[Paper Review] Some Algorithmic Problems in Polytope Theory

Volker Kaibel, Marc E. Pfetsch|ArXiv.org|Feb 20, 2002

Computational Geometry and Mesh Generation4 citations

TL;DR

This paper compiles and analyzes 35 algorithmic problems in polytope theory, focusing on computational complexity across five categories: coordinate descriptions, combinatorial structure, isomorphism, optimization, and realizability. It establishes that counting the number of $(n-1)$-dimensional faces of a simplicial complex derived from a Boolean formula is $#\mathcal{P}$-complete, demonstrating the inherent difficulty of face enumeration in polytopal and simplicial complexes.

ABSTRACT

This is a survey on algorithmic questions about combinatorial and geometric properties of convex polytopes. We give a list of 35 problems; for each the current state of knowledege on its theoretical complexity status is reported. The problems are grouped into the sections ``Coordinate Descriptions'', ``Combinatorial Structure'', ``Isomorphism'', ``Optimization'', ``Realizability'', and ``Beyond Polytopes''.

Motivation & Objective

• To systematize and classify fundamental algorithmic problems in polytope theory based on computational complexity.
• To determine the complexity status—especially NP-hardness or #P-completeness—of key problems in polyhedral and simplicial combinatorics.
• To distinguish between the complexity of problems when the dimension is part of the input versus fixed.
• To highlight open problems and provide references for further research, particularly in the context of computational geometry and combinatorics.

Proposed method

• Problem descriptions are structured with input, output, complexity status (general vs. fixed dimension), and comments with references.
• The authors use polynomial-time Turing reductions to analyze complexity, distinguishing between polynomial-time and polynomial-total-time algorithms.
• For #P-completeness, a reduction from the Boolean satisfiability problem (SAT) is constructed via simplicial complexes defined by minimal non-faces.
• The dual complex construction is used to relate face counts between a complex and its complement, enabling efficient transformation of face enumeration problems.
• Theoretical complexity is analyzed using foundational results from computational complexity (Garey and Johnson) and polyhedral theory (Ziegler, Schrijver).
• Homology and shellability problems are studied via boundary matrices and Smith normal forms, with complexity assessed based on input representation (facets vs. all simplices).

Experimental results

Research questions

• RQ1 What is the computational complexity of counting the number of $(n-1)$-dimensional faces in a simplicial complex derived from a CNF formula?
• RQ2 Can shellability of a pure simplicial complex be decided in polynomial time, especially in fixed dimension?
• RQ3 Is partitionability of a simplicial complex decidable in polynomial time, and how does this depend on input representation?
• RQ4 What is the complexity of computing the $i$-th homology group of a simplicial complex given by its facets?
• RQ5 How does the complexity of polyhedral problems change when the dimension is fixed versus part of the input?

Key findings

• Counting the number of $(n-1)$-faces in a simplicial complex constructed from a CNF formula is $#\mathcal{P}$-complete, as shown via a reduction from #SAT.
• The dimension of a simplicial complex given by its minimal non-faces is $#\mathcal{P}$-hard to compute, even though the complex is defined by a small number of circuits.
• Homology computation is polynomial-time solvable when the complex is given by all simplices, due to efficient computation of the Smith normal form of boundary matrices.
• Shellability and partitionability remain open problems in general, even for two-dimensional complexes, though both are in NP.
• For fixed dimension, homology computation and certain face enumeration problems can be solved in polynomial time, but the general case remains unresolved.
• The dual complex construction allows efficient transformation between face counts of a complex and its complement, enabling reduction of one problem to another with known complexity.