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Worst-case analysis of clique MIPs


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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    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 \(q=n-d+1\).

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


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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).

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

Mathematics Subject Classification

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