Positive Integer Weights Research Articles

Computational Methods for Reliability and Importance Measures of Weighted-Consecutive-System

We consider the two reliability systems, namely, the weighted-consecutive- k-out-of- n:F (Cω(k,n:F)), and weighted-m-consecutive- k-out-of- n:F system (Cmω(k,n:F)). The weighted-systems have in general n components, each one having a positive integer weight ωi, i=1,2,...n such that total weight of all components of the system is ω=Σi=1nωi. The Cω(k,n:F) fails iff the total weight of failed consecutive components is at least k. We propose Cmω(k,n:F) as a generalization of Cω(k,n:F) which fails iff there are at least m non-overlapping groups of consecutive failed components with a total weight of at least k. Here we study the reliability, Birnbaum reliability importance, and improvement potential importance of a weighted-consecutive system based on the distribution of the failure run statistic in the sequence of weighted Bernoulli trials. We develop a simplified, efficient formula for the evaluation of reliability and importance measures of the systems under consideration, and demonstrate the results numerically.

Vertex Nim played on graphs

Given a graph G with positive integer weights on the vertices, and a token placed on some current vertex u, two players alternately remove a positive integer weight from u and then move the token to a new current vertex adjacent to u. When the weight of a vertex is set to 0, it is removed and its neighborhood becomes a clique. The player making the last move wins. This adaptation of Nim on graphs is called Vertexnim, and slightly differs from the game Vertex NimG introduced by Stockman in 2004. Vertexnim can be played on both directed or undirected graphs. In this paper, we study the complexity of deciding whether a given game position of Vertexnim is winning for the first or second player. In particular, we show that for undirected graphs, this problem can be solved in quadratic time. Our algorithm is also available for the game Vertex NimG, thus improving Stockmanʼs exptime algorithm. In the directed case, we are able to compute the winning strategy in polynomial time for several instances, including circuits or digraphs with self loops.

‘Truncate, replicate, sample’: A method for creating integer weights for spatial microsimulation

Iterative proportional fitting (IPF) is a widely used method for spatial microsimulation. The technique results in non-integer weights for individual rows of data. This is problematic for certain applications and has led many researchers to favour combinatorial optimisation approaches such as simulated annealing. An alternative to this is ‘integerisation’ of IPF weights: the translation of the continuous weight variable into a discrete number of unique or ‘cloned’ individuals. We describe four existing methods of integerisation and present a new one. Our method – ‘truncate, replicate, sample’ (TRS) – recognises that IPF weights consist of both ‘replication weights’ and ‘conventional weights’, the effects of which need to be separated. The procedure consists of three steps: (1) separate replication and conventional weights by truncation; (2) replication of individuals with positive integer weights; and (3) probabilistic sampling. The results, which are reproducible using supplementary code and data published alongside this paper, show that TRS is fast, and more accurate than alternative approaches to integerisation.

Bandwidth consecutive multicolorings of graphs

Let G be a simple 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 of G assigns each vertex v a specified number b(v) of consecutive positive integers 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 a vertex is called the span of the coloring. In the paper, we first investigate fundamental properties of such a coloring. We then obtain a pseudo polynomial-time exact algorithm and a fully polynomial-time approximation scheme for the problem of finding such a coloring of a given series-parallel graph with the minimum span. We finally extend the results to the case where a given graph G is a partial k-tree, that is, G has a bounded tree-width.

Faster optimal algorithms for segment minimization with small maximal value

The segment minimization problem consists of finding a smallest set of binary matrices (segments), where non-zero values in each row of each matrix are consecutive, each matrix is assigned a positive integer weight (a segment-value), and the weighted sum of the matrices corresponds to the given input intensity matrix. This problem has direct applications in intensity-modulated radiation therapy, an effective form of cancer treatment.We study here the special case when the largest value H in the intensity matrix is small. We show that for an intensity matrix with one row, this problem is fixed-parameter tractable (FPT) in H; our algorithm obtains a significant asymptotic speedup over the previous best FPT algorithm. We also show how to solve the full-matrix problem faster than all previously known algorithms. Finally, we address a closely related problem that deals with minimizing the number of segments subject to a minimum beam-on time, defined as the sum of the segment-values, and again improve the running time of previous algorithms. Our algorithms have running time O(mn) in the case that the matrix has only entries in {0,1,2}.

Fast Polynomial-Space Algorithms Using Inclusion-Exclusion

Given a graph with n vertices, k terminals and positive integer weights not larger than c, we compute a minimum Steiner Tree in $\mathcal{O}^{\star}(2^{k}c)$ time and $\mathcal{O}^{\star}(c)$ space, where the $\mathcal{O}^{\star}$ notation omits terms bounded by a polynomial in the input-size. We obtain the result by defining a generalization of walks, called branching walks, and combining it with the Inclusion-Exclusion technique. Using this combination we also give $\mathcal{O}^{\star}(2^{n})$ -time polynomial space algorithms for Degree Constrained Spanning Tree, Maximum Internal Spanning Tree and #Spanning Forest with a given number of components. Furthermore, using related techniques, we also present new polynomial space algorithms for computing the Cover Polynomial of a graph, Convex Tree Coloring and counting the number of perfect matchings of a graph.

Algorithm for Cost Non-preemptive Scheduling of Partial k-Trees

Let G be a graph, in which each vertex (job) v has a positive integer weight (processing time) p(v) and eachedge (u,v) represented that the pair of jobs u and v cannot be processed in the same slot. In this paper we assume that every job is non-preemptive. Let C={1,2,...} be a color set. A multicoloring (scheduling) F of G is to assign each job v a set of p(v) consecutive positive integers (processing consecutive time slots) in C so that any pair of adjacent vertices receive disjoint sets. Such a multicoloring is called a non-preemptive scheduling. The cost non-preemptive scheduling problem is to find an optimal multicoloring of G.

All-pairs shortest paths with a sublinear additive error

We show that, for every 0 ≤ p ≤ 1, there is an O ( n 2.575− p /(7.4−2.3 p ) )-time algorithm that given a directed graph with small positive integer weights, estimates the length of the shortest path between every pair of vertices u , v in the graph to within an additive error δ p ( u , v ), where δ( u , v ) is the exact length of the shortest path between u and v . This algorithm runs faster than the fastest algorithm for computing exact shortest paths for any 0 < p ≤ 1. Previously the only way to “beat” the running time of the exact shortest path algorithms was by applying an algorithm of Zwick [2002] that approximates the shortest path distances within a multiplicative error of (1 + ϵ). Our algorithm thus gives a smooth qualitative and quantitative transition between the fastest exact shortest paths algorithm, and the fastest approximation algorithm with a linear additive error. In fact, the main ingredient we need in order to obtain the above result, which is also interesting in its own right, is an algorithm for computing (1 + ϵ) multiplicative approximations for the shortest paths, whose running time is faster than the running time of Zwick's approximation algorithm when ϵ ≪ 1 and the graph has small integer weights.

An iterative approach to graph irregularity strength

An assignment of positive integer weights to the edges of a simple graph G is called irregular if the weighted degrees of the vertices are all different. The irregularity strength, s ( G ) , is the maximal edge weight, minimized over all irregular assignments, and is set to infinity if no such assignment is possible. In this paper, we take an iterative approach to calculating the irregularity strength of a graph. In particular, we develop a new algorithm that determines the exact value s ( T ) for trees T in which every two vertices of degree not equal to two are at distance at least eight.

On a reduction of the interval coloring problem to a series of bandwidth coloring problems

Given a graph G=(V,E) with strictly positive integer weights ? i on the vertices i?V, an interval coloring of G is a function I that assigns an interval I(i) of ? i consecutive integers (called colors) to each vertex i?V so that I(i)?I(j)=? for all edges {i,j}?E. The interval coloring problem is to determine an interval coloring that uses as few colors as possible. Assuming that a strictly positive integer weight ? ij is associated with each edge {i,j}?E, a bandwidth coloring of G is a function c that assigns an integer (called a color) to each vertex i?V so that |c(i)?c(j)|?? ij for all edges {i,j}?E. The bandwidth coloring problem is to determine a bandwidth coloring with minimum difference between the largest and the smallest colors used. We prove that an optimal solution of the interval coloring problem can be obtained by solving a series of bandwidth coloring problems. Computational experiments demonstrate that such a reduction can help to solve larger instances or to obtain better upper bounds on the optimal solution value of the interval coloring problem.

About equivalent interval colorings of weighted graphs

Given a graph G = ( V , E ) with strictly positive integer weights ω i on the vertices i ∈ V , a k -interval coloring of G is a function I that assigns an interval I ( i ) ⊆ { 1 , … , k } of ω i consecutive integers (called colors) to each vertex i ∈ V . If two adjacent vertices x and y have common colors, i.e. I ( i ) ∩ I ( j ) ≠ 0̸ for an edge [ i , j ] in G , then the edge [ i , j ] is said conflicting. A k -interval coloring without conflicting edges is said legal. The interval coloring problem (ICP) is to determine the smallest integer k , called interval chromatic number of G and denoted χ i n t ( G ) , such that there exists a legal k -interval coloring of G . For a fixed integer k , the k -interval graph coloring problem ( k -ICP) is to determine a k -interval coloring of G with a minimum number of conflicting edges. The ICP and k -ICP generalize classical vertex coloring problems where a single color has to be assigned to each vertex (i.e., ω i = 1 for all vertices i ∈ V ). Two k -interval colorings I 1 and I 2 are said equivalent if there is a permutation π of the integers 1 , … , k such that ℓ ∈ I 1 ( i ) if and only if π ( ℓ ) ∈ I 2 ( i ) for all vertices i ∈ V . As for classical vertex coloring, the efficiency of algorithms that solve the ICP or the k -ICP can be increased by avoiding considering equivalent k -interval colorings, assuming that they can be identified very quickly. To this purpose, we define and prove a necessary and sufficient condition for the equivalence of two k -interval colorings. We then show how a simple tabu search algorithm for the k -ICP can possibly be improved by forbidding the visit of equivalent solutions.

Edge searching weighted graphs

In traditional edge searching one tries to clean all of the edges in a graph employing the least number of searchers. It is assumed that each edge of the graph initially has a weight equal to one. In this paper we modify the problem and introduce the Weighted Edge Searching Problem by considering graphs with arbitrary positive integer weights assigned to its edges. We give bounds on the weighted search number in terms of related graph parameters including pathwidth. We characterize the graphs for which two searchers are sufficient to clear all edges. We show that for every weighted graph the minimum number of searchers needed for a not-necessarily-monotonic weighted edge search strategy is enough for a monotonic weighted edge search strategy, where each edge is cleaned only once. This result proves the NP-completeness of the problem.

Point-weight designs with design conditions on t points

This paper examines some of the properties of point-weight incidence structures, i.e. incidence structures for which every point is assigned a positive integer weight. In particular it examines point-weight designs with a design condition that stipulates that any two “identical” sets of t points must lie on the same number of blocks. We introduce a new class of designs with this property: row-sum designs, and examine the basic properties of row-sum point-weight designs and their similarities to classical (non-point-weight) designs and the point-weight designs of Horne [On point-weighted designs, Ph.D. Thesis, Royal Holloway, University of London, 1996].

Sparse connectivity certificates via MA orderings in graphs

For an undirected multigraph G = ( V , E ) , let α be a positive integer weight function on V. For a positive integer k, G is called ( k , α ) -connected if any two vertices u , v ∈ V remain connected after removal of any pair ( Z , E ′ ) of a vertex subset Z ⊆ V - { u , v } and an edge subset E ′ ⊆ E such that ∑ v ∈ Z α ( v ) + | E ′ | < k . The ( k , α ) -connectivity is an extension of several common generalizations of edge-connectivity and vertex-connectivity. Given a ( k , α ) -connected graph G, we show that a ( k , α ) -connected spanning subgraph of G with O ( k | V | ) edges can be found in linear time by using MA orderings. We also show that properties on removal cycles and preservation of minimum cuts can be extended in the ( k , α ) -connectivity.

An exact algorithm for the channel assignment problem

A channel assignment problem is a triple ( V , E , w ) where V is a vertex set, E is an edge set and w is a function assigning edges positive integer weights. An assignment c of integers between 1 and K to the vertices is proper if | c ( u ) - c ( v ) | ⩾ w ( uv ) for each uv ∈ E ; the smallest K for which there is a proper assignment is called the span. The input problem is set to be l-bounded if the values of w do not exceed l. We present an algorithm running in time O ( n ( l + 2 ) n ) which outputs the span for l-bounded channel assignment problems with n vertices. An algorithm running in time O ( nl ( l + 2 ) n ) for computing the number of different proper assignments of span at most K is further presented.

Multicolorings of Series-Parallel Graphs

Let G be a graph, and let each vertex v of G have a positive integer weight ω(v). A multicoloring of G is to assign each vertex v a set of ω(v) colors so that any pair of adjacent vertices receive disjoint sets of colors. This paper presents an algorithm to find a multicoloring of a given series-parallel graph G with the minimum number of colors in time O(nW), where n is the number of vertices and W is the maximum weight of vertices in G.

On graph irregularity strength

AbstractAn assignment of positive integer weights to the edges of a simple graph G is called irregular, if the weighted degrees of the vertices are all different. The irregularity strength, s(G), is the maximal weight, minimized over all irregular assignments. In this study, we show that s(G) ≤ c1 n/δ, for graphs with maximum degree Δ ≤ n1/2 and minimum degree δ, and s(G) ≤ c2(log n)n/δ, for graphs with Δ &gt; n1/2, where c1 and c2 are explicit constants. To prove the result, we are using a combination of deterministic and probabilistic techniques. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 120–137, 2002

The Lucas property of a number array

For all nonnegative integers i, j let w(i,j | a,b,c) denote the number of all paths in the plane from (0,0) to ( i, j) with steps (1,0), (0,1), (1,1), and with positive integer weights a, b, c, respectively. The divisibility property of the array w(i,j | a,b,c) is studied. The notation of the Lucas property is introduced. Let p be a prime and let w ̄ (i,j | a,b,c) denote the remainders of dividing w(i,j | a,b,c) by p where 0⩽ w ̄ (i,j | a,b,c)<p . The Lucas property of w(i,j | a,b,c) explains the self-similar pattern of w ̄ (i,j | a,b,c) . The principal clusters of higher orders are generated by the tensor powers of the principal cell containing w ̄ (i,j | a,b,c) where 0⩽ i< p and 0⩽ j< p.

An Analogue of Covering Space Theory for Ranked Posets

Suppose $P$ is a partially ordered set that is locally finite, has a least element, and admits a rank function. We call $P$ a weighted-relation poset if all the covering relations of $P$ are assigned a positive integer weight. We develop a theory of covering maps for weighted-relation posets, and in particular show that any weighted-relation poset $P$ has a universal cover $\tilde P\to P$, unique up to isomorphism, so that 1. $\tilde P\to P$ factors through any other covering map $P'\to P$; 2. every principal order ideal of $\tilde P$ is a chain; and 3. the weight assigned to each covering relation of $\tilde P$ is 1. If $P$ is a poset of "natural" combinatorial objects, the elements of its universal cover $\tilde P$ often have a simple description as well. For example, if $P$ is the poset of partitions ordered by inclusion of their Young diagrams, then the universal cover $\tilde P$ is the poset of standard Young tableaux; if $P$ is the poset of rooted trees ordered by inclusion, then $\tilde P$ consists of permutations. We discuss several other examples, including the posets of necklaces, bracket arrangements, and compositions.

A Decomposition Theorem for Maximum Weight Bipartite Matchings

Let G be a bipartite graph with positive integer weights on the edges and without isolated nodes. Let n, N, and W be the node count, the largest edge weight, and the total weight of G. Let k(x, y) be log x / log (x2/y). We present a new decomposition theorem for maximum weight bipartite matchings and use it to design an $O(\sqrt{n}W / k(n, W/N))$-time algorithm for computing a maximum weight matching of G. This algorithm bridges a long-standing gap between the best known time complexity of computing a maximum weight matching and that of computing a maximum cardinality matching. Given G and a maximum weight matching of G, we can further compute the weight of a maximum weight matching of G - {u} for all nodes u in O(W) time.