts from random graph theory, combinatorial optimization, semidefinite relaxations, and robust nonconvex programming.

1. Formal Definitions and Model Variants

Let $G = (V, E)$, $|V| = n$, denote an undirected graph. The classical planted clique problem samples $G' \sim G(n, p)$ (Erdős–Rényi) and plants a clique of size $k$ on a uniform random $K \subset V$, yielding $G(n,p,k)$. In the adversarially planted clique model $\mathrm{AG}(n, p, k)$, a computationally unbounded adversary selects an arbitrary $k$-subset $K \subseteq V$ after observing $G'$, then adds all missing edges on $K$ to form $G$ (Feige et al., 2020).

Separately, the adversarial maximum clique problem in the robust optimization sense introduces an uncertainty set $\mathcal{U} = \{U^1, \ldots, U^m\}$ of perturbed adjacency matrices, each $U^k = A(G_k) + I_n$. The clique seeker selects $x \in \Delta^V$ (the $n$-simplex), and the adversary chooses the worst-case $U \in \mathcal{U}$; the objective is to maximize $\min_{U \in \mathcal{U}} x^\top U x$ (Bomze et al., 9 Jan 2026).

Key goals include:

• Recovering the planted clique $K$ (or a maximum clique in the adversarial ensemble).
• Extracting a clique of size at least $k$.
• Identifying cliques appearing universally in all adversarially generated graphs ("common cliques").

2. Algorithmic and Complexity Results in the Adversarially Planted Clique Model

For $G \sim \mathrm{AG}(n, \frac{1}{2}, k)$ with $k \geq \sqrt{n}$, there exists an explicit polynomial-time algorithm, based on semidefinite programming (SDP) relaxations via Lovász's $\theta$-function, which finds a maximum clique with high probability, uniformly over all adversarial choices of $K$. The main steps are:

1. Solve the $\theta$-function SDP for the complement graph $\hat{G}$ to high precision.
2. Identify the set $H = \{i : (h \cdot s_i)^2 \geq 3/4\}$, where $(s_i, h)$ are obtained from the SDP (handle and vectors).
3. Define $F = \{v : \deg_H(v) \geq \frac{3}{4}k\}$.
4. Find a maximum clique in $G[F]$ by recursively branching on non-edges (of depth $O(\log n)$).

This approach leverages the result that $\theta(\hat{G})$ is tightly concentrated, with $k \leq \theta(\hat{G}) \leq k + O(\log n)$ even under arbitrary adversarial planting, and the final combinatorial search is polynomial-time for $k \geq 10\sqrt{n}$ (Feige et al., 2020).

For sparser regimes, notably in $\mathrm{A}\bar{G}(n, n^{\delta-1}, k)$ with $0 < \delta < 1$ and $k \gtrsim n^{1-\delta} \log n$, the problem becomes provably NP-hard unless NP = RP. The reduction constructs an adversarial instance embedding an NP-hard maximum independent set instance as a subgraph, then leverages randomized versus adversarial planting indistinguishability, establishing that polynomial-time algorithms cannot exist unless randomized polynomial time can solve NP-hard problems (Feige et al., 2020).

3. Robust Min–Max and Common Clique Formulation

In the robust uncertainty framework (Bomze et al., 9 Jan 2026), the problem is formalized as

$$\max_{x \in \Delta^V} \min_{U \in \mathcal{U}} x^\top U x,$$

with $\mathcal{U}$ a finite set of adjacency matrices encoding adversarial perturbations. This can be equivalently recast (by sign negation) as a nonconvex, nonsmooth minimax problem:

$$\min_{x \in \Delta^V} \max_{U \in \mathcal{U}} -x^\top U x.$$

To enforce integrality (characteristic vectors of cliques), variables $y \in [0,1]^n$ are introduced with constraints $\varepsilon y_i \leq x_i \leq y_i$, $y_i \in \{0,1\}$, collectively forming a feasible set $\Delta^V_\varepsilon$. An exact penalization method replaces the hard clique constraint with a continuous surrogate, such that stable global minima correspond exactly to the maximum common cliques—those cliques found in all graphs in the ensemble. For an appropriate choice of penalty parameter $\beta$, the continuous optimum is integral and yields the largest common clique (Bomze et al., 9 Jan 2026).

4. Algorithms: Semidefinite Relaxation and Projection-Free Optimization

Semidefinite Programming for Adversarial Planted Clique

The core algorithm for $\mathrm{AG}(n, \frac{1}{2}, k)$ involves:

• Solving the Lovász $\theta$-function SDP.
• Thresholding handle-inner-products to get near-planted candidate sets.
• Cleaning false positives via recursive search in the induced subgraph.

The method is robust against adversarial hiding of planted cliques within denser localities and does not rely on degree heuristics or spectral purity. It leverages a uniform upper bound on $\theta(\hat{G})$ and the structure of the solution space to ensure all maximum cliques are found amongst a small candidate set (Feige et al., 2020).

Frank–Wolfe-Type Scheme in Robust Min–Max Setting

For the adversarial uncertainty model, a projection-free Frank–Wolfe algorithm utilizing Clarke and Goldstein generalized subdifferential calculus is constructed. At each step:

1. Active sets $\mathcal{U}^\delta(x, y)$ and local sets $\mathcal{U}_\eta(x, y)$ are computed for subdifferential approximation.
2. The Linear Minimization Oracle (LMO) solves a small LP over these sets to produce search directions.
3. Armijo step-size selection and update is performed.
4. The process repeats until the Frank–Wolfe gap goals are met.

Rigorous analysis shows global sublinear rates: for $k$ iterations, the optimality gap $c_\eta = O(1/\sqrt{k})$, and by decreasing $\eta$, cluster points converge to Clarke-stationary points of the nonsmooth objective (Bomze et al., 9 Jan 2026).

5. Empirical Performance and Benchmarks

Numerical experiments evaluate the robust min–max approach on ensembles of perturbed DIMACS benchmark graphs. A fraction $b$ ("backbone") of edges is fixed across the ensemble, and the rest are perturbed independently. Results indicate:

• For small backbones ($b=0.25$), all large common cliques are small (size $4$–$5$), and the method reliably finds or matches the true maximum.
• With larger backbones ($b=0.75$) and moderate additional noise, the method recovers $90\%$–$95\%$ of the true maximum clique size; in many runs, it matches the maximum exactly.
• Standard deviation across multiple random initializations is low (≤1.5). No direct prior competitor exists for the adversarial clique min-max setting, and the results confirm the effectiveness of the Frank–Wolfe scheme (Bomze et al., 9 Jan 2026).

6. Open Questions and Theoretical Outlook

Several open problems remain central:

• In the planted clique/independent set model, efficient algorithms in the sparse regime ($p = d/n$, $d$ constant) for adversarially planted independent sets with $k \approx c\sqrt{n}$ are unresolved.
• Determining precise thresholds in $p$ for the transition from algorithmic solvability to NP-hardness in adversarially planted independent set models remains open.
• The possibility of establishing stronger hardness results under the Exponential Time Hypothesis (ETH) for semi-random models with moderate sparsity.
• A plausible implication is that connections between robust convex programming, semidefinite relaxations, and robust min-max combinatorial structures warrant further investigation to delineate the boundaries of tractability in adversarial environments (Feige et al., 2020, Bomze et al., 9 Jan 2026).