Coloring of mixed graphs that contain both directed arcs and undirected edges is relevant for scheduling of unit-length jobs with precedence constraints and conflicts. The classic GHRV theorem (attributed to Gallai, Hasse, Roy, and Vitaver) relates graph coloring to longest paths. It can be extended to mixed graphs. In the present paper we further extend the GHRV theorem to...

We consider an online interval scheduling problem on two related machines. If one machine is at least as twice as fast as the other machine, we say the machines are distinct; otherwise the machines are said to be similar. Each job \(j \in J\) is characterized by a length \(p_j\), and an arrival time \(t_j\); the question is to determine whether there exists a feasible schedule...

Correlation clustering is an approach for clustering a set of objects from given pairwise information. In this approach, the given pairwise information is usually represented by an undirected graph with nodes corresponding to the objects, where each edge in the graph is assigned a nonnegative weight, and either the positive or negative label. Then, a clustering is obtained by...

We study the problem of scheduling a set of jobs with release dates, deadlines and processing requirements (or works) on parallel speed scalable processors so as to minimize the total energy consumption. We consider that both preemptions and migrations of jobs are allowed. For this problem, there exists an optimal polynomial-time algorithm which uses as a black box an algorithm...

We study a multidimensional hyperbox packing with one active bin. The items (d-dimensional hyperboxes of edge length not greater than 1) arrive one by one. Each item must be packed online into a hypercube bin of edge 1 and \(90^{\circ }\)-rotations are allowed. If it is impossible to pack an item into an active bin, we close the bin and open a new active bin to pack that item. In...

Hospital-acquired infection threatens the patients’ health and life and also impacts medical quality by decreasing the bed turnover rate, prolonging hospitalization, increasing hospital costs and bringing the patients the huge economic losses. Therefore, hospital infection management is the focus of today’s hospital management and one of the most prominent public health problems...

The quadratic shortest path problem (QSPP) is the problem of finding a path with prespecified start vertex s and end vertex t in a digraph such that the sum of weights of arcs and the sum of interaction costs over all pairs of arcs on the path is minimized. We first consider a variant of the QSPP known as the adjacent QSPP. It was recently proven that the adjacent QSPP on cyclic...

In this paper we consider a variant of graph partitioning consisting in partitioning the vertex set of a graph into the minimum number of sets such that each of them induces a graph in hereditary class of graphs \({\mathcal {P}}\) (the problem is also known as \({\mathcal {P}}\)-coloring). We focus on the computational complexity of several problems related too greedy...

TU games with two-level communication structure, in which a two-level communication structure relates fundamentally to the given coalition structure and consists of a communication graph on the collection of the a priori unions in the coalition structure, as well as a collection of communication graphs within each union, are considered. For such games we introduce two families of...

An arc in \(\mathbb Z^2_n\) is defined to be a set of points no three of which are collinear. We describe some properties of arcs and determine the maximum size of arcs for some small n.

We study uniqueness of Nash equilibria in atomic splittable congestion games and derive a uniqueness result based on polymatroid theory: when the strategy space of every player is a bidirectional flow polymatroid, then equilibria are unique. Bidirectional flow polymatroids are introduced as a subclass of polymatroids possessing certain exchange properties. We show that important...

We address a variant of the single item lot sizing problem affected by proportional storage (or inventory) losses and uncertainty in the product demand. The problem has applications in, among others, the energy sector, where storage losses (or storage deteriorations) are often unavoidable and, due to the need for planning ahead, the demands can be largely uncertain. We first...

This paper deals with the recoverable robust spanning tree problem under interval uncertainty representations. A strongly polynomial time, combinatorial algorithm for the recoverable spanning tree problem is first constructed. This problem generalizes the incremental spanning tree problem, previously discussed in literature. The algorithm built is then applied to solve the...

Given integers \(1\le k<n\), the Gusein-Zade version of a generalized secretary problem is to choose one of the k best of n candidates for a secretary, which are interviewing in random order. The stopping rule in the selection is based only on the relative ranks of the successive arrivals. It is known that the best policy can be described by a non-decreasing sequence \((s_1...

We study the classical 0–1 knapsack problem with additional restrictions on pairs of items. A conflict constraint states that from a certain pair of items at most one item can be contained in a feasible solution. Reversing this condition, we obtain a forcing constraint stating that at least one of the two items must be included in the knapsack. A natural way for representing...

In several areas like global optimization using branch-and-bound methods for mixture design, the unit n-simplex is refined by longest edge bisection (LEB). This process provides a binary search tree. For \(n>2\), simplices appearing during the refinement process can have more than one longest edge (LE). The size of the resulting binary tree depends on the specific sequence of...

In this paper, we show that sequence pair (SP) representation, primarily applied to the rectangle packing problems appearing in the VLSI industry, can be a solution representation of precedence constrained scheduling. We present three interpretations of sequence pair, which differ in complexity of schedule evaluation and size of a corresponding solution space. For each...

A k-weighting w of a graph is an assignment of an integer weight \(w(e)\in \{1,...k\}\) to each edge e. Such an edge weighting induces a vertex coloring c defined by \(c(v)=\mathop {\displaystyle {\prod }}\limits _{v\in e}w(e).\) A k-weighting of a graph G is multiplicative vertex-coloring if the induced coloring c is proper, i.e., \(c(u)\ne c(v)\) for any edge \(uv\in E(G...