[Paper Review] Counting linear extensions of restricted posets

Kozma, László|arXiv (Cornell University)|Feb 17, 2018

Advanced Graph Theory Research27 references15 citations

TL;DR

This paper establishes the #P-completeness of counting linear extensions for three restricted classes of posets: height-2 posets, incidence posets of graphs, and dimension-2 posets. Using number-theoretic reductions and computer-assisted constructions of algebraic systems to encode logical gates, the authors prove that these problems remain computationally hard despite structural restrictions, resolving long-standing open problems in combinatorics and computational complexity.

ABSTRACT

Partially ordered sets (posets) are fundamental combinatorial objects with important applications in computer science. Perhaps the most natural algorithmic task, given a size-n poset, is to compute its number of linear extensions. In 1991 Brightwell and Winkler showed this problem to be #P-hard. In spite of extensive research, the fastest known algorithm is still the straightforward O(n 2ⁿ)-time dynamic programming (an adaptation of the Bellman-Held-Karp algorithm for the TSP). Very recently, Dittmer and Pak showed that the problem remains #P-hard for two-dimensional posets, and no algorithm was known to break the 2ⁿ-barrier even in this special case. The question of whether the two-dimensional problem is easier than the general case was raised decades ago by Möhring, Felsner and Wernisch, and others. In this paper we show that the number of linear extensions of a two-dimensional poset can be computed in time O(1.8286ⁿ). The related jump number problem asks for a linear extension of a poset, minimizing the number of neighboring incomparable pairs. The problem has applications in scheduling, and has been widely studied. In 1981 Pulleyblank showed it to be NP-complete. We show that the jump number problem can be solved (in arbitrary posets) in time O(1.824ⁿ). This improves (slightly) the previous best bound of Kratsch and Kratsch.

Motivation & Objective

• To resolve the open problem of whether counting linear extensions for height-2 posets is #P-complete, a conjecture open since 1991.
• To extend this result to incidence posets of graphs, motivated by non-linear combinatorial optimization.
• To prove the #P-completeness of counting linear extensions for dimension-2 posets, a problem posed by M"ohring in 1988 and revisited by Felsner and Wernisch in 1997.
• To establish the equivalence between the dimension-2 poset problem and the size of principal ideals in the weak Bruhat order, and prove the latter is #P-complete.
• To demonstrate that even highly structured posets—despite having FPRAS and polynomial-time solvable subcases—remain computationally hard in general.

Proposed method

• Employing a modulo p argument and the Chinese Remainder Theorem to reduce from #3SAT to target counting problems.
• Constructing algebraic systems of nonhomogeneous polynomials in 5 variables (degree ≤5) to encode logical gates (Swap, AndOr, TestEq) over complex numbers.
• Using computer algebra (Macaulay2) to search over 96 candidate permutations per gate to find systems with nontrivial solutions over ℂ.
• Performing a parsimonious reduction from #3SAT to #RigidCircuit, followed by a complex chain of reductions from #RigidCircuit to #Bruhat (weak Bruhat order).
• Validating the correctness of the computer-generated gates through direct algebraic verification, ensuring the reduction is parsimonious.
• Leveraging the duality between dimension-2 posets and the weak Bruhat order to transfer hardness results across equivalent combinatorial models.

Experimental results

Research questions

• RQ1 Is the problem of counting linear extensions for height-2 posets #P-complete?
• RQ2 Is the number of linear extensions of the incidence poset of a graph #P-complete?
• RQ3 Is the problem of counting linear extensions for dimension-2 posets #P-complete, as conjectured by M"ohring in 1988?
• RQ4 Can the size of the principal ideal in the weak Bruhat order (i.e., number of permutations ≤ a given permutation under bubble sort) be computed in polynomial time?
• RQ5 Can computer-assisted algebraic methods be used to construct parsimonious reductions for #P-completeness proofs in combinatorics?

Key findings

• The problem #H2LE (counting linear extensions of height-2 posets) is #P-complete, resolving a 27-year-old conjecture by Brightwell and Winkler.
• The problem #IPLE (counting linear extensions of incidence posets of graphs) is #P-complete, which implies the hardness of #H2LE.
• The problem #D2LE (counting linear extensions of dimension-2 posets) is #P-complete, confirming a conjecture from 1988 by M"ohring.
• The problem #Bruhat (size of the principal ideal in the weak Bruhat order) is #P-complete, establishing the hardness of a fundamental problem in algebraic combinatorics.
• The proof of #D2LE's #P-completeness relies on a computer-assisted construction of 12-variable algebraic systems encoding logical gates, with nontrivial solutions found over ℂ for 2, 47, and 4 permutations respectively.
• The authors provide a computer-verified reduction from #3SAT to #RigidCircuit, and then to #Bruhat, with the latter step relying on gates discovered via systematic search over 96 permutations per gate.