Positive Integer Weights Research Articles

On Pareto optimal balanced exchanges

We investigate a market without money in which every agent offers indivisible goods in multiple copies, in exchange for goods of other agents. The exchange must be balanced in the sense that each agent should receive a quantity of good(s) equal to the one she transfers to others. We describe the market in graph-theoretic terms hence we use the notion of circulations to describe a balanced exchange of goods. Each agent has strict preferences over the agents from which she will receive goods and an upper bound on the quantity of each transaction, while a positive integer weight reflects the social importance of each unit exchanged. In this paper, we propose a simple variant of the Top Trading Cycles mechanism that finds a Pareto optimal circulation. We then offer necessary and sufficient conditions for a circulation to be Pareto optimal and, as a consequence, a easy recognition procedure. Last, we show that finding a maximum weight Pareto optimal circulation is NP-hard but becomes polynomial if weights are concordant with preferences.

Heuristic approach for bandwidth consecutive multi-coloring problem

Bandwidth consecutive multicoloring problem is also known as a b-coloring problem. Let G = (V, E) be a graph where V is the set of vertices and E is the set of edges. Let each vertex v of V has a positive integer weight b (v) and each edge (v, w) of E has a non-negative integer weight b (v, w). A bandwidth consecutive multicoloring of a graph is a problem of assigning b (v) consecutive positive integers to each vertex v of V in such a manner that the difference between all the integers of vertex v and all the integers of vertex w is greater than b (v, w). The maximum integer assigned in this coloring is called the span of the coloring. The b-coloring is the problem of minimizing this span. No metaheuristic is proposed for general graphs so far for this problem till date as it is strongly NP-hard. In this paper, we proposed three heuristics for the problem including a greedy randomized adaptive search procedure (GRASP). The efficiency of these algorithms is tested on benchmark graphs and their performance is compared among themselves. Experimental results showed that among all three proposed heuristics, GRASP performed well for the given problem.

The maximum spectral radius of the weighted bicyclic hypergraphs

Let B n , k , a be the set of weighted k-uniform connected bicyclic hypergraphs of order n with positive integer weights and fixed total weight sum a, where n, k and a are positive integers with k ≥ 4 and a ≥ n + 1 k − 1 ≥ 3 . An α-normal labelling method is proposed for comparing the spectral radii of the weighted hypergraphs under consideration. The weighted hypergraph having the maximum spectral radius among the hypergraps in B n , k , a is obtained.

Diameter, Eccentricities and Distance Oracle Computations on H-Minor Free Graphs and Graphs of Bounded (Distance) Vapnik–Chervonenkis Dimension

Under the strong exponential-time hypothesis, the diameter of general unweighted graphs cannot be computed in truly subquadratic time (in the size of the input), as shown by Roditty and Williams. Nevertheless there are several graph classes for which this can be done such as bounded-treewidth graphs, interval graphs, and planar graphs, to name a few. We propose to study unweighted graphs of constant distance Vapnik–Chervonenkis (VC)-dimension as a broad generalization of many such classes—where the distance VC-dimension of a graph is defined as the VC-dimension of its ball hypergraph whose hyperedges are the balls of all possible radii and centers in . In particular for any fixed , the class of -minor free graphs has distance VC-dimension at most . Our first main result is a Monte Carlo algorithm that on graphs of distance VC-dimension at most , for any fixed , either computes the diameter or concludes that it is larger than in time , where only depends on and the notation suppresses polylogarithmic factors. We thus obtain a truly subquadratic-time parameterized algorithm for computing the diameter on such graphs. Then as a byproduct of our approach, we get a truly subquadratic-time randomized algorithm for constant diameter computation on all the nowhere dense graph classes. The latter classes include all proper minor-closed graph classes, bounded-degree graphs, and graphs of bounded expansion. Before our work, the only known such algorithm was resulting from an application of Courcelle’s theorem; see Grohe, Kreutzer, and Siebertz [J. ACM, 64 (2017), pp. 1–32]. For any graph of constant distance VC-dimension, we further prove the existence of an exact distance oracle in truly subquadratic space, that answers distance queries in truly sublinear time (in the number of vertices). The latter generalizes prior results on proper minor-closed graph classes to a much larger graph class. Finally, we show how to remove the dependency on for any graph class that excludes a fixed graph as a minor. More generally, our techniques apply to any graph with constant distance VC-dimension and polynomial expansion (or equivalently having strongly sublinear balanced separators). As a result for all such graphs one obtains a truly subquadratic-time deterministic algorithm for computing all the eccentricities, and thus both the diameter and the radius. Our approach can be generalized to the -minor free graphs with bounded positive integer weights. We note that all our algorithms for the diameter problem can be adapted for computing the radius, and more generally all the eccentricities. Our approach is based on the work of Chazelle and Welzl who proved the existence of spanning paths with strongly sublinear stabbing number for every hypergraph of constant VC-dimension. We show how to compute such paths efficiently by combining known algorithms for the stabbing number problem with a clever use of -nets, region decomposition, and other partition techniques.

Dimension of tensor network varieties

The tensor network variety is a variety of tensors associated to a graph and a set of positive integer weights on its edges, called bond dimensions. We determine an upper bound on the dimension of the tensor network variety. A refined upper bound is given in cases relevant for applications such as varieties of matrix product states and projected entangled pairs states. We provide a range (the “supercritical range”) of the parameters where the upper bound is sharp.

The weighted hypergraph with the maximum spectral radius

Let Γn,k,a and Un,k,a denote respectively the classes of weighted k-uniform hypertrees and connected weighted k-uniform linear and nonlinear unicyclic hypergraphs of order n with the positive integer weights and the fixed total weight sum a, where n, k (k≥3), and a are positive integers with a≥n−1k−1≥2 for Γn,k,a while a≥nk−1≥2 for Un,k,a. Four new and useful operations are proposed for comparing the spectral radii of the weighted hypergraphs under consideration. The weighted hypergraphs having the maximum spectral radii among Γn,k,a and Un,k,a are characterized.

Random Gap Processes and Asymptotically Complete Sequences

We study a process of generating random positive integer weight sequences \(\{ W_n \}\) where the gaps between the weights \(\{ X_n = W_n - W_{n-1} \}\) are i.i.d. positive integer-valued random variables. The main result of the paper is that if the gap distribution has a moment generating function with large enough radius of convergence, then the weight sequence is almost surely asymptotically m-complete for every \(m\ge 2\), i.e. every large enough multiple of the greatest common divisor (gcd) of gap values can be written as a sum of m distinct weights for any fixed \(m \ge 2\). Under the weaker assumption of finite \(\frac{1}{2}\)-moment for the gap distribution, we also show the simpler result that, almost surely, the resulting weight sequence is asymptotically complete, i.e. all large enough multiples of the gcd of the possible gap values can be written as a sum of distinct weights.

Time-dependent reliability of (weighted) k-out-of-n systems with identical component deterioration

The performance of civil infrastructure systems is vital in supporting a community’s functionalities. Reliability assessment of these systems is a powerful approach to evaluate whether the system performance is desirably safe under the impacts of resistance degradation and non-stationary loads. A k-out-of-n system is a widely-used logic model for a system with n components, which survives (works) if at least k components work. Its special cases include a series or a parallel system. Furthermore, a weighted k-out-of-n system has components with positive integer weights and the system survives if the total weight of working components reaches the predefined threshold k. This paper proposes a method for estimating the time-dependent reliability of both ordinary and weighted k-out-of-n systems, taking into account the effects of resistance deterioration, resistance correlation and load non-stationarity, for which a mathematical solution is derived. The applicability of the proposed method is illustrated through reliability evaluation of a representative k-out-of-n system.

A Modified Decomposition Algorithm for Maximum Weight Bipartite Matching and Its Experimental Evaluation

Let G be an undirected bipartite graph with positive integer weights on the edges. We refine the existing decomposition theorem originally proposed by Kao et al., for computing maximum weight bipartite matching. We apply it to design an efficient version of the decomposition algorithm to compute the weight of a maximum weight bipartite matching of G in O(|V |W /k(|V |, W /N))-time by employing an algorithm designed by Feder and Motwani as a subroutine, where |V | and N denote the number of nodes and the maximum edge weight of G, respectively and k(x, y) = log x/ log(x 2 /y). The parameter W is smaller than the total edge weight W, essentially when the largest edge weight differs by more than one from the second largest edge weight in the current working graph in any decomposition step of the algorithm. In best case W = O(|E|) where |E| be the number of edges of G and in worst case W = W, that is, |E| ≤ W ≤ W. In addition, we talk about a scaling property of the algorithm and research a better bound of the parameter W. An experimental evaluation on randomly generated data shows that the proposed improvement is significant in general.

Distributed Integer Balancing Under Weight Constraints in the Presence of Transmission Delays

We consider the distributed weight balancing problem in networks of nodes that are interconnected via directed edges, each of which admits a positive integer weight. A digraph with positive integer weights on its edges is weight balanced if, for each node, the sum of the weights of the incoming edges equals the sum of the weights of the outgoing edges. In this article, we develop a distributed iterative algorithm, which solves the integer weight balancing problem in the presence of arbitrary (time-varying and inhomogeneous) time delays that might affect transmissions at particular links. We assume that each positive weight is constrained to lie within a certain interval, captured by individual lower and upper limits and that communication between neighboring nodes is bidirectional. We show that even when different transmissions on communication links are affected from bounded delays, the proposed distributed algorithm allows nodes to obtain a set of weights that solves the integer weight balancing problem, after a finite number of iterations, as long as a feasible solution exists. Finally, we provide examples to illustrate the operation and performance of the proposed algorithm.

Distributed Balancing With Constrained Integer Weights

We consider the distributed integer-weight-balancing problem in networks of nodes that are interconnected via directed edges, each able to admit a positive integer weight (or flow) within a certain interval, captured by lower and upper limits. We propose and analyze distributed iterative algorithms for obtaining admissible and balanced integer weights, i.e., integer weights within the given interval constraints, such that, for each node, the sum of the weights of the incoming edges equals the sum of the weights of the outgoing edges. The proposed algorithms assume that communication among pairs of nodes that are interconnected is bidirectional and are shown to lead to a set of admissible and balanced integer weights after a finite number of time steps (as long as the necessary and sufficient integer circulation conditions are satisfied on the given digraph).

Reliability and Reliability Importance of Weighted-$r$ -Within-Consecutive-$k$-out-of-$n: F$ System

In this paper, we consider a weighted- $r$ -within-consecutive- $ k$ -out-of- $n:F$ system. The weighted system has in general $n$ components, each one having a positive integer weight ${w_i}, i= 1,2, \ldots,n$ . The weighted- $r$ -within-consecutive- $ k$ -out-of- $n:F$ system fails if and only if the total weight of failed components among $k$ consecutive components is at least $r$ . We introduce a binomial-type weighted scan statistic and study the reliability, the Birnbaum, improvement potential importance and Bayesian reliability importance of the system taken into consideration. We develop an explicit closed-form formula for the evaluation of reliability and reliability importance measures of a weighted- $r$ -within-consecutive- $ k$ -out-of- $n:F$ system and demonstrate the results numerically. We present a study showing the effectiveness of the method in terms of CPU time.

Reconfiguration of maximum-weight b-matchings in a graph

Consider a graph such that each vertex has a nonnegative integer capacity and each edge has a positive integer weight. Then, a b-matching in the graph is a multi-set of edges (represented by an integer vector on edges) such that the total number of edges incident to each vertex is at most the capacity of the vertex. In this paper, we study a reconfiguration variant for maximum-weight b-matchings: For two given maximum-weight b-matchings in a graph, we are asked to determine whether there exists a sequence of maximum-weight b-matchings in the graph between them, with subsequent b-matchings obtained by removing one edge and adding another. We show that this reconfiguration problem is solvable in polynomial time for instances with no integrality gap. Such instances include bipartite graphs with any capacity function on vertices, and 2-matchings in general graphs. Thus, our result implies that the reconfiguration problem for maximum-weight matchings can be solved in polynomial time for bipartite graphs.

Covering Triangles in Edge-Weighted Graphs

Let G = (V, E) be a simple graph and \(\mathbf {w}\in \mathbb {Z}^{E}_{>0}\) assign each edge e ∈ E a positive integer weight w(e). A subset of E that intersects every triangle of G is called a triangle cover of (G, w), and its weight is the total weight of its edges. A collection of triangles in G (repetition allowed) is called a triangle packing of (G, w) if each edge e ∈ E appears in at most w(e) members of the collection. Let τ t (G, w) and ν t (G, w) denote the minimum weight of a triangle cover and the maximum cardinality of a triangle packing of (G, w), respectively. Generalizing Tuza’s conjecture for unit weight, Chapuy et al. conjectured that τ t (G, w)/ν t (G, w) ≤ 2 holds for every simple graph G and every \(\mathbf {w}\in \mathbb {Z}^{E}_{>0}\). In this paper, using a hypergraph approach, we design polynomial-time combinatorial algorithms for finding triangle covers of small weights. These algorithms imply new sufficient conditions for the conjecture of Chapuy et al. More precisely, given (G, w), suppose that all edges of G are covered by the set \({\mathscr {T}}_{G}\) consisting of edge sets of triangles in G. Let \(|E|_{w}={\sum }_{e\in E}w(e)\) and \(|{\mathscr {T}}_{G}|_{w}={\sum }_{\{e,f,g\}\in {\mathscr {T}}_{G}}w(e)w(f)w(g)\) denote the weighted numbers of edges and triangles in (G, w), respectively. We show that a triangle cover of (G, w) of weight at most 2ν t (G, w) can be found in strongly polynomial time if one of the following conditions is satisfied: (i) \(\nu _{t}(G,\mathbf {w})/|{\mathscr {T}}_{G}|_{w}\ge \frac {1}{3}\), (ii) \(\nu _{t}(G,\mathbf {w})/|E|_{w}\ge \frac {1}{4}\), (iii) \(|E|_{w}/|{\mathscr {T}}_{G}|_{w}\ge 2\).

A linear weighted (n, f, k) system for non-homogeneous Markov-dependent components

ABSTRACTWe study a linear weighted (n, f, k) system, denoted by L(n, f, k, w) system and consider the situation where components are non-homogeneous Markov-dependent. An L(n, f, k, w) system consists of n components ordered in a line, and each component u has a positive integer weight wu for u = 1, 2, …, n and w = (w1, w2, …, wn). The L(n, f, k, w):F (G) system fails (works) if the total weight of failed (working) components is at least f or the total weight of consecutive failed (working) components is at least k. For the L(n, f, k, w):F system with non-homogeneous Markov-dependent components, we derive closed-form formulas for the system reliability, the marginal reliability importance measure of a single component, and the joint reliability importance measure of multiple components using a conditional probability generating function method. We extend these results to the L(n, f, k, w):G systems, the weighted consecutive-k-out-of-n systems, and the weighted f-out-of-n systems. Our numerical examples and a case study on a bridge system demonstrate the use of derived formulas and provide the insights on the L(n, f, k, w) systems and the importance measures. In addition, the two failure modes associated with the L(n, f, k, w):F systems are analyzed by comparing to the single failure mode associated with the weighted consecutive-k-out-of-n:F systems and the single failure mode associated with the weighted f-out-of-n:F systems.

Generalized edge-colorings of weighted graphs

Let [Formula: see text] be a graph with a positive integer weight [Formula: see text] for each vertex [Formula: see text]. One wishes to assign each edge [Formula: see text] of [Formula: see text] a positive integer [Formula: see text] as a color so that [Formula: see text] for any vertex [Formula: see text] and any two edges [Formula: see text] and [Formula: see text] incident to [Formula: see text]. Such an assignment [Formula: see text] is called an [Formula: see text]-edge-coloring of [Formula: see text], and the maximum integer assigned to edges is called the span of [Formula: see text]. The [Formula: see text]-chromatic index of [Formula: see text] is the minimum span over all [Formula: see text]-edge-colorings of [Formula: see text]. In the paper, we present various upper and lower bounds on the [Formula: see text]-chromatic index, and obtain three efficient algorithms to find an [Formula: see text]-edge-coloring of a given graph. One of them finds an [Formula: see text]-edge-coloring with span smaller than twice the [Formula: see text]-chromatic index.

Lattice points in orthotopes and a huge polynomial Tutte invariant of weighted gain graphs

A gain graph is a graph whose edges are orientably labeled from a group. A weighted gain graph is a gain graph with vertex weights from an abelian semigroup, where the gain group is lattice ordered and acts on the weight semigroup. For weighted gain graphs we establish basic properties and we present general dichromatic and forest-expansion polynomials that are Tutte invariants (they satisfy Tutte's deletion-contraction and multiplicative identities). In order to do that we develop a relative of the Tutte polynomial of a semimatroid. Our dichromatic polynomial includes the classical graph one by Tutte, Zaslavsky's two for gain graphs, Noble and Welsh's for graphs with positive integer weights, and that of rooted integral gain graphs by Forge and Zaslavsky. It is not a universal Tutte invariant of weighted gain graphs; that remains to be found.An evaluation of one example of our polynomial counts proper list colorations of the gain graph from a color set with a gain-group action. When the gain group is Zd, the lists are order ideals in the integer lattice Zd, and there are specified upper bounds on the colors, then there is a formula for the number of bounded proper colorations that is a piecewise polynomial function of the upper bounds, of degree nd where n is the order of the graph.This example leads to graph-theoretical formulas for the number of integer lattice points in an orthotope but outside a finite number of affinographic hyperplanes, and for the number of n×d integral matrices that lie between two specified matrices but not in any of certain subspaces defined by simple row equations.

CONSTRUCTIVE APPROXIMATION BY NEURAL NETWORKS WITH POSITIVE INTEGER WEIGHTS

In this paper, we study a constructive approximation by neural networks with positive integer weights. Like neural networks with real weights, we show that neural networks with positive integer weights can even approximate arbitrarily well for any continuous functions on compact subsets of $\mathbb{R}$. We give a numerical result to justify our theoretical result.

Efficient approximation algorithms for bandwidth consecutive multicolorings of graphs

Let G be a graph in which each vertex v has a positive integer weight b(v) and each edge (v,w) has a nonnegative integer weight b(v,w). A bandwidth consecutive multicoloring, simply called a b-coloring of G, assigns each vertex v a specified number b(v) of consecutive positive integers as colors of v so that, for each edge (v,w), all integers assigned to vertex v differ from all integers assigned to vertex w by more than b(v,w). The maximum integer assigned to vertices is called the span of the coloring. The b-coloring problem asks to find a b-coloring of a given graph G with the minimum span. In the paper, we present four efficient approximation algorithms for the problem, which have theoretical performance guarantees for the computation time, the span of a found b-coloring and the approximation ratio. We also obtain several upper bounds on the minimum span, expressed in terms of the maximum b-degrees, one of which is an extension of Brooks' theorem on an ordinary coloring.

RELIABILITY AND IMPORTANCE MEASURES OF WEIGHTED-k-OUT-OF-n SYSTEM

Recently many popular reliability systems are being extended to weighted systems where weight associated with each component refers to load/capacity of that component. The simplest k-out-of-n system extended under this concept is known as weighted-k-out-of-n system. A weighted-k-out-of-n: G(F) system consists of n components each one having a positive integer weight wi, i = 1, 2, …, n such that the total system weight is w and the system works (fails) if and only if the accumulated weight of working (failed) components is at least k. In this paper, we introduce the concept of Weighted Markov Bernoulli Trial (WMBT) {Xi, i ≥ 0} and study the distribution of [Formula: see text], the total weight of successes in the sequence of n WMBT and conditional distribution of [Formula: see text] given that uth trial results in failure/success. We refer the distribution of [Formula: see text] as Weighted Markov Binomial Distribution (WMBD). Further we discuss application of WMBD and conditional WMBD in evaluation of reliability, Birnbaum reliability importance (B-importance) and improvement potential importance of weighted-k-out-of-n: G/F system. The numerical work is included to demonstrate the computational simplicity of the developed results. Further we compare our study with the existing results in terms of efficiency and find that our results are efficient for large values of k.