Combinatorial Optimization and Extension Complexity of Polytopes

Combinatorial optimisation uses discrete structures to model decision‐making problems such as scheduling, network design and facility location. Many such problems can be expressed as optimisation over a polytope, the convex hull of feasible solutions. Direct linear descriptions often require an exponential number of inequalities; extended formulations overcome this by representing the original polytope as a linear projection of a higher‐dimensional one with fewer facets. The extension complexity of a polytope is the minimal size of such a representation, measured by the number of facets in the extended polytope. Fundamental results relate extension complexity to the nonnegative rank of the polytope’s slack matrix, yielding both upper and lower bounds. Seminal work has demonstrated exponential lower bounds on the extension complexity of the travelling‐salesman and matching polytopes, while other classes admit polynomial‐size formulations. Hierarchies of relaxations, including Sherali–Adams, Lasserre and theta‐body constructions, offer systematic refinements that trade off size and approximation quality. These advances have broad significance for designing compact linear and semidefinite programmes with practical applications in logistics, cryptography and machine learning.

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Recent efforts have refined the computation and application of relaxation complexity using efficient mixed‐integer programming techniques. These methods employ row and column generation, symmetry handling and propagation algorithms to compute robust variants of relaxation complexity for sets of integer points, enabling the solution of instances previously out of reach. Another line of work has explored the role of irrational coordinates in linear relaxations, showing that for the standard simplex the minimal number of facets in an extended formulation can be reduced below the rational bound and that asymptotically the ratio of irrational to rational relaxation complexities tends to zero. There has also been progress on explicit lifts of geometric structures: for Voronoi cells of certain lattices, worst‐case constructions exhibit exponentially large extension complexity, whereas root lattices and their duals admit near‐linear‐size semidefinite or polyhedral lifts. These studies deepen understanding of structural limits and guide the design of compact formulations for discrete and geometric optimisation problems.

Combinatorial Optimization and Extension Complexity of Polytopes publication trend

The graph below shows the total number of articles in combinatorial optimization and extension complexity of polytopes across all publications each year (not limited to Nature Index journals).

Technical terms

Polytope: A convex set defined as the solution set of finitely many linear inequalities or as the convex hull of points.

Extended formulation: A representation of a polytope as the linear projection of a higher-dimensional polytope.

Extension complexity: The minimum number of facets in any extended formulation of a given polytope.

Slack matrix: A matrix recording the ‘slack’ or residuals of each vertex with respect to each defining inequality.

Nonnegative rank: The smallest number of nonnegative factors needed to factorise a matrix, related to extension complexity.

Relaxation complexity: The minimal number of facets among all polyhedral relaxations of a set of integer points without auxiliary variables.

Lift: A higher-dimensional polytope whose projection yields the original polytope, often leading to smaller or simpler descriptions.

References

1. Efficient MIP techniques for computing the relaxation complexity. Mathematical Programming Computation (2023).
2. The role of rationality in integer-programming relaxations. Mathematical Programming (2023).
3. Lifts for Voronoi Cells of Lattices. Discrete & Computational Geometry (2023).