Nonnegative Integer Vectors Research Articles

Asymptotic enumeration of dense 0–1 matrices with specified line sums

Let s = ( s 1 , s 2 , … , s m ) and t = ( t 1 , t 2 , … , t n ) be vectors of non-negative integers with ∑ i = 1 m s i = ∑ j = 1 n t j . Let B ( s , t ) be the number of m × n matrices over { 0 , 1 } with jth row sum equal to s j for 1 ⩽ j ⩽ m and kth column sum equal to t k for 1 ⩽ k ⩽ n . Equivalently, B ( s , t ) is the number of bipartite graphs with m vertices in one part with degrees given by s , and n vertices in the other part with degrees given by t . Most research on the asymptotics of B ( s , t ) has focused on the sparse case, where the best result is that of Greenhill, McKay and Wang (2006). In the case of dense matrices, the only precise result is for the case of equal row sums and equal column sums (Canfield and McKay, 2005). This paper extends the analytic methods used by the latter paper to the case where the row and column sums can vary within certain limits. Interestingly, the result can be expressed by the same formula which holds in the sparse case.

Membrane division, restricted membrane creation and object complexity in P systems

We improve, by using register machines, some existing universality results for specific models of P systems. P systems with membrane creation are known to generate all recursively enumerable sets of vectors of non-negative integers, even when no region (except the environment) contains more than one object of the same kind. We show here that they generate all recursively enumerable languages, and that two membrane labels are sufficient (the same result holds for accepting all recursively enumerable vectors of non-negative integers). Moreover, at most two objects are present inside the system at any time in the generative case. We then prove that 10+m symbols are sufficient to generate any recursively enumerable language over m symbols. P systems with active membranes without polarizations are known to generate all recursively enumerable sets of vectors of non-negative integers. We show that they generate all recursively enumerable languages; four starting membranes with three labels or seven starting membranes with two labels are sufficient. P systems with active membranes and two polarizations are known to generate/accept all recursively enumerable sets of vectors of non-negative integers, using only rules of rewriting and sending objects out. We show that accepting can be done by deterministic systems. Finally, we show that P systems with restricted membrane creation (the newly created membrane can only be of the same kind as the parent one) generate at least matrix languages, even when having at most one object in the configuration (except the environment). We conclude by presenting a summary of the main results obtained in this paper and a list of open questions.

One-dimensional optimal bounded-shape partitions for Schur convex sum objective functions

Consider the problem of partitioning n nonnegative numbers into p parts, where part i can be assigned n i numbers with n i lying in a given range. The goal is to maximize a Schur convex function F whose ith argument is the sum of numbers assigned to part i. The shape of a partition is the vector consisting of the sizes of its parts, further, a shape (without referring to a particular partition) is a vector of nonnegative integers (n 1,..., n p ) which sum to n. A partition is called size-consecutive if there is a ranking of the parts which is consistent with their sizes, and all elements in a higher-ranked part exceed all elements in the lower-ranked part. We demonstrate that one can restrict attention to size-consecutive partitions with shapes that are nonmajorized, we study these shapes, bound their numbers and develop algorithms to enumerate them. Our study extends the analysis of a previous paper by Hwang and Rothblum which discussed the above problem assuming the existence of a majorizing shape.

Decomposition design theory and methodology for arbitrary-shaped switch boxes

We consider the optimal design problem for arbitrary-shaped switch box, (r1...rk) which r, terminals are located on side i for i= 1...k and programmable switches are joining pairs of terminals from different sides. Previous investigations on switch box designs mainly focused on regular switch boxes in which all sides have the same number of terminals. By allowing different numbers of terminals on different sides, irregular switch boxes are more general and flexible for applications such as customized FPGAs and reconfigurable interconnection networks. The optimal switch box design problem is to design a switch box satisfying the given shape and routing capacity specifications with the minimum number of switches. We present a decomposition design method for a wide range of irregular switch boxes. The main idea of our method is to model a routing requirement as a nonnegative integer vector satisfying a system of linear equations and then derive a decomposition theory of routing requirements based on the theory of systems of linear Diophantine equations. The decomposition theory makes it possible to construct a large irregular switch box by combining small switch boxes of fixed sizes. Specifically, we can design a family of hyperuniversal (universal) (u-d + c)-SBs with B(h-) switches, where d and c are constant vectors and w is a scalar. We illustrate the design method by designing a class of optimal hyperuniversal irregular 3-sided switch boxes and a class of optimal rectangular universal switch boxes. Experimental results on the rectangular universal switch boxes with the VPR router show that the optimal design of irregular switch boxes does pay off.

On survivable network polyhedra

Given an undirected network G = ( V , E ) , a vector of nonnegative integers r = ( r ( v ) : v ∈ V ) associated with the nodes of G and weights on the edges of G, the survivable network design problem is to determine a minimum-weight subnetwork of G such that between every two nodes u , v of V , there are at least min { r ( u ) , r ( v ) } edge-disjoint paths. In this paper we study the polytope associated with the solutions to that problem. We show that when the underlying network is series–parallel and r ( v ) is even for all v ∈ V , the polytope is completely described by the trivial constraints and the so-called cut constraints. As a consequence, we obtain a polynomial time algorithm for the survivable network design problem in that class of networks. This generalizes and unifies known results in the literature. We also obtain a linear description of the polyhedron associated with the problem in the same class of networks when the use of more than one copy of an edge is allowed.

On Feasible Sets of Mixed Hypergraphs

A mixed hypergraph $H$ is a triple $(V,{\cal C},{\cal D})$ where $V$ is the vertex set and ${\cal C}$ and ${\cal D}$ are families of subsets of $V$, called ${\cal C}$-edges and ${\cal D}$-edges. A vertex coloring of $H$ is proper if each ${\cal C}$-edge contains two vertices with the same color and each ${\cal D}$-edge contains two vertices with different colors. The spectrum of $H$ is a vector $(r_1,\ldots,r_m)$ such that there exist exactly $r_i$ different colorings using exactly $i$ colors, $r_m\ge 1$ and there is no coloring using more than $m$ colors. The feasible set of $H$ is the set of all $i$'s such that $r_i\ne 0$. We construct a mixed hypergraph with $O(\sum_i\log r_i)$ vertices whose spectrum is equal to $(r_1,\ldots,r_m)$ for each vector of non-negative integers with $r_1=0$. We further prove that for any fixed finite sets of positive integers $A_1\subset A_2$ ($1\notin A_2$), it is NP-hard to decide whether the feasible set of a given mixed hypergraph is equal to $A_2$ even if it is promised that it is either $A_1$ or $A_2$. This fact has several interesting corollaries, e.g., that deciding whether a feasible set of a mixed hypergraph is gap-free is both NP-hard and coNP-hard.

A finite basis characterization of α-split colorings

Fix t>1, a positive integer, and a=(a 1,…,a t) a vector of nonnegative integers. A t-coloring of the edges of a complete graph is called a -split if there exists a partition of the vertices into t sets V 1,…, V t such that every set of a i +1 vertices in V i contains an edge of color i, for i=1,…, t. We combine a theorem of Deza with Ramsey's theorem to prove that, for any fixed a , the family of a -split colorings is characterized by a finite list of forbidden induced subcolorings. A similar hypergraph version follows from our proofs. These results generalize previous work by Kézdy et al. (J. Combin. Theory Ser. A 73(2) (1996) 353) and Gyárfás (J. Combin. Theory Ser. A 81(2) (1998) 255). We also consider other notions of splitting.

An exact algorithm for IP column generation

An exact column generation algorithm for integer programs with a large (implicit) number of columns is presented. The family of problems that can be treated includes not only standard partitioning problems such as bin packing and certain vehicle routing problems in which the columns generated have 0–1 compenents and a right-hand side vector of 1's, but also the cutting stock problem in which the columns and right-hand side are nonnegative integer vectors. We develop a combined branching and subproblem modification scheme that generalizes existing approaches, and also describe the use of lower bounds to reduce tailing-off effects.

A characterization of matroidal systems of inequalities

We consider the polyhedron Conv{ xϵ{0,1} n : Mx⩽ b}, where M is a p × n matrix of zeroes and ones and b is a nonnegative integer vector. We give a characterization of such polyhedra whose extreme points are the incidence vectors of the family of independent sets of a matroid and extend our result to polyhedra which are the convex hull of integral polymatroids. We also introduce some new classes of integral matroid polyhedra which extend a result of Edmonds.

Counting binary matrices with given row and column sums

This paper is concerned with the calculation of certain numbers sb(p), b(p,q) related to combinatorial problems and graph theory, p, q are vectors of nonnegative integers, and sb(p) is the number of labelled graphs with vertex degree sequence p, or equivalently, the number of 0-diagonal, symmetric, binary matrices with row sum p. Similarly, b(p, q) is the number of binary rectangular matrices with row sum p and column sum q. The numbers also appear as coefficients in the expansions 11(1 + xixX 11(1 + xiyi). Explicit (i.e., nonrecursive) formulas for sb(p), b(p,q) are developed, together with an analysis of their complexity. Properties of p (or q), such as max{/>,}, k » ?Zpi, n = #{/>, # 0}, are incorporated into a numerical invariant which measures the total cost (or computing time). The performance of the theory for practical calculations has been thor- oughly tested. For example, with suitable restrictions on p one may obtain sb(p) for, say, n = 20 or k = 30.

Counting Binary Matrices with Given Row and Column Sums

This paper is concerned with the calculation of certain numbers $sb(\bar p)$, $b(\bar p,\bar q)$ related to combinatorial problems and graph theory. $\bar p,\bar q$ are vectors of nonnegative integers, and $sb(\bar p)$ is the number of labelled graphs with vertex degree sequence $\bar p$, or equivalently, the number of 0-diagonal, symmetric, binary matrices with row sum $\bar p$. Similarly, $b(\bar p,\bar q)$ is the number of binary rectangular matrices with row sum $\bar p$ and column sum $\bar q$. The numbers also appear as coefficients in the expansions $\Pi (1 + {x_i}{x_j})$, $\Pi (1 + {x_i}{y_j})$. Explicit (i.e., nonrecursive) formulas for $sb(\bar p)$, $b(\bar p,\bar q)$ are developed, together with an analysis of their complexity. Properties of $\bar p$ (or $\bar q$), such as $\max \{ {p_i}\}$, $k = \Sigma {p_i}$, $n = \# \{ {p_i} \ne 0\}$, are incorporated into a numerical invariant which measures the total "cost" (or computing time). The performance of the theory for practical calculations has been thoroughly tested. For example, with suitable restrictions on $\bar p$ one may obtain $sb(\bar p)$ for, say, $n = 20$ or $k = 30$.

A periodicity lemma in linear diophantine analysis

Let g = ( g 1,…, g r ) ≥ 0 and h = ( h 1,…, h r ) ≥ 0, g ϱ , h ϱ ∈ J , be two vectors of nonnegative integers and let λ ϵ J , λ ≥ 0, λ ≡ 0 mod d, where d denotes g.c.d. ( g 1,…, g r ). Define Δ(λ)=Δ(λg,h):= min ∑ ϱ=1 r x ϱh ϱ:x ϱ⩾0,x ϱ∈J, ∑ ϱ=1 ϱ x ϱg ϱ=λ It is shown in this paper that Λ(λ) is periodic in λ with constant jump. If i ϵ {1,…, r} is such that det g i h i g ϱ h ϱ ⩾ (ϱ1,…r) then Δ(λ)+g iΔ(λ)+h i holds true for all sufficiently large λ, λ ≡ 0 mod d.