Improved formulations and branch-and-cut algorithms for the angular constrained minimum spanning tree problem

Journal of Combinatorial Optimization - The Angular Constrained Minimum Spanning Tree Problem ( $$\alpha $$ -MSTP) is defined in terms of a complete undirected graph $$G=(V,E)$$ and an angle...     

The minimum vulnerability problem on specific graph classes

Suppose that each edge e of an undirected graph G is associated with three nonnegative integers \(\mathsf{cost}(e)\), \(\mathsf{vul}(e)\) and \(\mathsf{cap}(e)\), called the cost, vulnerability and capacity of e, respectively. Then, we consider the problem of finding \(k\) paths in G between two prescribed vertices with the minimum total cost; each edge e can be shared without any cost by at most \(\mathsf{vul}(e)\) paths, and can be shared by more than \(\mathsf{vul}(e)\) paths if we pay \(\mathsf{cost}(e)\), but cannot be shared by more than \(\mathsf{cap}(e)\) paths even if we pay the cost for e. This problem generalizes the disjoint path problem, the minimum shared edges problem and the minimum edge cost flow problem for undirected graphs, and it is known to be NP-hard. In this paper, we study the problem from the viewpoint of specific graph classes, and give three results. We first show that the problem is NP-hard even for bipartite outerplanar graphs, 2-trees, graphs with pathwidth two, complete bipartite graphs, and complete graphs. We then give a pseudo-polynomial-time algorithm for bounded treewidth graphs. Finally, we give a fixed-parameter algorithm for chordal graphs when parameterized by the number \(k\) of required paths.     

Capacity inverse minimum cost flow problem

Given a directed graph G=(N,A) with arc capacities u _ij and a minimum cost flow problem defined on G, the capacity inverse minimum cost flow problem is to find a new capacity vector [^(u)]\hat{u} for the arc set A such that a given feasible flow [^(x)]\hat{x} is optimal with respect to the modified capacities. Among all capacity vectors [^(u)]\hat{u} satisfying this condition, we would like to find one with minimum ||[^(u)]-u||\|\hat{u}-u\| value. We consider two distance measures for ||[^(u)]-u||\|\hat{u}-u\| , rectilinear (L ₁) and Chebyshev (L _∞) distances. By reduction from the feedback arc set problem we show that the capacity inverse minimum cost flow problem is NP\mathcal{NP} -hard in the rectilinear case. On the other hand, it is polynomially solvable by a greedy algorithm for the Chebyshev norm. In the latter case we propose a heuristic for the bicriteria problem, where we minimize among all optimal solutions the number of affected arcs. We also present computational results for this heuristic.     

Some further results on minimum distribution cost flow problems

Manufacturing network flow (MNF) is a generalized network model that can model more complicated manufacturing scenarios, such as the synthesis of different materials to one product and/or the distilling of one material to many different products. Minimum distribution cost flow problem (MDCF) is a simplified version of MNF optimization problems, in which a general supplier wants to proportionally distribute certain amount of a particular product from a source node to several retailers at different destinations through a distribution network. A network simplex algorithm has been outlined in recent years for solving a special case of MDCF. In this paper, we characterize the network structure of the bases of the MDCF problem and develop a primal simplex algorithm that exploits the network structure of the problem. These results are extensions of those of the ordinary network flow problems. In conclusion, some related interesting problems are proposed for future research. This research is partially supported by the National Natural Science Foundation of China (No. 10371028) and a grant from Southern Yangtze University (No. 0003182).     

On the minimum routing cost clustered tree problem

For an edge-weighted graph \(G=(V,E,w)\), in which the vertices are partitioned into k clusters \(\mathcal {R}=\{R_1,R_2,\ldots ,R_k\}\), a spanning tree T of G is a clustered spanning tree if T can be cut into k subtrees by removing \(k-1\) edges such that each subtree is a spanning tree for one cluster. In this paper, we show the inapproximability of finding a clustered spanning tree with minimum routing cost, where the routing cost is the total distance summed over all pairs of vertices. We present a 2-approximation for the case that the input is a complete weighted graph whose edge weights obey the triangle inequality. We also study a variant in which the objective function is the total distance summed over all pairs of vertices of different clusters. We show that the problem is polynomial-time solvable when the number of clusters k is 2 and NP-hard for \(k=3\). Finally, we propose a polynomial-time 2-approximation algorithm for the case of three clusters.     

ILP formulation of the degree-constrained minimum spanning hierarchy problem

Given a connected edge-weighted graph G and a positive integer B, the degree-constrained minimum spanning tree problem (DCMST) consists in finding a minimum cost spanning tree of G such that the degree of each vertex in the tree is less than or equal to B. This problem, which has been extensively studied over the last few decades, has several practical applications, mainly in networks. However, some applications do not especially impose a subgraph as a solution. For this purpose, a more flexible so-called hierarchy structure has been proposed. Hierarchy, which can be seen as a generalization of trees, is defined as a homomorphism of a tree in a graph. In this paper, we discuss the degree-constrained minimum spanning hierarchy (DCMSH) problem which is NP-hard. An integer linear program (ILP) formulation of this new problem is given. Properties of the solution are analysed, which allows us to add valid inequalities to the ILP. To evaluate the difference of cost between trees and hierarchies, the exact solution of DCMST and z problems are compared. It appears that, in sparse random graphs, the average percentage of improvement of the cost varies from 20 to 36% when the maximal authorized degree of vertices B is equal to 2, and from 11 to 31% when B is equal to 3. The improvement increases as the graph size increases.     

Algorithms for the minimum diameter terminal Steiner tree problem

Given an undirected connected graph \(G=(V(G),E(G),d)\) with a function \(d(\cdot )\ge 0\) on edges and a subset \(S\subseteq V(G)\) of terminals, the minimum diameter terminal Steiner tree problem (MDTSTP) asks for a terminal Steiner tree in \(G\) of a minimum diameter. In the paper, the diameter of a tree refers to the longest of all the distances between two different leaves of the tree. When \(G\) is a complete graph and \(d(\cdot )\) is a metric function, we demonstrate that an optimal solution of MDTSTP is monopolar or dipolar and give an \(O(|S|\cdot |V(G)\setminus S|^2)\) -time exact algorithm. For the nonmetric version of MDTSTP, we present a simple 2-approximation algorithm with a time complexity of \(O(|V(G)\setminus S|\log |S|)\) , as well as two exact algorithms with a time complexity of \(O(|S|^3|V(G)|^2)\) and \(O(|S|\cdot |V(G)\setminus S|^2+|S|^2\cdot |V(G)\setminus S|)\) , respectively.     

The resource constrained single-item inventory problem with demand-dependent unit cost

In this paper, we present an economic order quantity (EOQ) with both demand-dependent unit cost and restrictions. An analytical solution of the EQO is derived using a recent and simple method, which isthe geometric programming approach. The EOQ inventory model with demand-dependent unit cost without any restriction and the classical EOQ inventory model are obtained.     

Some results on the target set selection problem

In this paper we consider a fundamental problem in the area of viral marketing, called Target Set Selection problem. We study the problem when the underlying graph is a block-cactus graph, a chordal graph or a Hamming graph. We show that if G is a block-cactus graph, then the Target Set Selection problem can be solved in linear time, which generalizes Chen’s result (Discrete Math. 23:1400–1415, 2009) for trees, and the time complexity is much better than the algorithm in Ben-Zwi et al. (Discrete Optim., 2010) (for bounded treewidth graphs) when restricted to block-cactus graphs. We show that if the underlying graph G is a chordal graph with thresholds θ(v)≤2 for each vertex v in G, then the problem can be solved in linear time. For a Hamming graph G having thresholds θ(v)=2 for each vertex v of G, we precisely determine an optimal target set S for (G,θ). These results partially answer an open problem raised by Dreyer and Roberts (Discrete Appl. Math. 157:1615–1627, 2009).     

Capacitated inverse optimal value problem on minimum spanning tree under bottleneck Hamming distance

Journal of Combinatorial Optimization - We consider the capacitated inverse optimal value problem on minimum spanning tree under Hamming distance. Given a connected undirected network $$G=(V,E)$$...     

A branch-and-bound algorithm for the minimum cut linear arrangement problem

Given an edge-weighted graph G of order n, the minimum cut linear arrangement problem (MCLAP) asks to find a one-to-one map from the vertices of G to integers from 1 to n such that the largest of the cut values c ₁,…,c _n?1 is minimized, where c _i , i∈{1,…,n?1}, is the total weight of the edges connecting vertices mapped to integers 1 through i with vertices mapped to integers i+1 through n. In this paper, we present a branch-and-bound algorithm for solving this problem. A salient feature of the algorithm is that it employs a dominance test which allows reducing the redundancy in the enumeration process drastically. The test is based on the use of a tabu search procedure developed to solve the MCLAP. We report computational results for both the unweighted and weighted graphs. In particular, we focus on calculating the cutwidth of some well-known graphs from the literature.     

Approximation algorithms for minimum weight partial connected set cover problem

In the Minimum Weight Partial Connected Set Cover problem, we are given a finite ground set \(U\), an integer \(q\le |U|\), a collection \(\mathcal {E}\) of subsets of \(U\), and a connected graph \(G_{\mathcal {E}}\) on vertex set \(\mathcal {E}\), the goal is to find a minimum weight subcollection of \(\mathcal {E}\) which covers at least \(q\) elements of \(U\) and induces a connected subgraph in \(G_{\mathcal {E}}\). In this paper, we derive a “partial cover property” for the greedy solution of the Minimum Weight Set Cover problem, based on which we present (a) for the weighted version under the assumption that any pair of sets in \(\mathcal {E}\) with nonempty intersection are adjacent in \(G_{\mathcal {E}}\) (the Minimum Weight Partial Connected Vertex Cover problem falls into this range), an approximation algorithm with performance ratio \(\rho (1+H(\gamma ))+o(1)\), and (b) for the cardinality version under the assumption that any pair of sets in \(\mathcal {E}\) with nonempty intersection are at most \(d\)-hops away from each other (the Minimum Partial Connected \(k\)-Hop Dominating Set problem falls into this range), an approximation algorithm with performance ratio \(2(1+dH(\gamma ))+o(1)\), where \(\gamma =\max \{|X|:X\in \mathcal {E}\}\), \(H(\cdot )\) is the Harmonic number, and \(\rho \) is the performance ratio for the Minimum Quota Node-Weighted Steiner Tree problem.     

A primal-dual algorithm for the minimum power partial cover problem

Journal of Combinatorial Optimization - In this paper, we study the minimum power partial cover problem (MinPPC). Suppose X is a set of points and $${\mathcal {S}}$$ is a set of sensors on the...     

Polyhedral combinatorics of the cardinality constrained quadratic knapsack problem and the quadratic selective travelling salesman problem

This paper considers the Cardinality Constrained Quadratic Knapsack Problem (QKP) and the Quadratic Selective Travelling Salesman Problem (QSTSP). The QKP is a generalization of the Knapsack Problem and the QSTSP is a generalization of the Travelling Salesman Problem. Thus, both problems are NP hard. The QSTSP and the QKP can be solved using branch-and-cut methods. Good bounds can be obtained if strong constraints are used. Hence it is important to identify strong or even facet-defining constraints. This paper studies the polyhedral combinatorics of the QSTSP and the QKP, i.e. amongst others we identify facet-defining constraints for the QSTSP and the QKP, and provide mathematical proofs that they do indeed define facets. Author now works at Motorola. (2005 onwards)     

Approximate solution of a profit maximization constrained virtual business planning problem

A virtual business problem is studied, in which a company-contractor outsources production to specialized subcontractors. Finances of the contractor and resource capacities of subcontractors are limited. The objective is to select subcontractors and distribute a part of the demanded production among them so that the profit of the contractor is maximized. A generalization of the knapsack problem, called Knapsack-of-Knapsacks (K-of-K), is used to model this situation, in which items have to be packed into small knapsacks and small knapsacks have to be packed into a large knapsack. A fully polynomial time approximation scheme is developed to solve the problem K-of-K.     

Online pickup and delivery problem with constrained capacity to minimize latency

The online pickup and delivery problem is motivated by the takeaway order delivery on crowdsourcing delivery platform, which is a newly emerged online to offline business model based on sharing economy. Considering the features of crowdsourcing delivery, an online pickup and delivery problem with constrained capacity is proposed, whose objective is to route a delivery man with constrained capacity to serve requests released over time so as to minimize the total latency. We consider online point-to-point requests with single pickup location where each request has to be picked up at the single pickup location and delivered to its destination, and each request become available at its release time, which is not known in advance. The lower bound of this problem for various capacities is proved. Two online algorithms WR and WI are presented, the competitive ratios on a half line and on general metric space are proved respectively. Further, a computational study is conducted to compare the performance of these two online algorithms on random instances of general metric space. The result shows algorithm WR performs better than WI in random cases but not in the worst case.

     

Technology adoption and training practices as a constrained shortest path problem

Technology adoption is not a new venue for research. Much work of decision modeling, diffusion of new technology and statistical analysis of survey data has been done. Some studies focus on finding the optimal forms of technology to adopt within a complementarity framework, but there is no mention of finding an optimal path from a firm's current state to its optimal state. This represents a significant gap in the literature. The paper applies a constrained shortest path problem to training and technology adoption decisions by firms. Given the current set of training and technology adoption the method solves for what technology/practice should be adopted or removed from the complete set of combinations and in what order so as to maximize performance subject to budget constraints. To the authors' knowledge, this is the first application of the constrained shortest path problem to technology adoption decisions.