The Parameterized Complexity of Finding Minimum Bounded Chains

Abstract

Finding the smallest d-chain with a specific (d − 1)-boundary in a simplicial complex is known as the Minimum Bounded Chain (MBC d ) problem. The MBC d problem is NP-hard for all d ≥ 2. In this paper, we prove that it is also W[1]-hard for all d ≥ 2, if we parameterize the problem by solution size. We also give an algorithm solving the MBC 1 problem in polynomial time and introduce and implemented two fixed parameter tractable (FPT) algorithms solving the MBC d problem for all d. The first algorithm is a generalized version of Dijkstra's algorithm and is parameterized by solution size and coface degree. The second algorithm is a dynamic programming approach based on treewidth, which has the same runtime as a lower bound we prove under the exponential time hypothesis.

ETH-tight algorithms for finding surfaces in simplicial complexes of bounded treewidth

“…The most commonly used graph is the dual graph of combinatorial d-manifolds [4,9,10,11]. Other graphs that have been used are level d of the Hasse diagram [10,6,5], the adjacency graph of the d-simplices [6], and the 1-skeleton [4]. Our algorithm uses a tree decomposition of the entire Hasse diagram.…”

“…The pair (T, C) is a tree decomposition of H. Moreover, the width of (T, C) is O(k). 5 Proof. We first verify that (T, C) satisfies the definition of being a tree decomposition of H. The first two conditions of a tree decomposition are that all vertices and all edges of H are contained in some bag C t ; indeed, this follows from the fact that each vertex and edge of H are contained in some bag X t , and X t ⊂ C t for each node t ∈ T…”

“…If a and a −1 appear consecutively in some face, we would like to use move (5) to simplify (aa −1 X) to (X); however, we need to take an additional step to retain information on vertex links. Assume that a = {v, w} for some vertices v and w, and that a enters v (i.e.…”

“…We will keep a record of v with interior dummy edges. Before removing a, we first add in dummy edges (add −1 a −1 X) using move (5). We can imagine these edges as connecting v to a dummy vertex v d , where the edge d = (v, v d ).…”

“…v / ∈ B By Lemma 32, there is a single interior dummy edge d that enters v that forms a cyclic sequence of successors. We conclude there must be a face of the form (dd −1 X) in Σ, so we can simplify this face to (X) with move (5).…”