Computing Optimal Morse Matchings

1. I n t roduc t i on Discrete Morse theory was developed by Forman [9, 11] as a combinatorial analog to the classical smooth Morse theory. Applications to questions in combinatorial topology and related fields are numerous: e.g., Babson et al. [3], Forman [10], Batzies and Welker [4], and Jonsson [20]. I t turns out that the topologically relevant information of a discrete Morse function f on a simplicial complex can be encoded as a (partial) matching in its Hasse diagram (considered as a graph), the Morse matching of f . A matching in the Hasse diagram is Morse if i t satisfies a certain, entirely combinatorial, acyclicity condition. Unmatched k-dimensional faces are called critical; they correspond to the critical points of rank k of a smooth Morse function. The total number of non-critical faces equals twice the number of edges in the Morse matching. The purpose of this paper is to study algorithms which compute maximum Morse matchings of a given finite simplicial complex. This is equivalent to finding a Morse matching with as few critical faces as possible. A Morse matching M can be interpreted as a discrete flow on a simplicial complex . The flow indicates how can be deformed into a more compact description as a CW complex with one cell for each critical face of M . Naturally one is interested in a most compact description, which leads to the combinatorial optimization problem described above. This way optimal (or even sufficiently good) Morse matchings of can help to recognize the topological type of a space given as a finite simplicial complex. The latter problem is known to be undecidable even for highly structured classes of topological spaces, such as smooth 4-manifolds. Optimization of discrete Morse matchings has been studied by Lewiner, Lopes, and Tavares [24]. Hersh [18] investigated heuristic approaches to the maximum Morse matching problem with applications to combinatorics. Morse matchings can also be interpreted as pivoting strategies for homology computations; see [21]. Furthermore, the set of all Morse matchings of a given simplicial complex itself has the structure of a simplicial complex; see [7]. Since its beginnings in Lov ́asz’ proof [25] of the Kneser Conjecture, combinatorial topology seeks to solve combinatorial problems with techniques from (primarily algebraic) topology. And, conversely, such applications to combinatorics frequently shed light on basic concepts in topology. Recent years saw a still growing influx of ideas from differential geometry to the subject. Besides discrete Morse theory this concerns, e.g., various notions of discrete curvature which have been applied to problems in computational geometry and computer graphics. This is paralleled in the Date: 8/23/2004. 2000 Mathematics Subject Classification. Primary 90C27; Secondary 06A07, 52B99, 57Q05, 57R70.