Chamber Complex

When an integer program \(\{Ax = b,\, x \geq 0,\, x \in \mathbb{Z}^n\}\) depends on a parameter vector \(b \in \mathbb{R}^d\), the solution set and its generating function change as \(b\) varies. However, they remain combinatorially equivalent within regions of parameter space called chambers: open polyhedral cones separated by hyperplanes where the combinatorial type changes.

The collection of these chambers and their bounding hyperplanes is the chamber complex of the problem.

Why it matters

• Parametric counting. Algorithms such as Barvinok’s (1994) and Polyhedral Omega produce a piecewise rational generating function — one rational formula per chamber. Computing the chamber complex is a prerequisite for producing this output in closed form.
• Sensitivity analysis. Each chamber boundary is a hyperplane where the optimum jumps discontinuously or a new set of constraints becomes active. Knowing the complex lets you answer sensitivity questions exactly.
• Ehrhart theory. The Ehrhart quasi-polynomial of a polytope \(tP \cap \mathbb{Z}^n\) (as a function of integer \(t\)) has a chamber structure driven by the denominators of Barvinok’s rational functions.

Key objects

Hyperplane arrangement \(\mathcal{H}\): the set of hyperplanes \(H_i = \{\,b \in \mathbb{R}^d \mid \langle a_i, b \rangle = 0\,\}\) induced by the rows of \(A\) and the sign conditions that define feasibility transitions.

Chambers: connected components of \(\mathbb{R}^d \setminus \bigcup_i H_i\).

Faces: lower-dimensional cells where several hyperplanes meet; a chamber complex is a polyhedral fan.

Computational challenges

• The number of chambers can be exponential in \(d\) (up to $$\binom{

  \mathcal{H}

  }{d}$$).

• Incremental algorithms add one hyperplane at a time, updating the current fan.
• Signed hyperplane traversal and reverse search (Avis–Fukuda style) enumerate all chambers without storing them all simultaneously.

Goal

Study the definition and properties of chamber complexes, implement a basic algorithm for low-dimensional parameter spaces, and apply it to small parametric knapsack instances using the Polyhedral Omega package.

Deliverables

• Julia implementation of chamber complex enumeration for low-dimensional parameter spaces
• Application to small parametric knapsack instances via Polyhedral Omega
• Short report (6–8 pages)

References

1. A. I. Barvinok. A Polynomial Time Algorithm for Counting Integral Points in Polyhedra When the Dimension is Fixed. Mathematics of Operations Research, 19(4):769–779, 1994. DOI 10.1287/moor.19.4.769

2. S. Verdoolaege, R. Seghir, K. Beyls, V. Loechner, and M. Bruynooghe. Counting Integer Points in Parametric Polytopes Using Barvinok’s Rational Functions. Algorithmica, 48(1):37–66, 2007. DOI 10.1007/s00453-006-1231-0

3. M. Köppe and S. Verdoolaege. Computing Parametric Rational Generating Functions with a Primal Barvinok Algorithm. Electronic Journal of Combinatorics, 15(1), 2008. combinatorics.org

4. F. Breuer and Z. Zafeirakopoulos. Polyhedral Omega: a New Algorithm for Solving Linear Diophantine Systems. Annals of Combinatorics, 21(2):211–280, 2017. DOI 10.1007/s00026-017-0349-x

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