Convoy movement problem: a civilian perspective

We study the convoy movement problem in peacetime from a civilian perspective by seeking to minimize civilian traffic disruptions. We develop an exact hybrid algorithm that combines the k-shortest path algorithm along with finding a minimum weighted k-clique in a k-partite graph. Through this coupling scheme, we are able to exactly solve large instances of the convoy movement problem without relaxing many of its complicating constraints. An experimental study is performed based on pseudo-transportation networks to illustrate the computational viability of the method as well as policy implications.     

A two-level graph partitioning problem arising in mobile wireless communications

In the k -partition problem (k-PP), one is given an edge-weighted undirected graph, and one must partition the node set into at most k subsets, in order to minimise (or maximise) the total weight of the edges that have their end-nodes in the same subset. Various hierarchical variants of this problem have been studied in the context of data mining. We consider a ‘two-level’ variant that arises in mobile wireless communications. We show that an exact algorithm based on intelligent preprocessing, cutting planes and symmetry-breaking is capable of solving small- and medium-size instances to proven optimality, and providing strong lower bounds for larger instances.     

An exact and heuristic approach for the <Emphasis Type="Italic">d</Emphasis>-minimum branch vertices problem

Given a connected graph \(G=(V,E)\), the d-Minimum Branch Vertices (d-MBV) problem consists in finding a spanning tree of G with the minimum number of vertices with degree strictly greater than d. We developed a Miller–Tucker–Zemlin based formulation with valid inequalities for this problem. The results obtained for different values of d show the effectiveness of the proposed method, which has solved several instances faster than previous methods. Also, an heuristic is proposed for this problem, that was tested on several instances of the Minimum Branch Vertices problem, which is the d-MBV problem, when \(d = 2\).     

Radar placement along banks of river

In this paper, we consider the Radar Placement and Power Assignment problem (RPPA) along a river. In this problem, a set of crucial points in the river are required to be monitored by a set of radars which are placed along the two banks. The goal is to choose the locations for the radars and assign powers to them such that all the crucial points are monitored and the total power is minimized. If each crucial point is required to be monitored by at least k radars, the problem is a k-Coverage RPPA problem (k-CRPPA). Under the assumption that the river is sufficiently smooth, one may focus on the RPPA problem along a strip (RPPAS). In this paper, we present an O(n ⁹) dynamic programming algorithm for the RPPAS, where n is the number of crucial points to be monitored. In the special case where radars are placed only along the upper bank, we present an O(kn ⁵) dynamic programming algorithm for the k-CRPPAS. For the special case that the power is linearly dependent on the radius, we present an O(n log n)-time \({2\sqrt 2}\)-approximation algorithm for the RPPAS.     

Reducing work-in-process movement for multiple products in one-dimensional layout problems

This research describes a method to assign M machines, which are served by a material handling transporter, to M equidistant locations along a track, so that the distance traveled by a given set of jobs is minimized. Traditionally, this problem (commonly known as a machine location problem) has been modeled as a quadratic assignment problem (QAP), which is NP-hard, thus motivating the need for efficient procedures to solve instances with several machines. In this paper we develop a branching heuristic to obtain sub-optimum solutions to the problem; a lower bound on the optimum solution has also been presented. Results obtained from the heuristics are compared with results obtained from other heuristics with similar objectives. It is observed that the results are promising, and justify the usage of developed methods.     

Design of balanced MBA student teams

In some schools and universities, students must sometimes be divided into several teams in such a way that each team provides a good representation of the classroom population. In this paper, two different ways of measuring the balance among teams are proposed: min-sum and min-max objective functions. For the first function and the L₁-norm used in the space of attributes, an exact solution method based on a set partitioning formulation and on the enumeration of all possible team patterns is presented. For the second objective function, a set partitioning formulation is also considered, but as an approximation. In order to solve large problem instances, we have also developed metaheuristics based on variable neighbourhood search. Models and methods are tested on data from an MBA programme.     

A branch-and-cut algorithm for the plant-cycle location problem

The Plant-Cycle Location Problem (PCLP) is defined on a graph G=(I∪J, E), where I is the set of customers and J is the set of plants. Each customer must be served by one plant, and the plant must be opened to serve customers. The number of customers that a plant can serve is limited. There is a cost of opening a plant, and of serving a customer from an open plant. All customers served by a plant are in a cycle containing the plant, and there is a routing cost associated to each edge of the cycle. The PCLP consists in determining which plants to open, the assignment of customers to plants, and the cycles containing each open plant and its customers, minimizing the total cost. It is an NP-hard optimization problem arising in routing and telecommunications. In this article, the PCLP is formulated as an integer linear program, a branch-and-cut algorithm is developed, and computational results on real-world data and randomly generated instances are presented. The proposed approach is able to find optimal solutions of random instances with up to 100 customers and 100 potential plants, and of instances on real-world data with up to 120 customers and 16 potential plants.     

The Local Metric Dimension of Strong Product Graphs

A vertex \(v\in V(G)\) is said to distinguish two vertices \(x,y\in V(G)\) of a nontrivial connected graph G if the distance from v to x is different from the distance from v to y. A set \(S\subset V(G)\) is a local metric generator for G if every two adjacent vertices of G are distinguished by some vertex of S. A local metric generator with the minimum cardinality is called a local metric basis for G and its cardinality, the local metric dimension of G. It is known that the problem of computing the local metric dimension of a graph is NP-Complete. In this paper we study the problem of finding exact values or bounds for the local metric dimension of strong product of graphs.     

Reformulation and solution approach for non-separable integer quadratic programs

We consider quadratic programs with pure general integer variables. The objective function is quadratic and convex and the constraints are linear. An exact solution approach is proposed. It is decomposed into two phases. In the first phase, the initial problem is reformulated into an equivalent problem with a separable objective function. This is done by use of a Gauss decomposition of the Hessian matrix of the initial problem and requires the addition of some continuous variables and constraints. In the second phase, the reformulated problem is linearized by an approximation of each squared term by a set of K linear functions that correspond to the tangents of a hyperbola in K points. We give a proof of the intuitive property that when K is large enough, the optimal value of the obtained linear program is very close to optimal value of the two previous problems, the initial problem and the reformulated separable problem. The reminder is dedicated to the implementation of a branch-and-bound algorithm for the solution of linearized problem, and its application to a set of instances. Several points are considered among which choice of the right value for parameter K and the implementation of a sophisticated heuristic solution algorithm. The numerical comparison is done with CPLEX 12.2 since, in this case, the initial problem as well as the problem reformulated by the first step can be solved by CPLEX. We show that with our approach, the total CPU time is divided by a factor ranging from 1.2 to 131.6 for instances with 40–60 variables.     

Asymptotically optimal approach to the approximate solution of several problems of covering a graph by nonadjacent cycles

We consider the m-Cycle Cover Problem of covering a complete undirected graph by m vertex-nonadjacent cycles of extremal total edge weight. The so-called TSP approach to the construction of an approximation algorithm for this problem with the use of a solution of the traveling salesman problem (TSP) is presented. Modifications of the algorithm for the Euclidean Max m-Cycle Cover Problem with deterministic instances (edge weights) in a multidimensional Euclidean space and the Random Min m-Cycle Cover Problem with random instances UNI(0,1) are analyzed. It is shown that both algorithms have time complexity O(n ³) and are asymptotically optimal for the number of covering cycles m = o(n) and \(m \leqslant \frac{{n^{1/3} }}{{\ln n}}\), respectively.     

Error estimates for a projection-difference method for a linear differential-operator equation

We study a projection-difference method for approximately solving the Cauchy problem u′(t) + A(t)u(t) + K(t)u(t) = h(t), u(0) = 0 for a linear differential-operator equation in a Hilbert space, where A(t) is a self-adjoint operator and K(t) is an operator subordinate to A(t). Time discretization is based on a three-level difference scheme, and space discretization is carried out by the Galerkin method. Under certain smoothness conditions on the function h(t), we obtain estimates for the convergence rate of the approximate solutions to the exact solution.     

An <Emphasis Type="Italic">L</Emphasis>-approach for packing (<Emphasis Type="Italic">ℓ</Emphasis>, <Emphasis Type="Italic">w</Emphasis>)-rectangles into rectangular and <Emphasis Type="Italic">L</Emphasis>-shaped pieces

This paper presents an approach using a recursive algorithm for packing (?, w)-rectangles into larger rectangular and L-shaped pieces. Such a problem has actual applications for non-guillotine cutting and pallet/container loading. Our motivation for developing the L-approach is based on the fact that it can solve difficult pallet loading instances. Indeed, it is able to solve all testing problems (more than 20 000 representatives of infinite equivalence classes of the literature), including the 18 hard instances unresolved by other heuristics. We conjecture that the L-approach always finds optimum packings of (?, w)-rectangles into rectangular pieces. Moreover, the approach may also be useful when dealing with cutting and packing problems involving L-shaped pieces.     

On Frankl and Füredi’s conjecture for 3-uniform hypergraphs

Frankl and Füredi in [1] conjectured that the r-graph with m edges formed by taking the first m sets in the colex ordering of N^(r) has the largest Lagrangian of all r-graphs with m edges. Denote this r-graph by C _r,m and the Lagrangian of a hypergraph by λ(G). In this paper, we first show that if \(\leqslant m \leqslant \left( {\begin{array}{*{20}{c}}t \\ 3 \end{array}} \right)\), G is a left-compressed 3-graph with m edges and on vertex set [t], the triple with minimum colex ordering in G ^c is (t ? 2 ? i)(t ? 2)t, then λ(G) ≤ λ(C _3,m ). As an implication, the conjecture of Frankl and Füredi is true for \(\left( {\begin{array}{*{20}{c}}t \\ 3\end{array}} \right) - 6 \leqslant m \leqslant \left( {\begin{array}{*{20}{c}}t \\ 3\end{array}} \right)\).     

The <Emphasis Type="Italic">p</Emphasis>-adic order of some fibonomial coefficients whose entries are powers of <Emphasis Type="Italic">p</Emphasis>

Let (F _n ) _n≥0 be the Fibonacci sequence. For 1 ≤ k ≤ m, the Fibonomial coefficient is defined as

$${\left[ {\begin{array}{*{20}{c}} m \\ k \end{array}} \right]_F} = \frac{{{F_{m - k + 1}} \cdots {F_{m - 1}}{F_m}}}{{{F_1} \cdots {F_k}}}$$

. In 2013, Marques, Sellers and Trojovský proved that if p is a prime number such that p ≡ ±2 (mod 5), then \(p{\left| {\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]} \right._F}\) for all integers a ≥ 1. In 2015, Marques and Trojovský worked on the p-adic order of \({\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]_F}\) for all a ≥ 1 when p ≠ 5. In this paper, we shall provide the exact p-adic order of \({\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]_F}\) for all integers a, b ≥ 1 and for all prime number p.

     

The Existence of a Solution to the Multi-Index Problem

The multi-index problem can be described as minimizing the cost of moving a set of p different commodities (k = 1, 2,..., p) from n origins (i = 1, 2,..., n) to m destinations (j = 1, 2,..., m). The equations then give rise to the conditions on the amount of the various types of combination that is available and required. Alternatively, the same set of restrictions arise when a single commodity has to be moved by different methods, e.g. road, rail, sea, canal, air, etc. Similarly the use of intermediate depots may require the use of a multi-index formulation. A third special type of problem where the method can be used is the capacitated transportation problem (each variable has an upper bound).     

On the phase transitions of (<Emphasis Type="Italic">k</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)-SAT

Call a sequence of k Boolean variables or their negations a k-tuple. For a set V of n Boolean variables, let T _k (V) denote the set of all 2 ^k n ^k possible k-tuples on V. Randomly generate a set C of k-tuples by including every k-tuple in T _k (V) independently with probability p, and let Q be a given set of q “bad” tuple assignments. An instance I = (C,Q) is called satisfiable if there exists an assignment that does not set any of the k-tuples in C to a bad tuple assignment in Q. Suppose that θ, q > 0 are fixed and ε = ε(n) > 0 be such that εlnn/lnlnn→∞. Let k ≥ (1 + θ) log₂ n and let \({p_0} = \frac{{\ln 2}}{{q{n^{k - 1}}}}\). We prove that

$$\mathop {\lim }\limits_{n \to \infty } P\left[ {I is satisfiable} \right] = \left\{ {\begin{array}{*{20}c} {1,} & {p \leqslant (1 - \varepsilon )p_0 ,} \\ {0,} & {p \geqslant (1 + \varepsilon )p_0 .} \\ \end{array} } \right.$$

     

The optimal solution set of the interval linear programming problems

Several methods exist for solving the interval linear programming (ILP) problem. In most of these methods, we can only obtain the optimal value of the objective function of the ILP problem. In this paper we determine the optimal solution set of the ILP as the intersection of some regions, by the best and the worst case (BWC) methods, when the feasible solution components of the best problem are positive. First, we convert the ILP problem to the convex combination problem by coefficients 0 ≤ λ _j , μ _ij , μ _i  ≤ 1, for i = 1, 2, . . . , m and j = 1, 2, . . . , n. If for each i, j, μ _ij  = μ _i  = λ _j  = 0, then the best problem has been obtained (in case of minimization problem). We move from the best problem towards the worst problem by tiny variations of λ _j , μ _ij and μ _i from 0 to 1. Then we solve each of the obtained problems. All of the optimal solutions form a region that we call the optimal solution set of the ILP. Our aim is to determine this optimal solution set by the best and the worst problem constraints. We show that some theorems to validity of this optimal solution set.     

Expanding operators for the independent set problem

The notion is introduced of an expanding operator for the independent set problem. This notion is a useful tool for the constructive formation of new cases with the efficient solvability of this problem in the family of hereditary classes of graphs and is applied to hereditary parts of the set Free({P ₅, C ₅}). It is proved that if for a connected graph G the problem is polynomial-time solvable in the class Free({P ₅, C ₅,G}) then it remains so in the class Free({P ₅, C ₅,G ○ \(\bar K_2 \), G ⊕ K _1,p ) for every p. We also find two new hereditary subsets of Free({P ₅, C ₅}) with polynomially solvable independent set problem that are not a corollary of applying the revealed operators.     

An Optimal Algorithm for the <Emphasis Type="Italic">k</Emphasis>-Fixed-Endpoint Path Cover on Proper Interval Graphs

In this paper we consider the k-fixed-endpoint path cover problem on proper interval graphs, which is a generalization of the path cover problem. Given a graph G and a set T of k vertices, a k-fixed-endpoint path cover of G with respect to T is a set of vertex-disjoint simple paths that covers the vertices of G, such that the vertices of T are all endpoints of these paths. The goal is to compute a k-fixed-endpoint path cover of G with minimum cardinality. We propose an optimal algorithm for this problem with runtime O(n), where n is the number of intervals in G. This algorithm is based on the Stair Normal Interval Representation (SNIR) matrix that characterizes proper interval graphs. In this characterization, every maximal clique of the graph is represented by one matrix element; the proposed algorithm uses this structural property, in order to determine directly the paths in an optimal solution.     

A new heuristic approach for the P-median problem

The Euclidean p-median problem is concerned with the decision of the locations for public service centres. Existing methods for the planar Euclidean p-median problems are capable of efficiently solving problems of relatively small scale. This paper proposes two new heuristic algorithms aiming at problems of large scale. Firstly, to reflect the different degrees of proximity to optimality, a new kind of local optimum called level-m optimum is defined. For a level-m optimum of a p-median problem, where m<p, each of its subsets containing m of the p partitions is a global optimum of the corresponding m-median subproblem. Starting from a conventional local optimum, the first new algorithm efficiently improves it to a level-2 optimum by applying an existing exact algorithm for solving the 2-median problem. The second new algorithm further improves it to a level-3 optimum by applying a new exact algorithm for solving the 3-median problem. Comparison based on experimental results confirms that the proposed algorithms are superior to the existing heuristics, especially in terms of solution quality.