# Worst-case analysis of clique MIPs

## Abstract

The usual integer programming formulation for the maximum clique problem has several undesirable properties, including a weak LP relaxation, a quadratic number of constraints and nonzeros when applied to sparse graphs, and poor guarantees on the number of branch-and-bound nodes needed to solve it. With this as motivation, we propose new mixed integer programs (MIPs) for the clique problem that have more desirable worst-case properties, especially for sparse graphs. The smallest MIP that we propose has just $$O(n+m)$$ nonzeros for graphs with n vertices and m edges. Nevertheless, it ensures a root LP bound of at most $$d+1$$, where d denotes the graph’s degeneracy (a measure of density), and is solved in $$O(2^d n)$$ branch-and-bound nodes. Meanwhile, the strongest MIP that we propose visits fewer nodes, $$O(1.62^d n)$$. Further, when a best-bound node selection strategy is used, $$O(2^g n)$$ nodes are visited, where $$g=(d+1)-\omega$$ is the clique-core gap. Often, g is so small that it can be treated as a constant in which case O(n) nodes are visited. Experiments are conducted to understand their performance in practice.

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## Notes

1. 1.

Actually, $$(n-d)+1$$ subproblems suffice: $$n-d$$ subproblems $$G[V_1]$$, $$G[V_2]$$, $$\dots$$, $$G[V_{n-d}]$$, and a final subproblem $$G[\{v_q, v_{q+1}, \dots , v_n\}]$$ where $$q=n-d+1$$.

2. 2.

Something like Proposition 1 holds for unbounded polyhedra, but we will not need this.

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## Acknowledgements

This material is based upon work supported by the National Science Foundation under Grant Nos. 1662757 and 1942065, and by the Office of Naval Research under Contract No. N00014-20-1-2242. We thank Hamidreza Validi for helpful comments.

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Naderi, M.J., Buchanan, A. & Walteros, J.L. Worst-case analysis of clique MIPs. Math. Program. (2021). https://doi.org/10.1007/s10107-021-01706-2

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### Keywords

• Integer program
• Branch-and-bound
• Fixed-parameter tractability
• Clique
• k-core
• Degeneracy
• Clique-core gap

• 90-10
• 90B10
• 90C06
• 90C10
• 90C27
• 90C35
• 68Q25
• 68Q27
• 68R10