Edge Interdiction Clique Problem

• Edge Interdiction Clique Problem is a graph optimization paradigm that minimizes the maximum clique size by strategically removing up to k edges.
• It employs bilevel optimization and MILP reformulations like EBCP-MILP to manage the combinatorial complexity inherent in clique reduction.
• Its Σ2^p-completeness underlines the problem's intractability, while advanced cutting-plane and branch-and-cut methods deliver significant empirical performance benefits.

The Edge Interdiction Clique Problem (EICP) is a graph-theoretic optimization paradigm focused on minimizing the maximum clique size of a graph by the strategic removal of up to $k$ edges. This interdiction model finds applications in domains such as protein function analysis and image matching, where the structural integrity of large cliques is critical. EICP is characterized by a min–max structure and involves both combinatorial complexity and challenging bilevel optimization, situating it among the most intractable classes of discrete optimization problems (Zhou et al., 5 Jan 2026, Martin, 2024, Grüne et al., 7 Feb 2025).

1. Formal Definition and Problem Structure

Given a simple undirected graph $G=(V,E)$ and an interdiction budget $k\in \mathbb{Z}_{\geq 0}$, the objective is to select an edge subset $F \subseteq E$ with $|F| \leq k$ such that removal of $F$ minimizes the clique number, $\omega(G')$, of the residual graph $G' = (V, E \setminus F)$. The formal optimization is:

$$\eta(G, k) = \min_{F \subseteq E : |F| \leq k}\ \omega((V, E \setminus F))$$

Equivalently, the decision version asks whether there exists $F \subseteq E$ with $|F| \leq k$ so that $\omega((V, E \setminus F)) \leq \alpha^{*}$ for given $\alpha^{*}$ (Zhou et al., 5 Jan 2026, Martin, 2024, Grüne et al., 7 Feb 2025).

2. Complexity Classification

EICP is fundamentally harder than the classical Maximum Clique problem. Whereas Maximum Clique is NP-complete, EICP resides at the second level of the polynomial hierarchy, formally classified as $\Sigma_2^p$-complete under polynomial-time many-one reductions (Grüne et al., 7 Feb 2025).

• In bilevel form: the leader selects a subset $F$ of up to $k$ edges to delete, the follower then solves a maximal clique problem in the resulting graph.
• $\Sigma_2^p$: EICP requires existential quantification over interdiction sets, followed by universal quantification over cliques, which cannot be collapsed into a single polynomial-time witness unless NP = $\Sigma_2^p$ (Zhou et al., 5 Jan 2026, Grüne et al., 7 Feb 2025).

This complexity implies the absence of compact MIP formulations and the practical infeasibility of standard certificate-based approaches found in NP problems.

3. Modeling and Mixed-Integer Programming Formulations

Given the problem’s bilevel nature and combinatorial explosion, direct solution via mixed-integer bilevel programming (MIBLP) is computationally prohibitive. Modern strategies recast EICP using Edge Blocker Clique Problems (EBCP) as single-level MILPs, which are then solved in a sequence (Zhou et al., 5 Jan 2026).

• EBCP: For fixed $p$, determine the minimum edges to remove so the residual clique number $\leq p$; denote the optimum by $\gamma(G, p)$.
• MILP Formulation: For candidate cliques $C$ of size $|C|\geq p+1$, constraints $$\sum_{e \in E(C)} x_e \geq \gamma_{\text{clq}}(|C|,p)$$ enforce deletion of enough edges to guarantee the required reduction.
• Maximal Clique Formulation: Tightening is achieved by focusing on maximal cliques and introducing permutation-based strengthening cuts.

The table below summarizes the key MILP features:

MILP Type	Constraints Focus	Strengthening Mechanism
EBCP-MILP	All cliques $\geq p+1$	Set-cover constraints
EBCP-MILP-2	Maximal cliques	Permutation-based $z$-cuts

EICP is then solved by sequentially searching for the smallest $p$ so that $\gamma(G,p)\leq k$.

4. Exact Algorithms and Computational Techniques

The RLCM algorithm represents the current state-of-the-art exact solver, integrating graph reduction, lower-bound estimation, and a branch-and-cut MILP engine (Zhou et al., 5 Jan 2026).

• Stage I—Preprocessing: Lower bound via disjoint clique packing, graph reductions based on clique local structures, and fast upper bounds via degree/coloring heuristics.
• Stage II—Branch-and-Cut MILP: For each candidate $p$, MILP is constructed using pool of maximal cliques and permutation cuts, iteratively strengthening the model by adding violated covering inequalities (from newly discovered cliques) and permutation-based cuts.

Empirical benchmarks:

• RLCM demonstrates superior optimal solution rates and speed compared to previous single-level MILP and general bilevel solvers, notably outperforming EDGE-INTER and BILEVEL (Zhou et al., 5 Jan 2026).
• Reduction rules provide the most significant performance gains, particularly in sparse and large-scale graphs.

5. Polyhedral Analysis and Cutting Planes

Polyhedral investigations highlight the underlying structure of valid inequalities and facets in the feasible polytope. For EICP and its vertex version (Martin, 2024):

• Clique-interdiction cuts for each maximal clique $K$: $\theta + \sum_{u \in K} w_u \geq |K|$ (facet-defining under specific conditions).
• Lifting techniques adjust coefficients for redundancy within clique covers.

Cutting-plane separation demands exact or high-quality maximal clique detections in the current residual graph, typically deployed via efficient bit-parallel clique solvers.

6. Computational Benchmarks and Experimental Findings

EICP solvers are evaluated on DIMACS2 benchmarks, real-world sparse networks, and random Erdős–Rényi graphs (Zhou et al., 5 Jan 2026, Martin, 2024).

• RLCM solves up to $87.5\%$ of DIMACS2 instances for $k=10$ versus $68.75\%$ for EDGE-INTER.
• RLCM resolves $664$ out of $695$ solvable instances faster than BILEVEL, with significant robustness in memory-intensive cases.
• Ablation: Graph reduction yields the largest empirical speedups, while permutation-based $z$-inequalities are decisive in dense regime.

7. Theoretical Implications, Parameterized and Approximation Complexity

$\Sigma_2^p$-completeness of EICP yields several profound consequences:

• No polynomial-time algorithms (unless major complexity collapses occur), and even parameterized approaches (by $k$) face W[2]-hardness (Grüne et al., 7 Feb 2025).
• Constant-factor approximation algorithms are unlikely for general instances.
• Polyhedral studies and cutting-plane enhancements remain essential for any practical exact or heuristic advancement.
• Open directions involve exploring approximation bounds for special graph classes, dynamic/stochastic interdiction settings, and bi-criteria formulations balancing budget vs. residual clique size (Martin, 2024).