Non-decreasing Order Research Articles

A successive approximation algorithm for the multiple knapsack problem

It is well-known that the multiple knapsack problem is NP-hard, and does not admit an FPTAS even for the case of two identical knapsacks. Whereas the 0-1 knapsack problem with only one knapsack has been intensively studied, and some effective exact or approximation algorithms exist. A natural approach for the multiple knapsack problem is to pack the knapsacks successively by using an effective algorithm for the 0-1 knapsack problem. This paper considers such an approximation algorithm that packs the knapsacks in the nondecreasing order of their capacities. We analyze this algorithm for 2 and 3 knapsack problems by the worst-case analysis method and give all their error bounds.

Computational experiments with a lazy version of a K quickest simple path ranking algorithm

The quickest path problem is related to the classical shortest path problem, but its objective function concerns the transmission time of a given amount of data throughout a path, which involves both cost and capacity. The K-quickest simple paths problem generalises the latter, by looking for a given number K of simple paths in non-decreasing order of transmission time.

Score Lists in Tripartite Hypertournaments

Given non-negative integers l, m, n, α, β and γ with l ≥ α ≥ 1, m ≥ β ≥ 1 and n ≥ γ ≥ 1, an [α,β,γ]-tripartite hypertournament on l + m + n vertices is a four tuple (U, V, W, E), where U, V and W are three sets of vertices with |U| = l , |V| = m and |W| = n, and E is a set of (α + β + γ)-tuples of vertices, called arcs, with exactly α vertices from U, exactly β vertices from V,and exactly γ vertices from W, such that any subset U1∪ V1∪ W1 of U∪ V∪ W, E contains exactly one of the (α + β + γ)! (α + β + γ) − tuples whose entries belong to U1∪ V1∪ W1. We obtain necessary and sufficient conditions for three lists of non-negative integers in non-decreasing order to be the losing score lists or score lists of some [α, β, γ]-tripartite hypertournament.

Mark sequences in multigraphs

An r -digraph is an orientation of a multigraph which has no loops and contains at most r edges between any pair of distinct vertices. We give a simple proof of necessary and sufficient conditions for a sequence of non-negative integers arranged in nondecreasing order to be a sequence of numbers, called marks or r -scores, attached to the vertices of an r -digraph.

Finding the K best policies in a finite-horizon Markov decision process

Directed hypergraphs represent a general modelling and algorithmic tool, which have been successfully used in many different research areas such as artificial intelligence, database systems, fuzzy systems, propositional logic and transportation networks. However, modelling Markov decision processes using directed hypergraphs has not yet been considered. In this paper we consider finite-horizon Markov decision processes ( MDPs) with finite state and action space and present an algorithm for finding the K best deterministic Markov policies. That is, we are interested in ranking the first K deterministic Markov policies in non-decreasing order using an additive criterion of optimality. The algorithm uses a directed hypergraph to model the finite-horizon MDP. It is shown that the problem of finding the optimal policy can be formulated as a minimum weight hyperpath problem and be solved in linear time, with respect to the input data representing the MDP, using different additive optimality criteria.

On-line scheduling of parallel jobs on two machines

We study the problem of on-line scheduling of parallel jobs on two machines. The jobs are parallel in the sense that each of them specifies the number of processors, in this case 1 or 2, required for simultaneous processing. The jobs are presented one by one. Upon receiving a job, we must assign the job to a time slot in the schedule before the next job is presented. No re-assignment is allowed. The goal is to minimize the makespan of the final schedule. There is a straightforward algorithm which achieves a competitive ratio of 2. In this paper we show that no on-line algorithm can have a competitive ratio less than 1 + 2 / 3 ( ≈ 1.816 ) . We also study two special cases of the problem: (i) Jobs arrive in a non-decreasing order of processing times where we give an optimal algorithm with competitive ratio 3/2; (ii) Jobs arrive in a non-increasing order of processing times where we show that no on-line algorithm has a competitive ratio less than 9/7 and give a greedy algorithm with a competitive ratio 4/3.

An automated reference point-like approach for multicriteria shortest path problems

In this paper we introduce a method of analysis for the automated ordering and selection of solutions of a multicriteria shortest path model. The method is based on a reference point approach, where the paths in a specific priority region are ranked by non-decreasing order of a Chebyshev metric. In order to list paths according with this objective function a labelling algorithm is proposed. The developed method is applied in a video-traffic routing context. Computational results are presented and analysed, for randomly generated networks of significant dimension.

Improved algorithms for the [formula omitted] maximum-sums problems

Given a sequence of n real numbers and an integer k , 1 ⩽ k ⩽ 1 2 n ( n - 1 ) , the k maximum-sum segments problem is to locate the k segments whose sums are the k largest among all possible segment sums. Recently, Bengtsson and Chen gave an O ( min { k + n log 2 n , n k } ) -time algorithm for this problem. Bae and Takaoka later proposed a more efficient algorithm for small k . In this paper, we propose an O ( n + k log ( min { n , k } ) ) -time algorithm for the same problem, which is superior to both of them when k is o ( n log n ) . We also give the first optimal algorithm for delivering the k maximum-sum segments in non-decreasing order if k ⩽ n . Then we develop an O ( n 2 d - 1 + k log min { n , k } ) -time algorithm for the d -dimensional version of the problem, where d > 1 and each dimension, without loss of generality, is of the same size n . This improves the best previously known O ( n 2 d - 1 C ) -time algorithm, also by Bengtsson and Chen, where C = min { k + n log 2 n , n k } . It should be pointed out that, given a two-dimensional array of size m × n , our algorithm for finding the k maximum-sum subarrays is the first one achieving cubic time provided that k is O ( m 2 n / log n ) .

Building k-connected neighborhood graphs for isometric data embedding

Isometric data embedding using geodesic distance requires the construction of a connected neighborhood graph so that the geodesic distance between every pair of data points can be estimated. This paper proposes an approach for constructing k-connected neighborhood graphs. The approach works by applying a greedy algorithm to add each edge, in a nondecreasing order of edge length, to a neighborhood graph if end vertices of the edge are not yet k-connected on the graph. The k-connectedness between vertices is tested using a network flow technique by assigning every vertex a unit flow capacity. This approach is applicable to a wide range of data. Experiments show that it gives better estimation of geodesic distances than other approaches, especially when the data are undersampled or nonuniformly distributed.

Majorization for Changes in Angles Between Subspaces, Ritz Values, and Graph Laplacian Spectra

Many inequality relations between real vector quantities can be succinctly expressed as “weak (sub)majorization” relations using the symbol ${\prec}_{w}$. We explain these ideas and apply them in several areas, angles between subspaces, Ritz values, and graph Laplacian spectra, which we show are all surprisingly related. Let $\Theta({\mathcal X},{\mathcal Y})$ be the vector of principal angles in nondecreasing order between subspaces ${\mathcal X}$ and ${\mathcal Y}$ of a finite dimensional space ${\mathcal H}$ with a scalar product. We consider the change in principal angles between subspaces ${\mathcal X}$ and ${\mathcal Z}$, where we let ${\mathcal X}$ be perturbed to give ${\mathcal Y}$. We measure the change using weak majorization. We prove that $|\cos^2\Theta({\mathcal X},{\mathcal Z})-\cos^2\Theta({\mathcal Y},{\mathcal Z})| {\prec}_{w} \sin\Theta({\mathcal X},{\mathcal Y})$, and give similar results for differences of cosines, i.e., $|\cos\Theta({\mathcal X},{\mathcal Z})-\cos\Theta({\mathcal Y},{\mathcal Z})| {\prec}_{w} \sin\Theta({\mathcal X},{\mathcal Y})$, and of sines and sines squared, assuming $\dim {\mathcal X} = \dim {\mathcal Y}$. We observe that $\cos^2\Theta({\mathcal X},{\mathcal Z})$ can be interpreted as a vector of Ritz values, where the Rayleigh–Ritz method is applied to the orthogonal projector on ${\mathcal Z}$ using ${\mathcal X}$ as a trial subspace. Thus, our result for the squares of cosines can be viewed as a bound on the change in the Ritz values of an orthogonal projector. We then extend it to prove a general result for Ritz values for an arbitrary Hermitian operator A, not necessarily a projector: let $\Lambda(P_{{\mathcal X}}A|_{{\mathcal X}})$ be the vector of Ritz values in nonincreasing order for A on a trial subspace ${\mathcal X}$, which is perturbed to give another trial subspace ${\mathcal Y}$; then $| \Lambda(P_{{\mathcal X}}A|_{{\mathcal X}})- \Lambda(P_{{\mathcal Y}}A|_{{\mathcal Y}})|\prec_w (\lmax-\lmin)~\sin\Theta({\mathcal X},{\mathcal Y})$, where the constant is the difference between the largest and the smallest eigenvalues of A. This establishes our conjecture that the root two factor in our earlier estimate may be eliminated. Our present proof is based on a classical but rarely used technique of extending a Hermitian operator in ${\mathcal H}$ to an orthogonal projector in the “double” space ${\mathcal H}^2$. An application of our Ritz values weak majorization result for Laplacian graph spectra comparison is suggested, based on the possibility of interpreting eigenvalues of the edge Laplacian of a given graph as Ritz values of the edge Laplacian of the complete graph. We prove that $|\cos^2\Theta({\mathcal X},{\mathcal Z})-\cos^2\Theta({\mathcal Y},{\mathcal Z})| {\prec}_{w} \sin\Theta({\mathcal X},{\mathcal Y})$ where $\lambda^1_k$ and $\lambda^2_k$ are all ordered elements of the Laplacian spectra of two graphs with the same n vertices and with l equal to the number of differing edges.

Chvátal's Condition cannot hold for both a graph and its complement

Chvatal’s Condition is a sufficient condition for a spanning cycle in an n-vertex graph. The condition is that when the vertex degrees are d1, . . . , dn in nondecreasing order, i i or dn−i ≥ n − i. We prove that this condition cannot hold in both a graph and its complement, and we raise the problem of finding its asymptotic probability in the random graph with edge probability 1/2. This note is motivated by a discussion in the book of Palmer [7, page 81–85]. A theorem is strong if the conclusion is satisfied only when the hypothesis is satisfied, because then the hypotheses cannot be weakened. Palmer defines the strength of a theorem to be the probability that its hypotheses hold divided by the probability that its conclusion holds. We use the standard random graph model for generating n-vertex simple graphs: the vertex set is {1, . . . , n}, and edge ij occurs with probability p, independently of other edges. Let Gn,p denote the random variable for the resulting graph. In general, “Gn,p almost always satisfies Q” means that the probability of Gn,p satisfying Q tends to 1 as n→∞. We restrict our attention to constant p. ∗Supported in part by the NSF under Award No. DMS-0099608. †Supported in part by the NSA under Award No. MDA904-03-1-0037. The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation.

Building k-edge-connected neighborhood graph for distance-based data projection

Neighborhood graph construction is the first step of data projection based on geodesic distances. This paper presents an approach for constructing a k-edge-connected neighborhood graph. The approach works by applying a greedy algorithm to add edges in a non-decreasing order of edge length to the neighborhood graph if end vertices of each edge are not yet k-edge-connected on the graph. The approach is applicable to a wide range of data including data distributed in clusters. It is compared with other approaches through experiments.

A generalization of the 0–1 principle for sorting

The traditional zero–one principle for sorting networks states that “if a network with n input lines sorts all 2 n binary sequences into nondecreasing order, then it will sort any arbitrary sequence of n numbers into nondecreasing order”. We generalize this to the situation when a network sorts almost all binary sequences and relate it to the behavior of the sorting network on arbitrary inputs. We also present an application to mesh sorting.

OPTIMALITY OF FOUR-THRESHOLD POLICIES IN INVENTORY SYSTEMS WITH CUSTOMER RETURNS AND BORROWING/STORAGE OPTIONS

Consider a single-commodity inventory system in which the demand is modeled by a sequence of independent and identically distributed random variables that can take negative values. Such problems have been studied in the literature under the namecash managementand relate to the variations of the on-hand cash balances of financial institutions. The possibility of a negative demand also models product returns in inventory systems. This article studies a model in which, in addition to standard ordering and scrapping decisions seen in the cash management models, the decision-maker can borrow and store some inventory for one period of time. For problems with back orders, zero setup costs, and linear ordering, scrapping, borrowing, and storage costs, we show that an optimal policy has a simple four-threshold structure. These thresholds, in a nondecreasing order, are order-up-to, borrow-up-to, store-down-to, and scrap-down-to levels; that is, if the inventory position is too low, an optimal policy is to order up to a certain level and then borrow up to a higher level. Analogously, if the inventory position is too high, the optimal decision is to reduce the inventory to a certain point, after which one should store some of the inventory down to a lower threshold. This structure holds for the finite and infinite horizon discounted expected cost criteria and for the average cost per unit time criterion. We also provide sufficient conditions when the borrowing and storage options should not be used. In order to prove our results for average costs per unit time, we establish sufficient conditions when the optimality equations hold for a Markov decision process with an uncountable state space, noncompact action sets, and unbounded costs.

Open constraint programming

Traditionally, constraint satisfaction problems (CSP) have assumed closed-world scenarios where all domains and constraints are fixed from the beginning. With the Internet, many of the traditional CSP applications in resource allocation, scheduling and planning pose themselves in open-world settings, where domains and constraints must be discovered from different sources in a network. To model this scenario, we define open constraint satisfaction problems (OCSP) as CSP where domains and constraints are incrementally discovered through a network. We then extend the concept to open constraint optimization (OCOP). OCSP can be solved without complete knowledge of the variable domains, and we give sound and complete algorithms. We show that OCOP require the additional assumption that variable domains and relations are revealed in non-decreasing order of preference. We present a variety of algorithms for solving OCOP in the possibilistic and weighted model. We compare the algorithms through experiments on randomly generated problems. We show that in certain cases, open constraint programming can require significantly less information than traditional methods where gathering information and solving the CSP are separated. This leads to a reduction in network traffic and server load, and improves privacy in distributed problem solving.

Scheduling Transportation of Live Animals to Avoid the Spread of Diseases

In the classical vehicle-routing problem (VRP) the objective is to service some geographically scattered customers with a given number of vehicles at the minimal cost. In the present paper, we consider a variant of the VRP where the vehicles should deliver some goods between groups of customers. The customers have an associated time window, a precedence number, and a quantity. Each vehicle should visit the customers within their time windows, in nondecreasing order of precedence respecting the capacity of the vehicle. The problem will be denoted the pickup-and-delivery problem with time windows and precedence constraints (PDPTWP). The PDPTWP has applications in the transportation of live animals where veterinary rules demand that the livestocks are visited in a given sequence in order not to spread specific diseases. We propose a tighter formulation of the PDPTWP based on Dantzig-Wolfe decomposition. The formulation splits the problem into a master problem, which is a kind of set-covering problem, and a subproblem that generates legal routes for a single vehicle. The LP-relaxation of the decomposed problem is solved through delayed column generation. Computational experiments show that the obtained bounds are less than 0.24% from optimum for the considered problems. As solving the pricing problems takes up the majority of the solution time, a reformulation of the problem is proposed that makes use of the precedence constraints. By merging customers having the same precedence number into “super nodes,” the pricing problem may be reformulated as a shortest-path problem defined on an acyclic layered graph. This makes it possible to solve the pricing problem in pseudopolynomial time through dynamic programming. The paper concludes with a comprehensive computational study involving real-life instances from the transportation of live pigs. It is demonstrated that instances with up to 580 nodes can be solved to optimality.

On smallest triangles

AbstractPick n points independently at random in ℝ2, according to a prescribed probability measure μ, and let Δ ≤ Δ ≤ … be the areas of the () triangles thus formed, in nondecreasing order. If μ is absolutely continuous with respect to Lebesgue measure, then, under weak conditions, the set {n3Δ : i ≥ 1} converges as n → ∞ to a Poisson process with a constant intensity κ(μ). This result, and related conclusions, are proved using standard arguments of Poisson approximation, and may be extended to functionals more general than the area of a triangle. It is proved in addition that if μ is the uniform probability measure on the region S, then κ(μ) ≤ 2/|S|, where |S| denotes the area of S. Equality holds in that κ(μ) = 2/|S| if S is convex, and essentially only then. This work generalizes and extends considerably the conclusions of a recent paper of Jiang, Li, and Vitányi. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 206–223, 2003

Exact formulas for moments of sums of classical parking functions

For given positive integers a and b, an [ a, b]-parking function of length n is a sequence ( x 1, x 2,…, x n ) of positive integers whose order statistics x ( i) (the sequence obtained by rearranging the original sequence in non-decreasing order) satisfy the inequalities x ( i) ⩽ a+( i−1) b for all i. Using elementary methods, we derive explicit formulas for higher moments of sums and reversed sums of parking functions of length n. These formulas are finite double sums with reasonably simple terms. They are obtained by solving a recursion based on a combinatorial decomposition which decomposes a sequence of positive integers into a “maximum” parking function and a subsequence all of whose terms have “high” values. Moments of reversed sums of ordinary parking functions (the case when a= b=1) have a surprising connection with the enumeration of “sparsely-edged” graphs. From this connection, we obtain exact formulas and derive, using routine methods, asymptotic formulas (due originally to E.M. Wright) for the number of connected labeled graphs on N vertices and N+ k edges. Our method yields a new formula for the Wright constants in terms of a sequence which satisfies a linear recursion.

Gonc̆arov polynomials and parking functions

Let u be a sequence of non-decreasing positive integers. A u-parking function of length n is a sequence ( x 1, x 2,…, x n ) whose order statistics (the sequence ( x (1), x (2),…, x ( n) ) obtained by rearranging the original sequence in non-decreasing order) satisfy x ( i) ⩽ u i . The Gonc̆arov polynomials g n ( x; a 0, a 1,…, a n−1 ) are polynomials defined by the biorthogonality relation: ε(a i)D ig n(x;a 0,a 1,…,a n−1)=n!δ in, where ε( a) is evaluation at a and D is the differentiation operator. In this paper we show that Gonc̆arov polynomials form a natural basis of polynomials for working with u-parking functions. For example, the number of u-parking functions of length n is (−1) n g n (0; u 1, u 2,…, u n ). Various properties of Gonc̆arov polynomials are discussed. In particular, Gonc̆arov polynomials satisfy a linear recursion obtained by expanding x n as a linear combination of Gonc̆arov polynomials, which leads to a decomposition of an arbitrary sequence of positive integers into two subsequences: a “maximum” u-parking function and a subsequence consisting of terms of higher values. Many counting results for parking functions can be derived from this decomposition. We give, as examples, formulas for sum enumerators, and a linear recursion and Appell relation for factorial moments of sums of u-parking functions.

Finding the first K shortest paths in a time-window network

The time-constrained shortest path problem is an important generalization of the classical shortest path problem and has attracted much research interest in recent years. We consider a time-window network, where every node in the network has a list of pre-specified windows and departure from a node may take place only in these window periods. The objective of this paper is to find the first K shortest paths in a time-window network. An algorithm of time complexity of O( Kr| V| 3) is developed to find the first K shortest paths, where | V| is the numbers of nodes in the network and r represents the maximum number of windows associated with a node. Scope and purpose Time window has been a common form of time constraint considered in the literature. Basically, a time window is a time period, defined by the earliest and latest times, when a node is available for traveling through. There are many practical situations where time windows can be used to describe the time constraints associated with the nodes and arcs on a network. For example, a time window in a transportation network may be the time period that a service or transition facility is available for the traveler to pass through. Although there are many researches on the transportation problem in time-window networks, no previous researches study how to find the first K shortest paths in a time-window network; hence, this paper studies this extended problem. The extension has many potential applications in practice. For example, there may be some complex constraints associated with the nodes and/or arcs of the network, which are difficult to incorporate into the model formulation. A solution strategy is to temporarily ignore these constraints and obtain a list of paths in nondecreasing order in terms of their total times. The optimal solution can be found by identifying the first path in the list that satisfies the constraints. Another use of the result is a quick selection of an alternative path when encountering temporary disconnection of the active shortest path.