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On the Computational Complexity of Betti Numbers: Reductions from Matrix Rank∗

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On the Computational Complexity of Betti Numbers: Reductions from Matrix Rank∗ On the Computational Complexity of Betti Numbers: Reductions from Matrix Rank∗ H. Edelsbrunnery and S. Parsaz Abstract bers in the case of field coefficients can be computed We give evidence for the difficulty of computing Betti by finding the ranks of these matrices. However, in numbers of simplicial complexes over a finite field. some special cases, Betti numbers can be computed We do this by reducing the rank computation for by other means and more efficiently. For instance, if sparse matrices with m non-zero entries to computing the complex triangulates a 2-manifold, then the Betti Betti numbers of simplicial complexes consisting of at numbers can be computed using the Euler relation. most a constant times m simplices. Together with the Another example is when the simplicial complex is known reduction in the other direction, this implies embedded in 3-dimensional Euclidean space and the that the two problems have the same computational complement space is also triangulated; see [4]. In complexity. this situation, the Betti numbers can be computed in a time that is linear in the size of the complex. One of 1 Introduction the motivations of this paper is to understand what can be said if the simplicial complex is embedded in The efficient computation of topological properties the 4-dimensional Euclidean space. of a space is one of the main goals of computational We use the term computational complexity to topology [8,18]. Homology groups are important such refer to the run-time of an optimal algorithm solving properties that encode the connectivity of a space. a problem. It is commonly described as a function of For example, the zeroth homology group represents the size of the problem instance. We reduce problems the connected components, and if the space is em- to each other using worst-case linear-time reductions bedded in 3-dimensional Euclidean space, then the on the common RAM model. However, our results second homology group represents the voids. The are valid when the underlying model of computation computation of homology groups, and in particular allows these reductions to be done in linear time. For of their ranks { the Betti numbers { is of practical example, this is the case for randomized algorithms importance. In most applications, homology groups with expected run-time as computational complexity. are computed using field coefficients, in which case Here is an overview of our results, which concern the groups are vector spaces and the Betti numbers themselves with computing the Betti numbers of are their dimensions. Computing these numbers and finite 2-dimensional simplicial complexes, that is: their more advanced persistent versions is central in the ranks of the homology groups defined for - applications of computational topology [9]. Conse- F2 coefficients. Recall that the Euler number of a quently, hardness results and evidence for the diffi- complex is the alternating sum of simplex counts culty of computing Betti numbers imply similar re- as well as the alternating sum of Betti numbers: sults for all of the applications. χ = β − β + β . For a 2-dimensional complex with For a general finite simplicial complex, the ho- 0 1 2 m simplices, we can compute χ as well as β in O(m) mology groups can be computed by reducing the 0 time. The complexity of computing the first Betti boundary or incidence matrices using standard row- number is therefore equal to that of the second Betti and column-operations. Similarly, the Betti num- number. We have two main results: ∗This research is partially supported by the FP7 Toposys I. The complexity of computing the Betti numbers project, by ESF under the ACAT Research Network Pro- of a 2-dimensional simplicial complex with m gramme, and by a mega grant of the Russian Government. simplices is the same as that of computing the yIST Austria (Institute of Science and Technology Austria), Klosterneuburg, Austria, email: [email protected]. rank of an m-by-m 0-1 matrix with m 1s. zDuke Univ., Durham, North Carolina, and IST Austria, Klosterneuburg, Austria, email: [email protected] II. The complexity of computing the Betti numbers 152 Copyright © 2014. by the Society for Industrial and Applied Mathematics. of a 2-dimensional simplicial complex with n This means that the sum is the chain that consists vertices is at least that of computing the rank of all i-simplices in the symmetric difference of the of an n-by-n 0-1 matrix. two chains: c + d = (c [ d) − (c \ d). With this notion of addition, the set of i-chains forms a group, Our first result says that computing the Betti num- denoted as C = C (K). More specifically, C is the bers of a 2-dimensional complex has the same com- i i i n -dimensional vector space over with basis S . plexity as computing the rank of a sparse matrix. To i F2 i The boundary of an i-simplex, denoted as prove this claim, we build a simplicial complex for a @ (∆ ), is the collection of its faces of dimension given matrix such that the second Betti number of i i;` i − 1. It is a chain in C . Since S is a basis for the complex is the nullity of the matrix. The sim- i−1 i C , this definition can be extended to a unique ho- plicial complex in this construction embeds in the 4- i momorphism between vector spaces, @ : C ! C dimensional Euclidean space, which thus provides an i i i−1 defined by answer to the question stated above. Our second re- sult says that if we ignore the number of simplices and X @ (c) = @ (∆ ); express run-time in terms of the number of vertices i i i;` ∆ 2c of the complex, then we cannot do better than for i;` computing the rank of an n-by-n matrix. We remark called the i-th boundary homomorphism. By defini- that we can harvest a sparsification of 0-1 matrices tion, the zeroth boundary homomorphism, @0, is the as a side-product of our reductions; see Section 4. zero map. A chain with empty boundary is called a The paper is organized as follows. Section 2 in- cycle. Hence, the i-cycles are the chains in the kernel troduces background material and definitions, includ- of @i, and we write Zi = Zi(K) ⊆ Ci for this kernel. ing a short review of the current best bounds for com- For example, if K is a triangulation of a 2-manifold puting the rank of a matrix. Section 3 presents the (without boundary), then the chain c that includes reductions proving our two main results. Section 4 all triangles in K is a 2-cycle. Indeed, every edge extends our results from F2 to other finite fields, and of K belongs to the boundary of exactly two trian- it relates them to matrix sparsification. gles, which implies that the sum of the boundaries of all triangles is empty. A chain that is the boundary 2 Background of another chain { necessarily of one higher dimen- In this section, we recall basic definitions and facts sion { is called a boundary. Hence, the i-boundaries about simplicial and singular homology for general are the chains in the image of @i+1, and we write d-dimensional simplicial complexes. We also define Bi = Bi(K) ⊆ Ci for this image. Note that the vertical and horizontal homology classes. Unless Zi and Bi are also vector spaces over F2, and that explicitly stated, we will limit ourselves to coefficients Bi ⊆ Zi because the boundary of a boundary is nec- in F2, the field of integer arithmetic modulo 2. All essarily empty. We can therefore form the quotient, homology groups will be finite rank vector spaces over Hi = Hi(K) = Zi=Bi, called the i-th homology group F2. They are determined up to isomorphism by their of K. This quotient is again a vector space over F2, ranks. and its dimension is called the i-th Betti number of K, denoted as βi = βi(K). Since Hi is a quotient, Simplicial homology. Let K be a simplicial com- its elements are not cycles but rather classes of ho- plex, and write ni for its number of i-simplices. The mologous cycles. If such a class contains a cycle c, P size of K is the total number of simplices: i ni. then we denote the class by [c], and we call c a repre- Assuming an ordering, we write ∆i;` for the `-th i- sentative of the class. Note that the sum of any two simplex, and we let representative cycles of the same class is a boundary. Si(K) = f∆i;` j ` = 1; 2; : : : ; nig Computing simplicial homology. We already have a basis for the vector space Ci, namely Si or, be the set of i-simplices in K. Any subset of Si = equivalently, the vectors of length ni with only a sin- Si(K) is called an i-chain of K. It can be written as gle 1 each. We need to find bases for Zi and Bi. an ni-vector of 0s and 1s: c = (c(1); c(2); : : : ; c(ni)), Let Di be the matrix of the i-th boundary homomor- where c(`) = 1 iff the simplex ∆i;` belongs to c, for phism, @i, with respect to the bases Si and Si−1. The 1 ≤ ` ≤ ni. Two i-chains, c and d, can be added by vector addition modulo 2: rows of Di are indexed by (i − 1)-simplices and the columns by i-simplices. The `-th column of Di cor- (c + d)(`) = c(`) + d(`) (mod 2): responds to ∆i;` and is the vector @i(∆i;`).
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