Nonnegative Integer Vectors Research Articles

On the $ \{2\} $-domination number of graphs

<abstract><p>Let $ G $ be a nontrivial graph and $ k\geq 1 $ an integer. Given a vector of nonnegative integers $ w = (w_0, \ldots, w_k) $, a function $ f: V(G)\rightarrow \{0, \ldots, k\} $ is a $ w $-dominating function on $ G $ if $ f(N(v))\geq w_i $ for every $ v\in V(G) $ such that $ f(v) = i $. The $ w $-domination number of $ G $, denoted by $ \gamma_{w}(G) $, is the minimum weight $ \omega(f) = \sum_{v\in V(G)}f(v) $ among all $ w $-dominating functions on $ G $. In particular, the $ \{2\} $-domination number of a graph $ G $ is defined as $ \gamma_{\{2\}}(G) = \gamma_{(2, 1, 0)}(G) $. In this paper we continue with the study of the $ \{2\} $-domination number of graphs. In particular, we obtain new tight bounds on this parameter and provide closed formulas for some specific families of graphs.</p></abstract>

Sparse representation of vectors in lattices and semigroups

We study the sparsity of the solutions to systems of linear Diophantine equations with and without non-negativity constraints. The sparsity of a solution vector is the number of its nonzero entries, which is referred to as the ell _0-norm of the vector. Our main results are new improved bounds on the minimal ell _0-norm of solutions to systems Avarvec{x}=varvec{b}, where Ain mathbb {Z}^{mtimes n}, {varvec{b}}in mathbb {Z}^m and varvec{x} is either a general integer vector (lattice case) or a non-negative integer vector (semigroup case). In certain cases, we give polynomial time algorithms for computing solutions with ell _0-norm satisfying the obtained bounds. We show that our bounds are tight. Our bounds can be seen as functions naturally generalizing the rank of a matrix over mathbb {R}, to other subdomains such as mathbb {Z}. We show that these new rank-like functions are all NP-hard to compute in general, but polynomial-time computable for fixed number of variables.

Integer packing sets form a well-quasi-ordering

An integer packing set is a set of non-negative integer vectors with the property that, if a vector x is in the set, then every non-negative integer vector y with y≤x is in the set as well. The main result of this paper is that integer packing sets, ordered by inclusion, form a well-quasi-ordering. This result allows us to answer a recently posed question: the k-aggregation closure of any packing polyhedron is again a packing polyhedron.

Vector Domination in split-indifference graphs

Given a graph G=(V,E) and a vector of nonnegative integers R[u]u∈V (the vertex requirements), a set S⊆V is an R-dominating set of G if each u∈V∖S has at least R[u] neighbors in S. The Vector Domination problem aims at finding a minimum R-dominating set S. In this work we describe an O(n)-time algorithm to solve Vector Domination in split-indifference graphs.

Multimodularity in the Stochastic Appointment Scheduling Problem with Discrete Arrival Epochs

We address the problem of designing appointment scheduling strategies in a stochastic environment accounting for patient no-shows, nonpunctuality, general stochastic service times, and unscheduled emergency walk-ins. A good appointment schedule seeks to help outpatient clinics to utilize their resources efficiently while containing patients’ waiting times. The task of identifying an optimal schedule is modeled as a nonlinear integer program, where the objective function is the outcome of stochastic analysis in transient state. We maintain the discrete nature of the appointment scheduling problem by considering arrival epochs with discrete supports. By looking at discrete-time snapshots of the random evolution of a single-server queueing model, we characterize probabilistically the system’s workload over time as a function of an appointment schedule, and we derive recursive expressions for the performance measures of interest. Subsequently, we unfold discrete convexity properties of the optimization problem. We prove that under general conditions the objective function is supermodular and componentwise convex. Under assumptions on patient punctuality, we prove that the optimal scheduling strategy minimizes a multimodular function, a property which guarantees that a locally optimal schedule is also globally optimal. The size of the local neighborhood, however, grows exponentially with the dimension of the problem. To the best of our knowledge, this study is the first to develop and implement an algorithm for minimizing locally a multimodular function over nonnegative integer vectors via submodular set-function minimization over ring families, a task that can be performed in polynomial time. This paper was accepted by David Simchi-Levi, operations management.

Identification of Conserved Moieties in Metabolic Networks by Graph Theoretical Analysis of Atom Transition Networks.

Conserved moieties are groups of atoms that remain intact in all reactions of a metabolic network. Identification of conserved moieties gives insight into the structure and function of metabolic networks and facilitates metabolic modelling. All moiety conservation relations can be represented as nonnegative integer vectors in the left null space of the stoichiometric matrix corresponding to a biochemical network. Algorithms exist to compute such vectors based only on reaction stoichiometry but their computational complexity has limited their application to relatively small metabolic networks. Moreover, the vectors returned by existing algorithms do not, in general, represent conservation of a specific moiety with a defined atomic structure. Here, we show that identification of conserved moieties requires data on reaction atom mappings in addition to stoichiometry. We present a novel method to identify conserved moieties in metabolic networks by graph theoretical analysis of their underlying atom transition networks. Our method returns the exact group of atoms belonging to each conserved moiety as well as the corresponding vector in the left null space of the stoichiometric matrix. It can be implemented as a pipeline of polynomial time algorithms. Our implementation completes in under five minutes on a metabolic network with more than 4,000 mass balanced reactions. The scalability of the method enables extension of existing applications for moiety conservation relations to genome-scale metabolic networks. We also give examples of new applications made possible by elucidating the atomic structure of conserved moieties.

Asymptotic Enumeration of Sparse Uniform Linear Hypergraphs with Given Degrees

A hypergraph is simple if it has no loops and no repeated edges, and a hypergraph is linear if it is simple and each pair of edges intersects in at most one vertex. For $n\geq 3$, let $r= r(n)\geq 3$ be an integer and let $\boldsymbol{k} = (k_1,\ldots, k_n)$ be a vector of nonnegative integers, where each $k_j = k_j(n)$ may depend on $n$. Let $M = M(n) = \sum_{j=1}^n k_j$ for all $n\geq 3$, and define the set $\mathcal{I} = \{ n\geq 3 \mid r(n) \text{ divides } M(n)\}$. We assume that $\mathcal{I}$ is infinite, and perform asymptotics as $n$ tends to infinity along $\mathcal{I}$. Our main result is an asymptotic enumeration formula for linear $r$-uniform hypergraphs with degree sequence $\boldsymbol{k}$. This formula holds whenever the maximum degree $k_{\max}$ satisfies $r^4 k_{\max}^4(k_{\max} + r) = o(M)$. Our approach is to work with the incidence matrix of a hypergraph, interpreted as the biadjacency matrix of a bipartite graph, enabling us to apply known enumeration results for bipartite graphs. This approach also leads to a new asymptotic enumeration formula for simple uniform hypergraphs with specified degrees, and a result regarding the girth of random bipartite graphs with specified degrees.

On explaining integer vectors by few homogeneous segments

We extend previous studies on “explaining” a nonnegative integer vector by sums of few homogeneous segments, that is, vectors where all nonzero entries are equal and consecutive. We study two NP-complete variants which are motivated by radiation therapy and database applications. In Vector Positive Explanation, the segments may have only positive integer entries; in Vector Explanation, the segments may have arbitrary integer entries. Considering several natural parameterizations such as the maximum vector entry γ and the maximum difference δ between consecutive vector entries, we obtain a refined picture of the computational (in-)tractability of these problems. For example, we show that Vector Explanation is fixed-parameter tractable with respect to δ, and that, unless NP⊆coNP/poly, there is no polynomial kernelization for Vector Positive Explanation with respect to the parameter γ. We also identify relevant special cases where Vector Positive Explanation is algorithmically harder than Vector Explanation.

Analysis of the Binary Asymmetric Joint Sparse Form

We consider redundant binary joint digital expansions of integer vectors. The redundancy is used to minimize the Hamming weight,i.e., the number of non-zero digit vectors. This leads to efficient linear combination algorithms in abelian groups, which are used in elliptic curve cryptography, for instance.If the digit set is a set of contiguous integers containing zero, a special syntactical condition is known to minimize the weight. We analyse the optimal weight of all non-negative integer vectors with maximum entry less thanN. The expectation and the variance are given with a main term and a periodic fluctuation in the second-order term. Finally, we prove asymptotic normality.

Shellability of the higher pinched Veronese posets

The pinched Veronese poset $${\mathcal {V}}^{\bullet }_n$$ V n ? is the poset with ground set consisting of all nonnegative integer vectors of length $$n$$ n such that the sum of their coordinates is divisible by $$n$$ n with exception of the vector $$(1,\ldots ,1)$$ ( 1 , ? , 1 ) . For two vectors $$\mathbf {a}$$ a and $$\mathbf {b}$$ b in $${\mathcal {V}}^{\bullet }_n$$ V n ? , we have $$\mathbf {a}\preceq \mathbf {b}$$ a ? b if and only if $$\mathbf {b}- \mathbf {a}$$ b - a belongs to the ground set of $${\mathcal {V}}^{\bullet }_n$$ V n ? . We show that every interval in $${\mathcal {V}}^{\bullet }_n$$ V n ? is shellable for $$n \ge 4$$ n ? 4 . In order to obtain the result, we develop a new method for showing that a poset is shellable. This method differs from classical lexicographic shellability. Shellability of intervals in $${\mathcal {V}}^{\bullet }_n$$ V n ? has consequences in commutative algebra. As a corollary, we obtain a combinatorial proof of the fact that the pinched Veronese ring is Koszul for $$n \ge 4$$ n ? 4 . (This also follows from a result by Conca, Herzog, Trung, and Valla.)

Asymptotic Enumeration of Sparse Multigraphs with Given Degrees

Let $J$ and $J^*$ be subsets of $\mathbb{N}$ such that $0,1\in J$ and $0\in J^*$. For infinitely many $n$, let ${\boldsymbol{k}}=(k_1,\ldots, k_n)$ be a vector of nonnegative integers whose sum $M$ is even. We find an asymptotic expression for the number of multigraphs on the vertex set $\{1,\ldots, n\}$ with degree sequence given by ${\boldsymbol{k}}$ such that every loop has multiplicity in $J^*$ and every nonloop edge has multiplicity in $J$. Equivalently, these are symmetric integer matrices with values $J^*$ allowed on the diagonal and $J$ off the diagonal. Our expression holds when the maximum degree $k_{\mathrm{max}}$ satisfies $k_{\mathrm{max}} = o(M^{1/3})$. We prove this result using the switching method, building on an asymptotic enumeration of simple graphs with given degrees [B. D. McKay and N. C. Wormald, Combinatorica, 11 (1991), pp. 369--382]. Our application of the switching method introduces a novel way of combining several different switching operations into a single computation.

On integer polytopes with few nonzero vertices

We provide a simple description in terms of linear inequalities of the convex hull of the nonnegative integer vectors x that satisfy a given linear knapsack covering constraint ∑aixi≥b and have sum of the components that does not exceed 2. This description allows the replacement of “weak” knapsack-type constraints by stronger ones in several ILP formulations for practical problems, including railway rolling stock scheduling. In addition, we provide a simple description of the packing counterpart of the considered polytope, i.e. for the case in which the knapsack inequality is ∑aixi≤b (and again the sum of the components of the nonnegative integer vectors that does not exceed 2).

A Semidefinite Approach for Truncated K-Moment Problems

A truncated moment sequence (tms) in n variables and of degree d is a finite sequence y=(yź) indexed by nonnegative integer vectors ź:=(ź1,ź,źn) such that ź1+ź+źn≤d. Let K⊆źn be a semialgebraic set. The truncated K-moment problem (TKMP) is: How can one check if a tms y admits a K-measure μ (a nonnegative Borel measure supported in K) such that $y_{\alpha}= \int_{K} x_{1}^{\alpha_{1}}\cdots x_{n}^{\alpha_{n}}\,\mathrm{d}\mu$ for every ź? This paper proposes a semidefinite programming (SDP) approach for solving TKMP. When K is compact, we get the following results: whether a tms admits a K-measure or not can be checked via solving a sequence of SDP problems; when y admits no K-measure, a certificate for the nonexistence can be found; when y admits one, a representing measure for y can be obtained from solving the SDP problems under some necessary and some sufficient conditions. Moreover, we also propose a practical SDP method for finding flat extensions, which in our numerical experiments always found a finitely atomic representing measure when it exists.

Counting loopy graphs with given degrees

Let d = ( d 1 , d 2 , … , d n ) be a vector of nonnegative integers. We study the number of symmetric 0–1 matrices whose row sum vector equals d . While previous work has focussed on the case of zero diagonal, we allow diagonal entries to equal 1. Specifically, for D ∈ { 1 , 2 } we define the set G D ( d ) of all n × n symmetric 0–1 matrices with row sums given by d , where each diagonal entry is multiplied by D when forming the row sum. We obtain asymptotically precise formulae for | G D ( d ) | in the sparse range (where, roughly, the maximum row sum is o ( n 1 / 2 ) ), and in the dense range (where, roughly, the average row sum is proportional to n and the row sums do not vary greatly). The case D = 1 corresponds to enumeration by the usual row sum of matrices. The case D = 2 corresponds to enumeration by degree sequence of undirected graphs with loops but no repeated edges, due to the convention that a loop contributes 2 to the degree of its incident vertex. We also analyse the distribution of the trace of a random element of G D ( d ) , and prove that it is well approximated by a binomial distribution in the dense range, and by a Poisson binomial distribution in the sparse range.

Subgraphs of Dense Random Graphs with Specified Degrees

Let d = (d1, d2, . . ., dn) be a vector of non-negative integers with even sum. We prove some basic facts about the structure of a random graph with degree sequence d, including the probability of a given subgraph or induced subgraph.Although there are many results of this kind, they are restricted to the sparse case with only a few exceptions. Our focus is instead on the case where the average degree is approximately a constant fraction of n.Our approach is the multidimensional saddle-point method. This extends the enumerative work of McKay and Wormald (1990) and is analogous to the theory developed for bipartite graphs by Greenhill and McKay (2009).

Variants of P Colonies with Very Simple Cell Structure

We study two very simple variants of P colonies: systems with only one object inside the cells, and systems with insertion-deletion programs, so called P colonies with senders and consumers. We show that both of these extremely simple types of systems are able to compute any recursively enumerable set of vectors of non-negative integers.

Matrix columns allocation problems

Orthogonal Frequency Division Multiple Access (OFDMA) transmission technique is gaining popularity as a preferred technique in the emerging broadband wireless access standards. Motivated by the OFDMA transmission technique we define the following problem: Let M be a matrix (over R ) of size a × b . Given a vector of non-negative integers C → = 〈 c 1 , c 2 , … , c b 〉 such that ∑ c j = a , we would like to allocate a cells in M such that (i) in each row of M there is a single allocation, and (ii) for each element c i ∈ C → there is a unique column in M which contains exactly c i allocations. Our goal is to find an allocation with minimal value, that is, the sum of all the a cells of M which were allocated is minimal. The nature of the suggested new problem is investigated in this paper. Efficient algorithms are suggested for some interesting cases. For other cases of the problem, NP-hardness proofs are given followed by inapproximability results.

Efficiently Computing Minimal-Support Nonnegative Integer Invariants of Petri Nets

Invariants of Petri nets are fundamental algebraic characteristics of Petri nets, and are used in various situations, such as checking (as necessity of) liveness, boundedness, periodicity and so on. Any given Petri net N has two kinds of invariants: a P-invariant is a |P|-dimensional vector Y with Yt·A = 0 and a T-invariant is a |T|-dimensional vector X with A · X =0 for the place-transition incidence matrix A of N. T-invariants are nonnegative integer vectors, while this is not always the case with P-invariants. This paper deals only with nonnegative integer invariants (invariants that are nonnegative vectors) and shows results common to the two invariants. For simplicity of discussion, only P-invariants are treated. The Fourier-Motzkin method is well-known for computing all minimal support integer invariants. This method, however, has a critical deficiency such that, even if a given Perti net N has any invariant, it is likely that no invariants are obtained because of an overflow in storing intermediate vectors as candidates for invariants. The subject of the paper is to give an overview and results known to us for efficiently computing minimal-support nonnegative integer invariants of a given Petri net by means of the Fourier-Motzkin method. Also included are algorithms for efficiently extracting siphon-traps of a Petri net.

Generalized exponents of primitive two-colored digraphs

A l-colored digraph D ( l) is primitive if there exists a nonnegative integer vector α such that for each ordered pair of vertices x and y (not necessarily distinct), there exists an α-walk in D ( l) from x to y. The exponent of the primitive l-colored digraph D ( l) is defined to be the minimum value of the sum of all coordinates of α taken over all such α. In this paper, we generalize the concept of exponent of a primitive l-colored digraph by introducing three types of generalized exponents. Further, we study the generalized exponents of primitive two-colored Wielandt digraphs.

Resonance Graphs and a Binary Coding for the 1-Factors of Benzenoid Systems

Applying the recently obtained distributive lattice structure on the set of 1-factors, we show that the resonance graphs of any benzenoid systems G, as well as of general plane (weakly) elementary bipartite graphs, are median graphs and thus extend greatly Klavžar et al.'s result. The n-dimensional vectors of nonnegative integers as a labelling for the 1-factors of G with n inner faces are described. The labelling preserves the partial ordering of the above-mentioned lattice and can be transformed into a binary coding for the 1-factors. A simple criterion for such a labelling being binary is given. In particular, Klavžar et al.'s algorithm is modified to generate this binary coding for the 1-factors of a cata-condensed benzenoid system.