Integer Weights Research Articles

An efficient algorithm for packing cuts and (2,3)-metrics in a planar graph with three holes

We consider a planar graph G in which the edges have nonnegative integer lengths such that the length of every cycle of G is even, and three faces are distinguished, called holes in G. It is known that there exists a packing of cuts and (2,3)-metrics with nonnegative integer weights in G which realizes the distances within each hole. We develop a purely combinatorial strongly polynomial-time algorithm to find such a packing.

Integer linear programming models for the weighted total domination problem

A total dominating set of a graph G=(V,E) is a subset D of V such that every vertex in V (including the vertices from D) has at least one neighbour in D. Suppose that every vertex v ∈ V has an integer weight w(v) ≥ 0 and every edge e ∈ E has an integer weight w(e) ≥ 0. Then the weighted total domination (WTD) problem is to find a total dominating set D which minimizes the cost f(D):=∑u∈Dw(u)+∑e∈E[D]w(e)+∑v∈V∖Dmin{w(uv)|u∈N(v)∩D}. In this paper, we put forward three integer linear programming (ILP) models with a polynomial number of constraints, and present some numerical results implemented on random graphs for WTD problem.

Parameterized Analysis of Multiobjective Evolutionary Algorithms and the Weighted Vertex Cover Problem.

Evolutionary multiobjective optimization for the classical vertex cover problem has been analysed in Kratsch and Neumann (2013) in the context of parameterized complexity analysis. This article extends the analysis to the weighted vertex cover problem in which integer weights are assigned to the vertices and the goal is to find a vertex cover of minimum weight. Using an alternative mutation operator introduced in Kratsch and Neumann (2013), we provide a fixed parameter evolutionary algorithm with respect to , the cost of an optimal solution for the problem. Moreover, we present a multiobjective evolutionary algorithm with standard mutation operator that keeps the population size in a polynomial order by means of a proper diversity mechanism, and therefore, manages to find a 2-approximation in expected polynomial time. We also introduce a population-based evolutionary algorithm which finds a -approximation in expected time .

Faster FPTASes for counting and random generation of Knapsack solutions

In the #P-complete problem of counting 0/1 Knapsack solutions, the input consists of a sequence of n nonnegative integer weights w1,…,wn and an integer C, and we have to find the number of subsequences (subsets of indices) with total weight at most C. We give faster and simpler fully polynomial-time approximation schemes (FPTASes) for this problem, and for its random generation counterpart. Our method is based on dynamic programming and discretization of large numbers through floating-point arithmetic. We improve both deterministic counting FPTASes from Gopalan et al. (2011) [9], Štefankovič et al. (2012) [6] and the randomized counting and random generation algorithms in Dyer (2003) [5].Our method is general, and it can be directly applied on top of combinatorial decompositions (such as dynamic programming solutions) of various problems. For example, we also improve the complexity of the problem of counting 0/1 Knapsack solutions in an arc-weighted DAG.

Laurent Positivity of Quantized Canonical Bases for Quantum Cluster Varieties from Surfaces

In 2006, Fock and Goncharov constructed a nice basis of the ring of regular functions on the moduli space of framed ${\rm PGL}_2$-local systems on a punctured surface $S$. The moduli space is birational to a cluster $\mathcal{X}$-variety, whose positive real points recover the enhanced Teichm\"uller space of $S$. Their basis is enumerated by integral laminations on $S$, which are collections of closed curves in $S$ with integer weights. Around ten years later, a quantized version of this basis, still enumerated by integral laminations, was constructed by Allegretti and Kim. For each choice of an ideal triangulation of $S$, each quantum basis element is a Laurent polynomial in the exponential of quantum shear coordinates for edges of the triangulation, with coefficients being Laurent polynomials in $q$ with integer coefficients. We show that these coefficients are Laurent polynomials in $q$ with positive integer coefficients. Our result was expected in a positivity conjecture for framed protected spin characters in physics and provides a rigorous proof of it, and may also lead to other positivity results, as well as categorification. A key step in our proof is to solve a purely topological and combinatorial ordering problem about an ideal triangulation and a closed curve on $S$. For this problem we introduce a certain graph on $S$, which is interesting in its own right.

Bit-Parallelism Score Computation with Multi Integer Weight

Bit-Parallelism Score Computation with Multi Integer Weight

Exact Algorithms via Monotone Local Search

We give a new general approach for designing exact exponential-time algorithms for subset problems . In a subset problem the input implicitly describes a family of sets over a universe of size n and the task is to determine whether the family contains at least one set. A typical example of a subset problem is W EIGHTED d -SAT. Here, the input is a CNF-formula with clauses of size at most d , and an integer W . The universe is the set of variables and the variables have integer weights. The family contains all the subsets S of variables such that the total weight of the variables in S does not exceed W and setting the variables in S to 1 and the remaining variables to 0 satisfies the formula. Our approach is based on “monotone local search,” where the goal is to extend a partial solution to a solution by adding as few elements as possible. More formally, in the extension problem, we are also given as input a subset X of the universe and an integer k . The task is to determine whether one can add at most k elements to X to obtain a set in the (implicitly defined) family. Our main result is that a c k n O(1) time algorithm for the extension problem immediately yields a randomized algorithm for finding a solution of any size with running time O ((2−1/ c ) n ). In many cases, the extension problem can be reduced to simply finding a solution of size at most k . Furthermore, efficient algorithms for finding small solutions have been extensively studied in the field of parameterized algorithms. Directly applying these algorithms, our theorem yields in one stroke significant improvements over the best known exponential-time algorithms for several well-studied problems, including d -H ITTING S ET , F EEDBACK V ERTEX S ET , N ODE U NIQUE L ABEL C OVER , and W EIGHTED d -SAT. Our results demonstrate an interesting and very concrete connection between parameterized algorithms and exact exponential-time algorithms. We also show how to derandomize our algorithms at the cost of a subexponential multiplicative factor in the running time. Our derandomization is based on an efficient construction of a new pseudo-random object that might be of independent interest. Finally, we extend our methods to establish new combinatorial upper bounds and develop enumeration algorithms.

A robust performance enhancement of primary H∞ controller based on auto-selection of adjustable fractional weights: Application on a permanent magnet synchronous motor

This paper proposes a new robustification strategy of a primary H∞ controller based on a systematic selection of adequate fractional weights. Its robust performance (RP) margin is well enhanced with respect to unstructured multiplicative uncertainties presented in linear feedback systems. The proposed robustification strategy provides a robustified H∞ controller when following proposed hierarchical control is well respected. First, the primary [Formula: see text] controller is synthesized from solving a weighted-mixed sensitivity H∞ problem using initial integer weights. Thus, an initial RP margin is obtained. Second, an automatic selection of adjustable fractional weights is performed by the particle swarm optimization algorithm, in which some proposed tuning rules are accordingly well satisfied. Third, frequency response data of these weights are computed and then fitted by corresponding approximated integer weights using a frequency identification technique. Finally, these weights reformulate a new weighted-mixed sensitivity problem. The optimal solution to this problem updates the previous initial RP margin. These last three steps are repeated as the updated RP margin is diminished. Otherwise, the proposed hierarchical control is achieved by selecting the best adjustable fractional weights, providing, therefore, the best approximated integer weights and leading, therefore, to the robustified H∞ controller. In order to confirm the effectiveness of our proposed hierarchical control, primary and robustified H∞ controllers are applied on a permanent magnet synchronous motor where its actual behavior is modeled by an unstructured multiplicative uncertain model. The results obtained are compared in frequency domains using the singular value plots of their sensitivity functions. Otherwise, the same results are compared in time domains using the Powersim® software.

Explicit calculations with Eisenstein series

We find explicit change-of-basis formulas between Eisenstein series attached to cusps, and newform Eisenstein series attached to pairs of primitive Dirichlet characters. As a consequence, we prove a Bruggeman–Kuznetsov formula for newforms of square-free level and trivial nebentypus.We also derive, in explicit form, the Fourier expansion, Hecke eigenvalues, Atkin–Lehner pseudo-eigenvalues, and other data associated to these Eisenstein series, with arbitrary integer weight, level, and nebentypus.

The unit acquisition number of a graph

Let G be a graph with nonnegative integer weights. A unit acquisition move transfers one unit of weight from a vertex to a neighbor that has at least as much weight. The unit acquisition number of a graph G, denoted au(G), is the minimum size that the set of vertices with positive weight can be reduced to via successive unit acquisition moves when starting from the configuration in which every vertex has weight 1.For a graph G with n vertices and minimum degree k, we prove au(G)≤(n−1)∕k, with equality for complete graphs and C5. Also au(G) is at most the minimum size of a maximal matching in G, with equality on an infinite family of graphs. Furthermore, au(G) is bounded by the maximum degree and by n−1 when G is an n-vertex tree with diameter at most 4. We also construct arbitrarily large trees with maximum degree 5 having unit acquisition number 1, obtain a linear-time algorithm to compute the unit acquisition number of a caterpillar, and show that au(G)=1 when G has diameter 2, except that the value is 2 for C5 and the Petersen graph.

A risk score for carotid plaque as an assessment risk of cardiovascular risk among patients with hypertension

This study aimed to describe the status of carotid plaques and develop a simple scoring system to predict the risk of carotid lesions in patients with hypertension. Basic testing for carotid plaques was carried out and used for risk score development (the training dataset, n = 2665) and validation (the test dataset, n = 1333). Independent predictors of carotid plaques from the multivariate model were assigned integer weights based on their coefficients and incorporated into a risk score. The discriminant ability of the score was tested by receiver operating characteristic analysis using the test dataset. A total of 1346 of 2665 patients were examined for carotid plaques, which were more frequent in men than in women, and increased with age. The final model included eight significant variables, and these variables were then used to develop a risk score for the prediction of carotid plaques. Receiver operating characteristic analysis demonstrated good discriminant power with a C-statistic of 0.732 (95% confidence interval: 0.713–0.751) and good calibration across quantiles of observed predicted risk (74.6%). We developed a simple risk score for the prediction of carotid plaques based on eight variables. The prediction model showed good discriminant power and calibration.

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.

Distributed Integer Weight Balancing in the Presence of Time Delays in Directed Graphs

A digraph with positive weights on its edges is weight-balanced if, for each node, the sum of the weights of the incoming edges is equal to the sum of the weights of the outgoing edges. Weight-balanced digraphs play an important role in a variety of cooperative control problems, including formation control, event-triggered coordination, distributed averaging, and optimization. Classical distributed algorithms for asymptotic average consensus typically assume timely and reliable exchange of information between neighboring components of a given multicomponent system. These assumptions are not necessarily valid in practice due to varying delays that might affect computations at different nodes and/or transmissions at different links. In this paper, we propose a distributed algorithm that solves the integer weight balancing problem in the presence of arbitrary (time-varying and inhomogeneous) time delays that might affect the transmission at a particular link at a particular time. The algorithm converges after a finite number of steps (that we explicitly bound) in the presence of bounded delays and converges in finite time (with probability one) in the case of unbounded delays (packet drops). Furthermore, we show that the resulting weight-balanced digraph is unique regardless of how time delays or packet drops manifest themselves. We also analyze the computational and communication complexity of the proposed algorithm, and provide examples to illustrate its operation and performance.

Correction to: On minimum sum representations for weighted voting games

A proposal in a weighted voting game is accepted if the sum of the (nonnegative) weights of the “yea” voters is at least as large as a given quota. Several authors have considered representations of weighted voting games with minimum sum, where the weights and the quota are restricted to be integers. Here we correct the classification of all weighted voting games consisting of 9 voters which do not admit a unique minimum sum integer weight representation.

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.

On Almost Monge All Scores Matrices

The all scores matrix of a grid graph is a matrix containing the optimal scores of paths from every vertex on the first row of the graph to every vertex on its last row. This matrix is commonly used to solve diverse string comparison problems. All scores matrices have the Monge property, and this was exploited by previous works that used all scores matrices for solving various problems. In this paper, we study an extension of grid graphs that contain an additional set of edges, called bridges. Our main result is to show several properties of the all scores matrices of such graphs. We also apply these properties to obtain an $$O(r(nm+n^2))$$ time algorithm for constructing the all scores matrix of an $$m\times n$$ grid graph with r bridges and bounded integer weights.

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) \(|E|_{w}/|{\mathscr {T}}_{G}|_{w}\ge 2\).

Quasi-periodic β-expansions and cut languages

Motivated by the analysis of neural net models between integer and rational weights, we introduce a so-called cut language over a real digit alphabet, which contains finite β-expansions (i.e. base-β representations) of the numbers less than a given threshold. We say that an infinite β-expansion is eventually quasi-periodic if its tail sequence formed by the numbers whose representations are obtained by removing leading digits, contains an infinite constant subsequence. We prove that a cut language is regular iff its threshold is a quasi-periodic number whose all β-expansions are eventually quasi-periodic, by showing that altogether they have a finite number of tail values. For algebraic bases β, we prove that there is an eventually quasi-periodic β-expansion with an infinite number of tail values iff there is a conjugate of β on the unit circle. For transcendental β combined with algebraic digits, a β-expansion is eventually quasi-periodic iff it has a finite number of tail values. For a Pisot base β and digits from the smallest field extension Q(β) over rational numbers including β, we show that any number from Q(β) is quasi-periodic. In addition, we achieve a dichotomy that a cut language is either regular or non-context-free and we show that any cut language with rational parameters is context-sensitive.

Computing Weighted Strength and Applications to Partitioning

We study the mathematical concept of the strength of a graph as defined by Cunningham to partition very large graphs. The strength is the objective value of an attack problem on a graph where the striker aims at dividing the graph into a maximum number of connected components removing as few edges as possible per component uncoupled. The strength is the optimal number of edges per component uncoupled or, in the weighted version, the sum of the weights of the edges removed per component uncoupled. Given a connected graph with $n$ vertices and $m$ weighted edges, we denote by $W$ the total sum of the integer weights, and we describe an algorithm that approximates within a factor of $1+\varepsilon$ the weighted strength in time $O(m\log(W)\log^3(n)/\varepsilon^2)$, and that can be implemented with $O(m)$ memory use if we allow a complexity of $O(m\log^2(W)\log^3(n)/\varepsilon^2)$. This result has several important applications, in particular for the detection of dense areas of a graph, in terms of weighted edges, which in turn can be used for spam detection, community identification, and partitioning.

Independence and Efficient Domination on P 6 -free Graphs

In the M aximum W eight I ndependent S et problem, the input is a graph G , every vertex has a non-negative integer weight, and the task is to find a set S of pairwise nonadjacent vertices, maximizing the total weight of the vertices in S . We give an n O (log 2 n ) time algorithm for this problem on graphs excluding the path P 6 on 6 vertices as an induced subgraph. Currently, there is no constant k known for which M aximum W eight I ndependent S et on P k -free graphs becomes NP-hard, and our result implies that if such a k exists, then k > 6 unless all problems in NP can be decided in quasi-polynomial time. Using the combinatorial tools that we develop for this algorithm, we also give a polynomial-time algorithm for M aximum W eight E fficient D ominating S et on P 6 -free graphs. In this problem, the input is a graph G , every vertex has an integer weight, and the objective is to find a set S of maximum weight such that every vertex in G has exactly one vertex in S in its closed neighborhood or to determine that no such set exists. Prior to our work, the class of P 6 -free graphs was the only class of graphs defined by a single forbidden induced subgraph on which the computational complexity of M aximum W eight E fficient D ominating S et was unknown.