[Paper Review] Compressed polytopes and statistical disclosure limitation

Seth Sullivant|ArXiv.org|Dec 30, 2004

Complexity and Algorithms in Graphs10 references4 citations

TL;DR

This paper characterizes compressed lattice polytopes via their facet-defining inequalities, proving they are affinely isomorphic to 0/1 polytopes and providing a condition for compressed marginal and cut polytopes. The key contribution is a necessary and sufficient condition for compressed polytopes based on lattice point distribution across facet hyperplanes, enabling polynomial-time solutions to integer programming problems in statistical disclosure limitation when the condition holds.

ABSTRACT

We provide a characterization of the compressed lattice polytopes in terms of their facet defining inequalities and we show that every compressed lattice polytope is affinely isomorphic to a 0/1-polytope. As an application, we characterize those graphs whose cut polytopes are compressed and discuss consequences for studying linear programming relaxations in statistical disclosure limitation.

Motivation & Objective

• To characterize compressed lattice polytopes using their facet-defining inequalities.
• To establish that every compressed polytope is affinely isomorphic to a 0/1 polytope.
• To apply the characterization to marginal and cut polytopes in statistical disclosure limitation.
• To determine when linear programming relaxations yield sharp integer bounds for cell entry maximization problems.
• To identify new families of marginals where integer programming gaps are minimized or zero.

Proposed method

• Use of pulling triangulations to define and analyze unimodularity in lattice polytopes.
• Derivation of a necessary and sufficient condition for compressed polytopes based on lattice points lying in at most one translate of each facet-defining hyperplane.
• Application of the main theorem to marginal polytopes of decomposable hierarchical models.
• Use of the covariance mapping to relate marginal polytopes to cut polytopes in the case of binary variables.
• Proof that compressed cut polytopes correspond to graphs without $K_4$ minors and induced cycles of length ≤4.
• Leveraging known results on containment in $K_4$-free graphs to show polynomial-time solvability of the linear program.

Experimental results

Research questions

• RQ1 What characterizes a compressed lattice polytope in terms of its facet-defining inequalities?
• RQ2 When does a linear programming relaxation yield sharp integer bounds for maximizing cell entries under marginal constraints?
• RQ3 Which graphs yield compressed cut polytopes, and how does this relate to statistical disclosure limitation?
• RQ4 How can the structure of compressed polytopes be used to identify families of marginals with minimal integer programming gaps?
• RQ5 What is the relationship between the lattice point distribution across facet hyperplanes and the unimodularity of pulling triangulations?

Key findings

• A lattice polytope is compressed if and only if, for each facet-defining inequality, the lattice points in the polytope lie in at most one translate of the hyperplane defined by the inequality.
• Every compressed lattice polytope is affinely isomorphic to a 0/1 polytope, specifically to the intersection of a unit hypercube with an affine subspace.
• The marginal polytope $P_{ riangle}$ is compressed if and only if the simplicial complex $ riangle$ is decomposable or satisfies specific degree and cycle constraints, such as at most two indices with $d_i > 2$.
• For graphs with $d = (2,2,...,2)$, the marginal polytope $P_{ riangle}$ is compressed if and only if $ riangle$ has no $K_4$ minors and all induced cycles have length at most 4.
• When $P_{ riangle}$ is compressed, the integer programming maximum value for cell entries can be computed in polynomial time in $n$ and the bit complexity of the marginals.
• The failure of the lattice point condition in the main theorem corresponds to the presence of non-zero integer programming gaps, suggesting where to search for large gaps in statistical disclosure limitation problems.