Nonnegative Weights Research Articles

A PTAS for minimum weighted connected vertex cover $$P_3$$ P 3 problem in 3-dimensional wireless sensor networks

Given a connected and weighted graph $$G=(V, E)$$G=(V,E) with each vertex v having a nonnegative weight w(v), the minimum weighted connected vertex cover $$P_{3}$$P3 problem $$(MWCVCP_{3})$$(MWCVCP3) is required to find a subset C of vertices of the graph with minimum total weight, such that each path with length 2 has at least one vertex in C, and moreover, the induced subgraph G[C] is connected. This kind of problem has many applications concerning wireless sensor networks and ad hoc networks. When homogeneous sensors are deployed into a three-dimensional space instead of a plane, the mathematical model for the sensor network is a unit ball graph instead of a unit disk graph. In this paper, we propose a new concept called weak c-local and give the first polynomial time approximation scheme (PTAS) for $$MWCVCP_{3}$$MWCVCP3 in unit ball graphs when the weight is smooth and weak c-local.

A note on weighted rooted trees

Let T be a tree rooted at r. Two vertices of T are related if one is a descendant of the other; otherwise, they are unrelated. Two subsets A and B of V(T) are unrelated if, for any a∈A and b∈B, a and b are unrelated. Let ω be a nonnegative weight function defined on V(T) with ∑v∈V(T)ω(v)=1. In this note, we prove that either there is an (r,u)-path P with ∑v∈V(P)ω(v)≥13 for some u∈V(T), or there exist unrelated sets A,B⊆V(T) such that ∑a∈Aω(a)≥13 and ∑b∈Bω(b)≥13. The bound 13 is tight. This answers a question posed in a very recent paper of Bonamy, Bousquet and Thomassé.

Complexity of the cluster deletion problem on subclasses of chordal graphs

We consider the following vertex-partition problem on graphs, known as the CLUSTER DELETION (CD) problem: given a graph with real nonnegative edge weights, partition the vertices into clusters (in this case, cliques) to minimize the total weight of edges outside the clusters. The decision version of this optimization problem is known to be NP-complete even for unweighted graphs and has been studied extensively. We investigate the complexity of the decision CD problem for the family of chordal graphs, showing that it is NP-complete for weighted split graphs, weighted interval graphs and unweighted chordal graphs. We also prove that the problem is NP-complete for weighted cographs. Some polynomial-time solvable cases of the optimization problem are also identified, in particular CD for unweighted split graphs, unweighted proper interval graphs and weighted block graphs.

Gauss-type quadrature rule with complex nodes and weights for integrals involving Daubechies scale functions and wavelets

This paper deals with derivation of a Gauss-type quadrature rule (named as Gauss–Daubechies quadrature rule) for numerical evaluation of integrals involving product of integrable function and Daubechies scale functions/wavelets. Some of the nodes and weights of the quadrature formula may be complex and appear with their conjugates. This is in contrast with the standard Gauss-type quadrature rule for integrals involving products of integrable functions and non-negative weight functions. This quadrature rule has accuracy as good as the standard Gauss-type quadrature rule and is also found to be stable. The efficacy of the quadrature rule derived here has been tested through some numerical examples.

Playing weighted Tron on trees

We consider the weighted version of the Tron game on graphs where two players, Alice and Bob, each build their own path by claiming one vertex at a time, starting with Alice. The vertices carry non-negative weights that sum up to 1 and either player tries to claim a path with larger total weight than the opponent. We show that if the graph is a tree then Alice can always ensure she receives 1/5 less than Bob or better, and this is best possible.

Vertex coloring edge-weighted digraphs

A coloring of a digraph with non-negative edge weights is a partition of the vertex set into independent sets, where a set is independent if the weighted in-degree of each node within the set is less than 1. We give constructive optimal bounds on the chromatic number in terms of maximum in-degree and inductiveness of the graph.

Balanced splitting on weighted intervals

We study a problem on splitting intervals. Let I be a set of n intervals on a line L, with non-negative weights. Given any integer k≥1, we want to find k points on L to partition L into k+1 segments, such that the maximum cost of these segments is minimized, where the cost of each segment s is the sum of the weights of the intervals in I intersecting s. We present an O(nlogn) time algorithm for this problem.

Semi-supervised local ridge regression for local matching based face recognition

In this paper, a novel algorithm named Semi-supervised Local Ridge Regression (SSLRR) is proposed for local matching based face recognition. Compared with other algorithms, the proposed algorithm possesses two advantages. Firstly, SSLRR utilizes a multiple graph based semi-supervised technique to propagate the class labels of labeled samples to the unlabeled ones. Thus, the information of both labeled and unlabeled data can be employed in our algorithm to improve its performance. Secondly, unlike most local matching based face recognition algorithms which assume different sub-images from the same face are independent, an adaptive non-negative weight vector is introduced into our SSLRR to combine the Laplacian matrices obtained by different sub-images. Therefore, the latent complementary information of multiple sub-patterns from the same face image can be taken into account. Moreover, a simple yet efficient iterative update scheme is also proposed to solve our SSLRR model. Extensive experiments are performed on five standard face databases (Yale, Extended YaleB, AR, CMU PIE and LFW) to demonstrate the efficiency of the proposed algorithm. Experimental results show that SSLRR obtains better recognition performance than some other state-of-the-art approaches.

Approximation algorithms for clique transversals on some graph classes

Given a graph G=(V,E) a clique is a maximal subset of pairwise adjacent vertices of V of size at least 2. A clique transversal is a subset of vertices that intersects the vertex set of each clique of G. Finding a minimum-cardinality clique transversal is NP-hard for the following classes: planar, line and bounded degree graphs. For line graphs we present a 3-approximation for the unweighted case and a 4-approximation for the weighted case with nonnegative weights; a ⌈(Δ(G)+1)/2⌉-approximation for bounded degree graphs and a 3-approximation for planar graphs.

Fast Best-Effort Search on Graphs with Multiple Attributes

We address the problem of search on graphs with multiple nodal attributes. We call such graphs weighted attribute graphs (WAGs). Nodes of a WAG exhibit multiple attributes with varying, non-negative weights. WAGs are ubiquitous in real-world applications. For example, in a co-authorship WAG, each author is a node; each attribute corresponds to a particular topic (e.g., databases, data mining, and machine learning); and the amount of expertise in a particular topic is represented by a non-negative weight on that attribute. A typical search in this setting specifies both connectivity between nodes and constraints on weights of nodal attributes. For example, a user's search may be: find three coauthors (i.e., a triangle) where each author's expertise is greater than 50 percent in at least one topic area (i.e., attribute). We propose a ranking function which unifies ranking between the graph structure and attribute weights of nodes. We prove that the problem of retrieving the optimal answer for graph search on WAGs is NP-complete. Moreover, we propose a fast and effective top-k graph search algorithm for WAGs. In an extensive experimental study with multiple real-world graphs, our proposed algorithm exhibits significant speed-up over competing approaches. On average, our proposed method is more than 7χ faster in query processing than the best competitor.

Approximation algorithm for the balanced 2-connected k-partition problem

For two positive integers m,k and a connected graph G=(V,E) with a nonnegative vertex weight function w, the balanced m-connected k-partition problem, denoted as BCmPk, is to find a partition of V into k disjoint nonempty vertex subsets (V1,V2,…,Vk) such that each G[Vi] (the subgraph of G induced by Vi) is m-connected, and min1≤i≤k⁡{w(Vi)} is maximized. The optimal value of BCmPk on graph G is denoted as βm⁎(G,k), that is, βm⁎(G,k)=max⁡min1≤i≤k⁡{w(Vi)}, where the maximum is taken over all m-connected k-partition of G. In this paper, we study the BC2Pk problem on interval graphs, and obtain the following results.(1) For k=2, a 4/3-approximation algorithm is given for BC2P2 on 4-connected interval graphs.(2) In the case that there exists a vertex v with weight at least W/k, where W is the total weight of the graph, we prove that the BC2Pk problem on a 2k-connected interval graph G can be reduced to the BC2Pk−1 problem on the (2k−1)-connected interval graph G−v. In the case that every vertex has weight at most W/k, we prove a lower bound β2⁎(G,k)≥W/(2k−1) for 2k-connected interval graph G.(3) Assuming that weight w is integral, a pseudo-polynomial time algorithm is obtained. Combining this pseudo-polynomial time algorithm with the above lower bound, a fully polynomial time approximation scheme (FPTAS) is obtained for the BC2Pk problem on 2k-connected interval graphs.

Delay Optimization Via Packet Scheduling for Multi-Path Routing in Wireless Networks

In this paper, the delay optimization problem of multi-path routing in wireless networks is studied. We propose a packet scheduling algorithm weighted shortest delay (WSD) for multi-path routing with the objective minimizing the total weighted delay of a set of packets. To solve the issue of differentiated quality of service, WSD algorithm assigns a nonnegative weight for each packet of every kind of wireless network application. At each network node along the path from source to destination, whenever a link of the node becomes idle, WSD algorithm transmits the packet with largest ratio of its weight to packet length among available packets which have arrived but not yet transmitted. Theoretical analysis proves that WSD algorithm is asymptotically optimal for the total weighted delay of transmitted packets if the arrival times, weights and transmission times of packets are bounded. Real experiment results further verify that WSD algorithm can enhance the delay performance of multi-path routing protocol profoundly.

Min st -Cut Oracle for Planar Graphs with Near-Linear Preprocessing Time

For an undirected n -vertex planar graph G with nonnegative edge weights, we consider the following type of query: given two vertices s and t in G , what is the weight of a min st -cut in G ? We show how to answer such queries in constant time with O ( n log 4 n ) preprocessing time and O ( n log n ) space. We use a Gomory-Hu tree to represent all the pairwise min cuts implicitly. Previously, no subquadratic time algorithm was known for this problem. Since all-pairs min cut and the minimum-cycle basis are dual problems in planar graphs, we also obtain an implicit representation of a minimum-cycle basis in O ( n log 4 n ) time and O ( n log n ) space. Additionally, an explicit representation can be obtained in O ( C ) time and space where C is the size of the basis. These results require that shortest paths are unique. This can be guaranteed either by using randomization without overhead or deterministically with an additional log 2 n factor in the preprocessing times.

Goal attainment method and Taylor series approximation to solve multi-objective linear fractional programming problem

This paper illustrates the use of goal attainment method with Taylor series approximation to produce a set of Pareto optimal solutions of a multi-objective linear fractional programming problem (MOLFPP). The relative gap in between the aspired goal and the feasible objective space is minimised along different search directions by introducing several non-negative and normalised weight vectors. To justify the effectiveness of our proposed method, a practical problem with numerical example is solved and the results are compared with that of existing fuzzy max-min operator method.

Learning Understandable Neural Networks With Nonnegative Weight Constraints

People can understand complex structures if they relate to more isolated yet understandable concepts. Despite this fact, popular pattern recognition tools, such as decision tree or production rule learners, produce only flat models which do not build intermediate data representations. On the other hand, neural networks typically learn hierarchical but opaque models. We show how constraining neurons' weights to be nonnegative improves the interpretability of a network's operation. We analyze the proposed method on large data sets: the MNIST digit recognition data and the Reuters text categorization data. The patterns learned by traditional and constrained network are contrasted to those learned with principal component analysis and nonnegative matrix factorization.

Scaling limits of random planar maps with a unique large face

We study random bipartite planar maps defined by assigning nonnegative weights to each face of a map. We prove that for certain choices of weights a unique large face, having degree proportional to the total number of edges in the maps, appears when the maps are large. It is furthermore shown that as the number of edges $n$ of the planar maps goes to infinity, the profile of distances to a marked vertex rescaled by $n^{-1/2}$ is described by a Brownian excursion. The planar maps, with the graph metric rescaled by $n^{-1/2}$, are then shown to converge in distribution toward Aldous' Brownian tree in the Gromov-Hausdorff topology. In the proofs, we rely on the Bouttier-di Francesco-Guitter bijection between maps and labeled trees and recent results on simply generated trees where a unique vertex of a high degree appears when the trees are large.

Asymptotics of Suprema of Weighted Gaussian Fields with Applications to Kernel Density Estimators

For centered stationary Gaussian fields $X(s), s\in {\bf R}^d, d>1,$ we show that properly centered and normalized $\sup_{s\in {\bf R}^d}w(s/T)|X(s)|$ with a nonnegative weight function $w$ converges in distribution to a double exponential law as $T\to\infty$, provided that $w$ and the covariance function of $X$ satisfy certain smoothness and regularity conditions. This limit theorem extends the results for compact sets without weight functions obtained earlier in [V. Piterbarg, Asymptotics Methods in the Theory of Gaussian Processes and Fields, Amer. Math. Soc., Providence, RI, 1996]. This new result for Gaussian fields is then applied to obtain necessary and sufficient conditions for the properly centered and normalized sequence $\sup_{x\in {\bf R}^d}|\Psi(x)(\widehat{f}_n(x)-\e\,\widehat{f}_n(x))|$ to converge in distribution to a double exponential law under natural smoothness conditions, where $\widehat{f}_n$ denotes the kernel density estimator of the bounded continuous density $f$ on ${\bf R}^d$ based on the sample of size $n$, and $\Psi$ is a positive continuous function such that $\sup_{x\in {\bf R}^d}|\Psi(x) f(x)^\beta|<\infty$ for some $\beta\in (0, 1/2)$. This paper extends results of the paper [E. Giné, V. Koltchinskii, and L. Sakhanenko, Probab. Theory Related Fields, 130 (2004), pp. 167--198] to the case of $d>1$. It also extends the results of [E. Rio, Probab. Theory Related Fields, 98 (1994), pp. 21--45] for $\Psi(\cdot)=f^{-1/2}(\cdot)I_S(\cdot)$ with some compact set $S$ in ${\bf R}^d$ to a general class of functions $\Psi$. An example of an application completes this work.

Inequalities for some systems of orthogonal polynomials using reproducing kernel functions

In this paper we are engaged with classical orthogonal polynomials. We deduce relations for the reproducing kernels for these polynomials without derivations. By the using this form of reproducing kernels of classical orthogonal polynomials we deduce some inequalities for polynomials with special weight function. This enabled us to derive some inequalities for orthonormal polynomial with special weight function. It allows us to deduce general statement for every orthonormal polynomial with a nonnegative weight function on a finite interval of orthogonality. Mathematics Subject Classification: 33C45, 33C47

NETWORK MODEL OF SPREADING OF SEVERAL ACTIVITY TYPES AMONG COMPLEX AGENTS AND ITS APPLICATIONS

The paper describes the main principles of a model that simulates distribution of several types of activity among the agents having internal structure. The potential interaction of agents is given by a weighted graph. The vertices of the graph are heterogeneous automata with infinitely many states, and edges correspond to the influence of vertices on each other. The activity is simulated by the propagation of the integer resource called chips. The vertices exchange chips of m different types along edges. Each edge of the network has m non-negative weights characterizing the capacity of each type. We define the set of set relationships and operations on chip types, so that the vertex can not only change its internal state, but also affect the activity of the network configuration in general. As an example, the model of the threshold interactions in the social network with two types of activity and five types of agents with different activation threshold is described.

A New Computational Method for the Sparsest Solutions to Systems of Linear Equations

The connection between the sparsest solution to an underdetermined system of linear equations and the weighted $\ell_1$-minimization problem is established in this paper. We show that seeking the sparsest solution to a linear system can be transformed to searching for the densest slack variable of the dual problem of weighted $\ell_1$-minimization with all possible choices of nonnegative weights. Motivated by this fact, a new reweighted $\ell_1$-algorithm for the sparsest solutions of linear systems, going beyond the framework of existing sparsity-seeking methods, is proposed in this paper. Unlike existing reweighted $\ell_1$-methods that are based on the weights defined directly in terms of iterates, the new algorithm computes a weight in dual space via certain convex optimization and uses such a weight to locate the sparsest solutions. It turns out that the new algorithm converges to the sparsest solutions of linear systems under some mild conditions that do not require the uniqueness of the sparsest so...