k-Sum Decomposition of Strongly Unimodular Matrices K. Papalamprou and L. Pitsoulis∗ Department of Electrical and Computer Engineering Aristotle University of Thessaloniki, Greece [email protected], [email protected] November 2, 2018 Abstract Networks are frequently studied algebraically through matrices. In this work, we show that networks may be studied in a more abstract level using results from the theory of matroids by establishing connections to networks by decomposition results of matroids. First, we present the implications of the decomposition of regular matroids to networks and related classes of matrices, and secondly we show that strongly unimodular matrices are closed under k-sums for k = 1, 2 implying a decomposition into highly connected network-representing blocks, which are also shown to have a special structure. 1 Introduction It is widely accepted that networks play an important role in many aspects of today’s life. To name a few, social networks play an important role in relationships, job hunting, and marketing, while economic networks usually determine the sustainability and development of various organisations. Moreover, the understanding of complex biological networks may be the key in answering important questions in the areas of medicine and biology. For many other applications as well as for an extended overview of the approaches related to complex networks, the interested reader is referred to [5, 6, 9] Networks are naturally modelled as graphs and results from graph theory have been employed to explore and attack problems in networks (see e.g. [15]). Graphs are known to be represented alge- braically via matrices and it is such a representation that has been extensively used in various problems arXiv:1103.4258v2 [math.CO] 16 Nov 2015 concerning networks. In this work, we examine special classes of matrices that are related to networks and furthermore have important implications in optimisation. Our primary result shows that these ma- trices and, therefore the related networks, are decomposed into highly connected blocks which represent networks with specific properties. To do so we employ results from matroid decomposition theory and, to the best of our knowledge, this is among the few works using such tools to study complex networks. The purpose of this work is twofold; at the one hand we would like to relate complex networks with optimisation problems via well-known classes of matrices and on the other hand to present decomposi- tion results for such classes of matrices and discuss their implication to networks. From our viewpoint, ∗work of this author was conducted at National Research University Higher School of Economics and supported by RSF grant 14-41-00039 1 the main implication is the possibility of finding a way to study complex networks via exploring the properties of the building blocks that arise from specific decompositions. The organization of the paper is as follows. In Section 2, we provide the relevant theory and some preliminary results regarding matrices and matroids in order to make this work more self-contained. In Section 3, we focus on strongly unimodular matrices and show that they are closed under the k-sum operations (k = 1, 2) and, based on that, how these matrices can be decomposed into smaller strongly unimodular matrices. The special structure of these smaller matrices is discussed in Section 4. In the last section, the final decomposition result is provided along with the description of the associated highly connected building blocks. 2 Special Matrices and Matroids 2.1 Network and Unimodular Matrices We assume that the reader is familiar with the basic notions of graph theory as they are presented in [4]. Totally unimodular (TU) matrices form an important class of matrices for integer and linear programming due to the integrality properties of the associated polyhedron. A matrix A is totally unimodular if each square submatrix of A has determinant 0, +1, or −1. The class of TU matrices has been studied extensively and combinatorial characterisations for these matrices can be found in [8, 12]. An important subclass of TU matrices is defined as follows. A matrix A is strongly unimodular (SU) if: (i) A is TU, and (ii) every matrix obtained from A setting a ±1 entry to 0 is also TU. Another well- known characterisation for SU matrices goes as follows: a matrix is strongly unimodular if any of its nonsingular submatrices is triangular, where a triangular matrix is a square matrix whose entries below or above the main diagonal can become zero by permutation of rows or column. Strongly unimodular matrices have appeared several times in the literature [1, 3, 7] since they were first introduced in [2]. Another subclass of TU matrices discussed in this paper is the class of network matrices. A network matrix may be viewed as an edge-path matrix of a directed graph with respect to a particular spanning tree of the graph; results regarding network matrices can be found in [8, 12]. Seymour has shown in [13] that network matrices and their transposes are the main building blocks for TU matrices. Moreover, in [11], it has been shown that the building blocks of TU matrices are matrices associated with bidirected graphs. In this paper we focus on SU matrices which stand between the classes of network and TU matrices and show the network structure of that class. 2.2 Matroid Theory The main reference for matroid theory is the book of Oxley [10]. Definition 1. A matroid M is an ordered pair (E, I) of a finite set E and a collection I of subsets of E satisfying the following three conditions: (I1) ∅ ∈ I (I2) If X ∈ I and Y ⊆ X then Y ∈ I (I3) If U and V are members of I with |U| < |V | then there exists x ∈ V − U such that U ∪ x ∈ I. 2 Given a matroid M = (E, I), the set E is called the ground set of M and the members of I are the independent sets of M. Furthermore, any subset of E not in I is called a dependent set of M while a minimal dependent set is called a circuit of M. Let E be a finite set of vectors from a vector space over a field F and let I be the collection of linearly independent subsets of E; then it can be proved that M = (E, I) is a matroid called vector matroid denoted by M[A] where A is a matrix whose columns are the vectors of the ground set. It can be easily shown that there is one-to-one correspondence between the linearly independent columns of A and the independent sets of M, so the matroid M can be fully characterised by matrix A. Matrix A is called a representation matrix of M and we also say that M is F-representable where F is the field that the elements of matrix A belong. Suppose now that we delete from A all the linearly dependent rows and from the matrix A′ so-obtained we choose a basis B. Clearly, linear F-independence of columns is not affected by such a deletion of rows. By pivoting on non-zero elements of B we can transform A′ to matrix [I B′]. Pivoting does not affect linear F-independence of a matrix and, thus, M = M[I B′]. The matrix B′ is called a compact representation matrix of M. Two matrices are projectively equivalent if one can be obtained from the other by elementary row operations and nonzero column scaling. A matroid M is called uniquely representable over some field F if and only if any two representation matrices of M (over F) are projectively equivalen. A matroid representable over every field is regular. Furthermore, there is a clear connection between regular matroids and TU matrices. Specifically, any TU matrix is the representation matrix of some regular matroid and any regular matroid has a TU representation matrix (in R). Let G be an ordinary graph and let I be the collection of edge sets inducing a acyclic subgraph of G. Then it can be shown that the pair (E(G), I) is a matroid called the graphic matroid of G and is denoted by M(G). If A is the incidence matrix of an orientation of G (i.e. the directed graph obtained from G by assigning a direction to each edge) then it can be shown that M(G) is isomorphic to M[A] and we write M(G) =∼ M[A]. Thus, for any network matrix N with respect to some spanning tree of G we have that M(G) =∼ M(N), since the way we obtain N from A is also the way we can obtain from A a compact representation matrix of M[A]. The ordered pair (E, {E − S : S∈ / I}) is a matroid called the dual matroid of M and is denoted by M ∗. It is clear that (M ∗)∗ = M. The prefix ’co’ is used to dualize a term; therefore, a matroid is called cographic if it is the dual of a graphic matroid. We should note that not all matroids are closed under duality; for example regular matroids are closed while graphic matroids are not. Any matroid which can be obtained from M by a series of operations called deletions and contractions is called a minor of M (see e.g. Section 3.1 in [10]). The rank of a matroid M, denoted by r(M), equals the cardinality of the maximal independent set of M. For some positive integer k, a partition (X, Y ) of E(M) is called a k-separation of M if the following two conditions are satisfied: (i) min{|X|, |Y |} ≥ k, and (ii) rM (X)+ rM (Y ) − r(M) ≤ k − 1.