The matching polytope has exponential extension complexity (1311.2369v4)

Abstract: A popular method in combinatorial optimization is to express polytopes P, which may potentially have exponentially many facets, as solutions of linear programs that use few extra variables to reduce the number of constraints down to a polynomial. After two decades of standstill, recent years have brought amazing progress in showing lower bounds for the so called extension complexity, which for a polytope P denotes the smallest number of inequalities necessary to describe a higher dimensional polytope Q that can be linearly projected on P. However, the central question in this field remained wide open: can the perfect matching polytope be written as an LP with polynomially many constraints? We answer this question negatively. In fact, the extension complexity of the perfect matching polytope in a complete n-node graph is 2^Omega(n). By a known reduction this also improves the lower bound on the extension complexity for the TSP polytope from 2^Omega(n^1/2) to 2^Omega(n).

• The paper establishes an exponential lower bound (2^Ω(n)) for the extension complexity of the perfect matching polytope.
• It refutes a polynomial-sized LP formulation for the matching problem using advanced hyperplane separation techniques.
• The work tightens the extension complexity bounds for the TSP polytope, impacting combinatorial optimization research.

Analysis of the Extension Complexity of the Matching Polytope

The paper "The matching polytope has exponential extension complexity," authored by Thomas Rothvoß, presents a notable advancement in the understanding of the extension complexity of polytopes, particularly the perfect matching polytope. The paper provides a rigorous mathematical argument to prove that the perfect matching polytope, denoted as $P_{PM}$, in a complete $n$-node graph, has an extension complexity of $2^{\Omega(n)}$. This work forms a critical contribution to the domain of combinatorial optimization and polyhedral theory, addressing a long-standing question about whether the perfect matching polytope could be described with a polynomial-sized linear program (LP).

Main Contributions

1. Exponential Lower Bound: The core contribution of this paper is establishing that $P_{PM}$ has an exponential extension complexity $2^{\Omega(n)}$. This result conclusively shows that there is no polynomial-sized LP formulation for the perfect matching problem, refuting any speculations or attempts positing such possibilities in prior literature.
2. Implications for TSP: The paper goes further to improve the bounds on the extension complexity for another important combinatorial problem, the Traveling Salesman Problem (TSP) polytope, from $2^{\Omega(\sqrt{n})}$ to $2^{\Omega(n)}$. This is achieved through a reduction from the matching polytope, leveraging the established hardness result.
3. Innovative Techniques: The analysis employs an advanced technique known as the hyperplane separation bound. This method contrasts potential rectangle coverings of the slack matrix with the actual coverage needed to satisfy slack conditions, revealing an inherent inefficiency under any polynomial-sized formulation.
4. Effect on Related Problems: Beyond specific polytopes, the results bear broader implications for the complexity involved in expressing efficient LP formulations for polynomial-time solvable problems. The methodology and findings provide a basis for assessing the complexity across a broader class of combinatorial polytopes.

Theoretical Implications

The findings underscore the limitations of LP formulations in solving certain combinatorial structures efficiently. Rothvoß’s result elucidates the inherent complexity in describing combinatorial polytopes compactly, extending insights into the geometry of high-dimensional polytopes such as the perfect matching polytope. This work also cements the role of extension complexity as a critical metric in evaluating the computational boundaries of polyhedral combinatorics.

Future Directions

The paper hints at potential research avenues, particularly in exploring the extension complexity of other fundamental combinatorial polytopes and their approximations. Furthermore, given the difficulty of exponential bounds, there remains a compelling challenge to find compact, perhaps non-linear, representations that either approximate or capture essential characteristics of these polytopes.

In summary, Rothvoß’s paper delivers a crucial lower bound result with significant implications for both theoretical exploration and practical algorithm design in optimization problems. It challenges researchers to continue pushing the boundaries of known techniques or explore novel paradigms, such as semidefinite programming, to tackle related complexity challenges effectively.