In this paper we consider the closest vector problem (CVP) for lattices $\Lambda \subseteq \mathbb{Z}^n$ given by a generator matrix $A\in \mathcal{M}_{n\times n}(\mathbb{Z})$. Let $b>0$ be the maximum of the absolute values of the entries of the matrix $A$. We prove that the CVP can be reduced in polynomial time to a quadratic unconstrained binary optimization (QUBO) problem in $O(n^2(\log(n)+\log(b)))$ binary variables, where the length of the coefficients in the corresponding quadratic form is $O(n(\log(n)+\log(b)))$.

Students from Master of Business Administration (MBA) programs are usually split into teams. In light of the generalistic nature of MBA programs, diversity within every team is desirable in terms of gender, major, age and other criteria. Many schools rotate the teams at the beginning of every term so that each student works with a different set of peers during every term, thus training his or her adaptation skills and expanding the peer network. Achieving diverse teams while avoiding–or minimizing—the repetition of student pairs is a complex and time-consuming task for MBA Directors. We introduce the Max-Diversity Orthogonal Regrouping (MDOR) problem to manage the challenge of splitting a group of people into teams several times, pursuing the goals of high diversity and few repetitions. We propose a hybrid Greedy Randomized Adaptive Search Procedure/Variable Neighborhood Descent (GRASP/VND) heuristic combined with tabu search and path relinking for its resolution, as well as an Integer Linear Programming (ILP) formulation. We compare both approaches through a set of real MBA cohorts, and the results show that, in all cases, the heuristic approach significantly outperforms the ILP and manually formed teams in terms of both diversity and repetition levels.

A way to associate unweighted graphs from weighted ones is presented, such that linear stable equilibria of the Kuramoto homogeneous model associated to both graphs coincide, i.e., equilibria of the system ˙ θ i = ∑ j ∼ i sin ( θ j − θ j ) , where i ∼ j means vertices i and j are adjacent in the corresponding graph. As a consequence, the existence of linearly stable equilibrium is proved to be NP-Hard as conjectured by R. Taylor in 2015 and a new lower bound for the minimum degree that ensures synchronization is found.

This article introduces a Mixed-Integer Linear Programming model for cost optimization in multi-product multi-line production scheduling. This model considers discrete time windows and includes realistic constraints. The NP completeness of the problem is proven. A novel scheme based on embedding bounds is applied to speed up convergence. The model is tested on 16 input configurations of a real case study from the top Uruguayan grain production facility. The numerical results show that the model significantly improves the outcome of the current ad hoc heuristic planning, reducing on average 10% the overall production costs; and that the introduction of the embedded bounds-based scheme reduces significantly the elapsed time, on average by 22%.

In market basket analysis (MBA), the goal is to understand human behavior to maximize sales. A clear behavior is buying correlated items. As a consequence, the determination of a set of items with a high correlation with others is a valuable tool for MBA. In this work, we address a combinatorial optimization problem with valuable applications to MBA, especially in marketing and product placement. The focus will be on the combinatorial problem and not on specific applications. For any given undirected graph (where the nodes are items and the edges represent correlation), we aim to find the clique such that the number of edges shared between and is maximized. This problem is known in the literature as the max cut-clique ( MCC ) problem. We can generalize this problem by considering the weights associated with each edge. In this context, we are interested in finding the clique such that the weighted sum associated with each edge shared between and is maximized. The weighted version of the MCC problem is known as the maximum edge-weight neighborhood clique ( MEWNC ) problem. The main contributions of this paper are threefold. First, the computational complexity of both the MCC and MEWNC problems is established. Specifically, we prove that the MCC and MEWNC problems belong to the class of -complete problems. Second, an exact integer linear programming (ILP) formulation for the MEWNC problem is proposed. Third, a full greedy randomized adaptive search procedure/variable neighborhood descent methodology enriched with a tabu search is here developed, where the main components are novel neighborhood structures and a restricted candidate list that trade greediness for randomization in a multistart fashion. A dynamic tabu list considers a bounding technique based on the previous analysis. Finally, a fair comparison between our hybrid algorithm and a globally optimal solution using the ILP formulation confirms that a globally optimal solution is found by our heuristic for graphs with hundreds of nodes, but it is more efficient than the exact solution in terms of time and memory requirements.

The family of trees with palindromic characteristic polynomials is characterized. Large families of graphs with this property are found as well.

A key issue in network reliability analysis. A graph with n nodes and whose e edges fail independently with probability p is an Uniformly Most Reliable Graph (UMRG) if it has the highest reliability among all graphs with the same order and size for every value of p. The all-terminal reliability is a polynomial in p which defines the probability of a network to remain connected if some of its components fail. If the coefficients of the reliability polynomial are maximized by a graph, that graph is called Strong Uniformly Most Reliable Graph (SUMRG) and it should be UMRG. An exhaustive computer search of the SUMRG with vertices up to 9 is done. Regular graphs with 10 to 14 vertices that maximize tree number are proposed as candidates to UMRG. As an outstanding result a UMRG with 9 vertices and 18 edges which has girth 3 is found, so smaller than the conjectured by Boesch in 1986. A new conjecture about UMRG's topology is posed here: the (n,e)-UMRG is (k-1)C_3∪ C_3+r whenever n=3k+r,n≥5 and e=n(n-3)/2. A reformulation of Boesch's conjecture is presented stating that if a (n, kn/2)-UMRG exists and it has girth g, then it has maximum girth among all k-regular (n,kn/2) graphs and minimum number of g-cycles among those k-regular (n,kn/2) graphs with girth g.

This article presents a Mixed-Integer Linear Programming model for cost optimization in multi-product multi-line production scheduling. The proposed model applies discrete time windows and includes realistic constraints. The model is validated on a specific case study from a real Uruguayan grain production facility. Results of the evaluation indicate that the proposed model improves over the current ad-hoc heuristic planning, reducing up to 10.4% the overall production costs.

This document analyses the problem of designing a minimum cost local wind turbine grid (or LWTG) for an onshore wind farm. The LWTG is responsible for adding up the power of the farm’s wind turbines to then deliver it to a high voltage network. To minimize disruption of agricultural activities, the cables should be laid along underground conduits, parallel to an existing map of roads in the terrain. Hence, connections of onshore wind farms are limited differently from those in offshore ones. This document presents: the technical constraints of such a grid; a combinatorial model to find cost optimal solutions; and also shows some concrete optimal layouts. The work is based upon a real-world project, the Parque Eólico Palomas , a wind farm in Uruguay, a leader country in the usage of renewable energies and environmental care. The results of this research contributed to reduce investment costs of that project, with savings exceeding 30% over manually crafted solutions.

A tromino tiling problem is a packing puzzle where we are given a region of connected lattice squares and we want to decide whether there exists a tiling of the region using trominoes with the shape of an L. In this work we study a slight variation of the tromino tiling problem where some positions of the region have pegs and each tromino comes with a hole that can only be placed on top of the pegs. We present a characterization of this tiling problem with pegs using flow networks and show that (i) there exists a linear-time parsimonious reduction to the maximum-flow problem, and (ii) counting the number of such tilings can be done in linear-time. The proofs of both results contain algorithms that can then be used to decide the tiling of a region with pegs in $O(n)$ time.

In this paper we address a metaheuristic for an combinatorial optimization problem. For any given graph \(\mathcal {G}=(V,E)\) (where the nodes represent items and edges correlations), we want to find the clique \(\mathcal {C} \subseteq V\) such that the number of links shared between \(\mathcal {C}\) and \(V - \mathcal {C}\) is maximized. This problem is known in the literature as the Max Cut-Clique (MCC).

In network design, the all-terminal reliability maximization is of paramount importance. In this classical setting, we assume a simple graph with perfect nodes but independent edge failures with identical probability $\rho$. The goal is to communicate $n$ terminals using $e$ edges, in such a way that the connectedness probability of the resulting random graph is maximum. A graph with $n$ nodes and $e$ edges that meets the maximum reliability property for all $\rho \in (0,1)$ is called uniformly most-reliable $(n, e)$-graph (UMRG). The discovery of these graphs is a challenging problem that involves an interplay between extremal graph theory and computational optimization. Recent works confirm the existence of special cubic UMRGs, such as Wagner, Petersen and Yutsis graphs, and a 4-regular graph $H=\overline{C_7}$. In a foundational work in the field, Boesch. et. al. state with no proof that the bipartite complete graph $K_{4,4}$ is UMRG. In this paper, we revisit the breakthroughs in the theory of UMRG. A simple methodology to determine UMRGs based on counting trivial cuts is presented. Finally, we test this methodology to mathematically prove that the complete bipartite graph $K_{4,4}$ is UMRG.

The physical support of the Internet infrastructure is fiber-optics network (FON). Since the global Internet traffic doubles each year, FONs are suffering the capacity crunch problem, and several proposals offer content-oriented solutions, combined with a smart spectral utilization. It is well known that the physical implementation of FONs is robust under single failures on its links, or 2-connected. Unfortunately, simultaneous failures occur, and the service is disrupted for some hours per year in average. In this paper we propose a smart network augmentation which solves the previous problems simultaneously. The resulting network is a locally most-reliable 3-connected augmentation. Our methodology to build 3-connected networks is supported by the theory of uniformly most-reliable networks, elementary properties of cubic networks and a careful selection of local searches in a Variable Neighborhood Descent (VND) metaheuristic.

There is a vast literature focused on network reliability evaluation. In the last decades, reliability optimization has been also addressed. Frank Boesch in 1986 introduced the concept of uniformly most reliable graph (UMRG). Later, Boesch \emph{et al.} presented the first UMRGs and conjectured that some special subdivisions of the bipartite complete graph $K_{3,3}$, as well as the bipartite complete graph $K_{4,4}$, are UMRGs. Wang proved that the first conjecture is true. Wendy Myrvold confirmed that $K_{4,4}$ is also UMRG, by means of computational tests. However, thus far, there is no mathematical proof in the literature. A trivial cut is an edge-set that includes all the incident edges of a fixed node. In this article we describe a methodology to determine UMRGs based on bounding the number of trivial cuts. As a proof-of-concept it is proved that both $K_{3,3}$ and $K_{4,4}$ are UMRGs.

A fundamental problem in network reliability analysis is to find the connectedness probability of a random graph subject to perfect nodes and independent link failures with identical probabilities. This connectedness probability is called the all-terminal reliability, and its determination is a challenging problem. In this paper we study the corresponding network design problem: which is the best way to connect q links among p nodes in order to maximize the all-terminal reliability? The optimal solutions to this problem are called uniformly most-reliable graphs. To the best of our knowledge, the literature offers a single reliability-improving network transformation called swing surgery, credited to Kelmans. Here, we offer two novel reliability-improving transformations which, even simple, they help to improve the reliability in a uniform sense (i.e., under all feasible link failure probability). Supported by the previous transformations, we prove that Yutsis graph is uniformly most-reliable, which is currently a computationally prohibitive problem. Conjectures and open problems are also discussed.

There is a strong interplay between network reliability and connectivity theory. In fact, previous studies show that the graphs with maximum reliability, called uniformly most-reliable graphs, must have the highest connectivity. In this paper, we revisit the underlying theory in order to build uniformly most-reliable cubic graphs. The computational complexity of the problem promotes the development of heuristics. The contributions of this paper are three-fold. In a first stage, we propose an ideal Variable Neighborhood Descent (VND) which returns the graph with maximum reliability. This VND works in exponential time. In a second stage, we propose a Greedy Randomized Adaptive Search Procedure (GRASP), that trades quality for computational effort. A construction phase enriched with a Restricted Candidate List (RCL) offers diversification. Our local search phase includes a globally optimum solution of an Integer Linear Programming (ILP) formulation. As a product of our research, we recovered previous optimal graphs from the related literature in the field. Additionally, we offer new candidates of uniformly most-reliable graphs with maximum connectivity and maximum number of spanning trees.

In Market Basket Analysis, the goal is to understand the human behavior in order to maximize sales. An evident behavior is to buy correlated items. As a consequence, the determination of a set of items with a large correlation with others is a valuable tool for Market Basket Analysis. In this paper we address a combinatorial optimization problem that formalizes the previous application. Given a simple graph 𝒢=(V,E) (where the nodes are items and links represent correlation), we want to find the clique 𝒞⊆ V such that the number of links shared between 𝒞 and V - 𝒞 is maximized. This problem is known in the literature as Max Cut-Clique (MCC). The contributions of this paper are three-fold. First, the computational complexity of the MCC is established. Second, a full GRASP/VND methodology enriched with a Tabu Search is here developed, where the main ingredients are novel local searches and a Restricted Candidate List that trades greediness for randomization in a multi-start fashion. A Tabu Search is also included in order to avoid locally optimum solutions. Finally, a fair comparison with respect to recent heuristics reveals that our proposal is competitive with state-of-the-art solutions.

There is a strong interplay between network reliability and connectivity theory. In fact, previous studies show that the graphs with maximum reliability, called uniformly most-reliable graphs, must have the highest connectivity. In this paper, we revisit the underlying theory in order to build uniformly most-reliable cubic graphs. The computational complexity of the problem promotes the development of heuristics. The contributions of this paper are three-fold. In a first stage, we propose an ideal Variable Neighborhood Descent (VND) which returns the graph with maximum reliability. This VND works in exponential time. In a second stage, we propose a hybrid GRASP/VND approach that trades quality for computational effort. A construction phase enriched with a Restricted Candidate List (RCL) offers diversification. Our local search phase includes a factor-2 algorithm for an Integer Linear Programming (ILP) model. As a product of our research, we recovered previous optimal graphs from the related literature in the field. Additionally, we offer new candidates of uniformly most-reliable graphs with maximum connectivity and maximum number of spanning trees.

An embedding of a guest graph G over a host graph H is an injective map Φ from the vertices of G to the vertices of H and a mapping ρ, which associates every edge e = {x, y} in G to a Φ(x)-Φ(y) path ρ(e) in H.Given an edge f in H, if ρ -1 is the set of those edges that cross f , i.e., {e : f ∈ ρ(e)}, then the cardinality of ρ -1 (f ) is the (edge) congestion cong ρ (f ) of f .The length of ρ(e) is called the dilatation dil(e) of e.The sum of all the dilatations is the cost of the embedding.The removal of an edge f of H gives rise to a surviving graph G f = G\ρ -1 (f ).Given positive integers n and b, and a fixed vertex v of the n-cycle C n , we are facing the problem of finding a guest graph G of n vertices with an embedding (Φ, ρ) over C n of minimum cost, such that for any surviving graph G f there is an embedding of the star S n = K 1,n-1 over G f that associates the center of the star to Φ -1 (v), with congestions not greater than b.This work presents the optimal cost as well as a family of optimal solutions.