Computing the sequence of k-cardinality assignments Amnon Rosenmann Institute of Discrete Mathematics Graz University of Technology, Graz, Austria [email protected] Abstract The k-cardinality assignment problem asks for finding a maximal (min- imal) weight of a matching of cardinality k in a weighted bipartite graph Kn,n, k ≤ n. The algorithm of Gassner and Klinz from 2010 for the para- metric assignment problem computes in time O(n3) the set of k-cardinality assignments for those integers k ≤ n which refer to ”essential” terms of a corresponding maxpolynomial. We show here that one can extend this algorithm and compute in a second stage the other ”semi-essential” terms in time O(n2), which results in a time complexity of O(n3) for the whole sequence of k = 1,...,n-cardinality assignments. The more there are as- signments left to be computed at the second stage the faster the two-stage algorithm runs. In general, however, there is no benefit for this two-stage algorithm on the existing algorithms, e.g. the simpler network flow algo- arXiv:2104.04037v1 [math.OC] 8 Apr 2021 rithm based on the successive shortest path algorithm which also computes all the k-cardinality assignments in time O(n3). Keywords: k-cardinality assignment problem; parametric assignment algorithm; max-plus algebra; full characteristic maxpolynomial 1 1 Introduction The linear assignment problem is a basic problem in combinatorial optimization. The time complexity for solving the k-cardinality assignment problem in a full 3 bipartite graph Kn,n with weights on its edges is, for general k, O(n ), for example through the Hungarian method. Gassner and Klinz [GK10] showed that one can compute the parametric assign- ment problem in time O(n3). It follows that the list of k-cardinality assignments that correspond to essential terms of the full characteristic maxpolynomial of the 3 matrix representing the weights on Kn,n can be computed in O(n ). We show here that one can, in fact, compute in time O(n3) all the k-cardinality assignments and not just the essential ones. The result is based on the full characteristic maxpoly- nomial having the property of being in full canonical form [RLP19], for which we give here a simple proof. We note, however, that this time complexity does not improve on the existing algorithms. The simpler network flow algorithm (see [AMO93]), based on succes- sive shortest paths, which augments one unit flow at each iteration, runs also in time complexity O(n3). In special cases where there are many assignments left to compute after the completion of the Gassner and Klinz algorithm then the fact that these left as- signments can be computed in time O(n2) may be of benefit. 2 The k-linear assignment problem We are given a complete bipartite graph (biclique) Kn,n = G(U, V ; E) consisting of two disjoint sets of vertices U = {u1,...,un} and V = {v1,...,vn}, and an edge set E, adjoining each vertex ui ∈ U with each vertex vj ∈ V by an edge eij. In addition, we are given a weight (”cost”) function w : E → IRmax = IR ∪ {−∞} which assigns each edge eij a weight wij. The k-cardinality Linear Assignment Problem, or k-LAP (see e.g. [DM97], [DLM01], [Vol04], [BJ16]), asks for finding a maximal (optimal) weight of a matching (independent edge set) of cardinality k, k ≤ n, that is, a set Mk ⊆ E of k pairwise non-adjacent edges in Kn,n which is of maximal total weight: k max Σl=1wiljl , (2.1) {i1,...,ik},{j1,...,jk}⊆[1..n] where the indices il as well as the indices jl are pairwise distinct. 2 Remarks 2.1. 1. An analogous and similarly solved problem asks for comput- ing the minimal assignment, and then one works over IRmin = IR ∪{+∞}. 2. If the problem refers to sets U and V of different cardinalities n and m < n, respectively, then we can always introduce n − m spurious (dummy) vertices and n(n−m) spurious edges of weights −∞ (+∞ in the minimal assignment case) so that U and V become of equal cardinality. 3. We refer to [BDM12] for a comprehensive treatment of linear and related assignment problems. Representing the weight function as an n × n matrix W = (wij) over IRmax, the k-LAP is about finding k elements of k different rows and k different columns, such that their sum is maximal. It can be formulated as an Integer Programming (IP) problem over the variables xi,j, for i, j =1,...,n, as follows: n n maximize w x (2.2) X X ij ij i=1 j=1 subject to the conditions n x ≤ 1 (i =1,...,n), X ij j=1 n x ≤ 1 (j =1,...,n), (2.3) X ij i=1 n n x = k, X X ij i=1 j=1 xij ∈{0, 1} (i, j =1,...,n). If we replace the last condition xij ∈{0, 1} in (2.3) by the inequalities xij ≥ 0 (i, j =1,...,n), (2.4) we obtain the corresponding Linear Programming (LP) problem. A LAP can also be described in the language of network flows or in terms of matroids. In what follows we omit the word ”linear” and use the term k-assignment for k-linear (optimal) assignment. 3 3 Max-plus algebra The values of the k-assignment problem, k = 1,...,n, appear as the coefficients of the full characteristic polynomial of the weight matrix W in the setting of max- plus algebra. In this section we present the relevant background about max-plus algebra. Max-plus algebra [But10] is an algebra over the semifield IRmax = IR ∪ {−∞}, equipped with the operations of addition a ⊕ b defined as max(a, b) and multipli- cation a ⊙ b defined as a + b in standard arithmetic, with the unit elements −∞ for addition and 0 for multiplication. For the sake of readability, we suppress the multiplication sign ⊙, writing ab instead of a ⊙ b and ax3 instead of a ⊙ x⊙3. Also, when an indeterminate x appears without a coefficient, as in xn, then its coefficient is naturally the multiplicative identity element, i.e. 0. A (formal) maxpolynomial of degree d over IRmax in the indeterminate x is an expression of the form d p(x)= a xk = max{a + kx : k =0, 1,...,d} (3.1) M k k k=0 with a0,...,ad ∈ IRmax. Each element of the generated algebra under the max and plus operations can be reduced to a unique expression of the form (3.1) (where normally the maxmonomials with coefficient −∞ are deleted) with x treated as an indeterminate. A maxpolynomial p(x) induces also a functionp ˆ(x) on IRmax, which is convex and piecewise-affine. Unlike the situation in standard arithmetic, two distinct for- mal maxpolynomials p1(x) and p2(x) may represent the same polynomial function, k that is,p ˆ1(x)=ˆp2(x) as functions. This happens because a maxmonomial akx of p(x) with the property that for every value of x there exists another maxmonomial l k l alx , l =6 k, of p(x), such that akx ≤ alx , does not contribute to the functionp ˆ(x) k and thus may be omitted. The term akx is said to be an essential term of p(x) k k when, at some interval of IR,p ˆ(x) = akx as functions. When akx < pˆ(x) for all k k values of x then akx is said to be inessential. Otherwise, whenp ˆ(x)= akx at a k single point, then we say that akx is semi-essential (in the literature this term is also called inessential). For each functionq ˆ(x) there exists a unique maxpolynomial representation p(x), k such that every power x of p(x) appears with the maximal possible coefficient ak, as long as the functional equalityp ˆ(x)=ˆq(x) holds. We then say that p(x) is in 4 full canonical form (in FCF) [CGM80]. Note that when p(x) is in FCF than all its maxmonomials are either essential or semi-essential. We adopt the convention −∞ − (−∞)= −∞. n k Proposition 3.1 ([RLP19]). Let p(x)= akx be a maxpolynomial of degree Lk=0 n ≥ 1 and let λ1 ≤ λ2 ≤···≤ λn be its roots. Then the following are equivalent characterizations of p(x) to be in full canonical form. 1. p(x)= an(x ⊕ s1) ··· (x ⊕ sn) formally, for some s1,...,sn. 2. p(x)= an(x ⊕ λ1) ··· (x ⊕ λn) formally. 3. λk = ak−1 − ak, k =1,...,n. 4. Concavity: ak ≥ (ak−1 + ak+1)/2, k =1,...,n − 1. k 5. pˆ(λk)= akλk (ak + kλk in standard arithmetic) for k =1,...,n. The max-plus roots (tropical roots) of a maxpolynomial p(x) are the points at whichp ˆ(x) is non-differentiable. The multiplicity of a root equals the change of the slope ofp ˆ(x) at that root. Equivalently, the roots of p(x) are the values k k+1 k+d λ =6 −∞ of x at which several maxmonomials akx , ak+1x ,...,ak+dx in the k k+1 k+d corresponding full canonical form satisfy akλ = ak+1λ = ··· = ak+dλ =p ˆ(λ) and the multiplicity of the root λ is then d, the difference between the largest and the smallest index of these maxmonomials. We also count −∞ as a root with multiplicity d whenever a0, a1,...,ad−1 are all equal to −∞ and ad =6 −∞. 3.1 The (full) characteristic maxpolynomial of a matrix IRn×n Given an n × n matrix A ∈ max , the max-plus permanent of A is maxperm(A)= a ··· a , M 1σ(1) nσ(n) σ∈Sn where Sn is the group of permutations on [n] = {1,...,n}.