Courcelle's theorem for triangulations

Abstract

In graph theory, Courcelle's theorem essentially states that, if an algorithmic problem can be formulated in monadic second-order logic, then it can be solved in linear time for graphs of bounded treewidth. We prove such a metatheorem for a general class of triangulations of arbitrary fixed dimension d, including all triangulated d-manifolds: if an algorithmic problem can be expressed in monadic second-order logic, then it can be solved in linear time for triangulations whose dual graphs have bounded treewidth.We apply our results to 3-manifold topology, a setting with many difficult computational problems but very few parameterised complexity results, and where treewidth has practical relevance as a parameter. Using our metatheorem, we recover and generalise earlier fixed-parameter tractability results on taut angle structures and discrete Morse theory respectively, and prove a new fixed-parameter tractability result for computing the powerful but complex Turaev-Viro invariants on 3-manifolds.

Analysis of Morse Matchings: Parameterized Complexity and Stable Matching

“…In Chapter 4, we designed a polynomial time algorithm to find optimal Morse matching in 2-dmensional simplicial complexes and 3-manifolds whose adjacent graph has bounded tree-width. Very recently Burton and Downey [13] were able to generalize our result for 3-manifolds from Chapter 4 in very encompassing way.…”

“…Furthermore, we show that finding optimal Morse matchings on triangulated 3-manifolds is also fixed-parameter tractable in the treewidth of the dual graph of the triangulation (Theorem 4.8), which is a common parameter when working with triangulated 3-manifolds [15]. Our result for 3-manifold has been generalized in a surprising way very recently [13].…”

The Parameterized Complexity of Finding a 2-Sphere in a Simplicial Complex

“…Our work is one of the few ones combining topology and fixed parameter tractability. In this direction there have been recent results focused on algorithms in 3-manifold topology [2,5,6,7,16]. The problem of finding a shortest 1-dimensional cycle Z 2homologous to a given cycle in a 2-dimensional cycle was shown to be NP-hard by Chao and Freedman [9].…”

Finding minimum bounded and homologous chains in simplicial complexes with bounded-treewidth 1-skeleton

“…Existing algorithms use tree decompositions of a variety of graphs associated with a simplicial complex. The most commonly used graph is the dual graph of combinatorial d-manifolds [4,9,10,11]. Other graphs that have been used are the incidence graph between the (d−1)-and d-simplices [10,5], the adjacency graph of the d-simplices [5], and the 1-skeleton [4].…”