On covering multiplicity

On Covering Multiplicity

Let A = {as +nsZ}s=1 be a system of arithmetic sequences which forms an m-cover of Z (i.e. every integer belongs at least to m members of A). In this paper we show the following surprising properties of A: (a) For each J ⊆ {1, · · · , k} there exist at least m subsets I of {1, · · · , k} with I 6= J such that ∑ s∈I 1/ns − ∑ s∈J 1/ns ∈ Z. (b) If A forms a minimal m-cover of Z, then for any t = 1...

On the Covering Multiplicity of Lattices

Let the lattice Λ have covering radius R, so that closed balls of radius R around the lattice points justcover the space. The covering multiplicity CM(Λ) is the maximal number of times the interiors of theseballs overlap. We show that the least possible covering multiplicity for an n-dimensional lattice is n ifn ≤ 8, and conjecture that it exceeds n in all other cases. We determine ...

Approximation Schemes for Deal Splitting and Covering Integer Programs with Multiplicity Constraints

We consider the problem of splitting an order for R goods, R ≥ 1, among a set of sellers, each having bounded amounts of the goods, so as to minimize the total cost of the deal. In deal splitting with packages (DSP), the sellers offer packages containing combinations of the goods; in deal splitting with price tables (DST), the buyer can generate such combinations using price tables. Our problem...

Covering on a Circle

On Covering Equivalence 3

An arithmetic sequence a(n) = {a+nx : x ∈ Z} (0 6 a < n) with weight λ ∈ C is denoted by 〈λ, a, n〉. For two finite systems A = {〈λs, as, ns〉}s=1 and B = {〈μt, bt,mt〉}t=1 of such triples, if ∑ ns|x−as λs = ∑ mt|x−bt μt for all x ∈ Z then we say that A and B are covering equivalent. In this paper we characterize covering equivalence in various ways, our characterizations involve the Γ-function, t...