The dispersion problem is a variant of facility location problems, that has been extensively studied. Given a polygon with n edges on a plane we want to find k points in the polygon so that the minimum pairwise Euclidean distance of the k points is maximized. We call the problem the k-dispersion problem in a polygon. Intuitively, for an island, we want to locate k drone bases far away from each other in flying distance to avoid congestion in the sky. In this paper, we give a polynomial-time approximation scheme (PTAS) for this problem when k is a constant and ε < 1 (where ε is a positive real number). Our proposed algorithm runs in O(((1/ε)2 + n/ε)k) time with 1/(1 + ε) approximation, the first PTAS developed for this problem. Additionally, we consider three variations of the dispersion problem and design a PTAS for each of them.

Emergency Medical Services (EMS) play a crucial role in healthcare systems, managing pre-hospital or out-of-hospital emergencies from the onset of an emergency call to the patient’s arrival at a healthcare facility. The design of an efficient ambulance location model is pivotal in enhancing survival rates, controlling morbidity, and preventing disability. Key factors in the classical models typically include travel time, demand zones, and the number of stations. While urban EMS systems have received extensive examination due to their centralized populations, rural areas pose distinct challenges. These include lower population density and longer response distances, contributing to a higher fatality rate due to sparse population distribution, limited EMS stations, and extended travel times. To address these challenges, we introduce a novel mathematical model that aims to optimize coverage and equity. A distinctive feature of our model is the integration of equity within the objective function, coupled with a focus on practical response time that includes the period required for personal protective equipment procedures, ensuring the model’s applicability and realism in emergency response scenarios. We tackle the proposed problem using a tailored genetic algorithm and propose a greedy algorithm for solution construction. The implementation of our tailored Genetic Algorithm promises efficient and effective EMS solutions, potentially enhancing emergency care and health outcomes in rural communities.

This paper proposes a migration model for distributed server allocation. In distributed server allocation, each user is assigned to a server to minimize the communication delay. In the conventional model, a user cannot migrate to another server to avoid instability. We develop a model where each user can migrate to another server while receiving services. We formulate the proposed model as an integer linear programming problem. We prove that the considered problem is NP-complete. We introduce a heuristic algorithm. Numerical result shows that the proposed model reduces the average communication delay by 59% compared to the conventional model at most.

This paper presents a distributed server allocation model with preventive start-time optimization against a single server failure. The presented model preventively determines the assignment of servers to users under each failure pattern to minimize the largest maximum delay among all failure patterns. We formulate the proposed model as an integer linear programming (ILP) problem. We prove the NP-completeness of the considered problem. As the number of users and that of servers increase, the size of ILP problem increases; the computation time to solve the ILP problem becomes excessively large. We develop a heuristic approach that applies simulated annealing and the ILP approach in a hybrid manner to obtain the solution. Numerical results reveal that the developed heuristic approach reduces the computation time by 26% compared to the ILP approach while increasing the largest maximum delay by just 3.4% in average. It reduces the largest maximum delay compared with the start-time optimization model; it avoids the instability caused by the unnecessary disconnection permitted by the run-time optimization model.

This paper proposes two algorithms, namely Server-User Matching (SUM) algorithm and Extended Server-User Matching (ESUM) algorithm, for the distributed server allocation problem. The server allocation problem is to determine the matching between servers and users to minimize the maximum delay, which is the maximum time to complete user synchronization. We analyze the computational time complexity. We prove that the SUM algorithm obtains the optimal solutions in polynomial time for the special case that all server-server delay values are the same and constant. We provide the upper and lower bounds when the SUM algorithm is applied to the general server allocation problem. We show that the ESUM algorithm is a fixed-parameter tractable algorithm that can attain the optimal solution for the server allocation problem parameterized by the number of servers. Numerical results show that the computation time of ESUM follows the analyzed complexity while the ESUM algorithm outperforms the approach of integer linear programming solved by our examined solver.

The dispersion problem is a variant of the facility location problem. Given a set P of n points and an integer k, we intend to find a subset S of P with |S|=k such that the cost minp∈S{cost(p)} is maximized, where cost(p) is the sum of the distances from p to the nearest c points in S. We call the problem the dispersion problem with partial c sum cost, or the PcS-dispersion problem. In this paper we present two algorithms to solve the P2S-dispersion problem(c=2) if all points of P are on a line. The running times of the algorithms are O(kn2 log n) and O(n log n), respectively. We also present an algorithm to solve the PcS-dispersion problem if all points of P are on a line. The running time of the algorithm is O(knc+1).

An r-gathering of customers C to facilities F is an assignment A of C to open facilities F' ⊂ F such that r (≥ 2) or more customers are assigned to each open facility. (Each facility needs enough number of customers for its opening.) Then the r-gathering problem finds an r-gathering minimizing a designated cost. Armon gave a simple 3-approximation algorithm for the r-gathering problem and proved that with assumption P ≠ NP the problem cannot be approximated within a factor of less than 3 for any r ≥ 3. The running time of the 3-approximation algorithm is O(|C||F|+r|C|+|C|log|C|)). In this paper we improve the running time of the algorithm by (1) removing the sort in the algorithm and (2) designing a simple but efficient data structure.

We consider the capacitated multi-source multicast tree routing problem (CMMTR) in an undirected graph G=(V,E) with a vertex set V, an edge set E and an edge weight w(e) ≥ 0, e ∈ E. We are given a source set S ⊆ V with a weight g(e) ≥ 0, e ∈ S, a terminal set M ⊆ V-S with a demand function q : M → R+, and a real number κ > 0, where g(s) means the cost for opening a vertex s ∈ S as a source in a multicast tree. Then the CMMTR asks to find a subset S′⊆ S, a partition {Z1,Z2,...,Zl} of M, and a set of subtrees T1,T2,...,Tl of G such that, for each i, ∑t∈Ziq(t) ≤ κ and Ti spans Zi∪{s} for some s ∈ S′. The objective is to minimize the sum of the opening cost of S′and the constructing cost of {Ti}, i.e., ∑s∈S′g(s)+w(Ti), where w(Ti) denotes the sum of weights of all edges in Ti. In this paper, we propose a (2ρUFL+ρST)-approximation algorithm to the CMMTR, where ρUFL and ρST are any approximation ratios achievable for the uncapacitated facility location and the Steiner tree problems, respectively. When all terminals have unit demands, we give a ((3/2)ρUFL+(4/3)ρST)-approximation algorithm.

Given a graph G=(V,E), a set of vertices S ⊆ V covers v ∈ V if the edge connectivity between S and v is at least a given number k. Vertices in S are called sources. The source location problem is a problem of finding a minimum-size source set covering all vertices of a given graph. This paper presents a new variation of the problem, called maximum-cover source-location problem, which finds a source set S with a given size p, maximizing the sum of the weight of vertices covered by S. It presents an O(np + m + nlog n)-time algorithm for k=2, where n=|V| and m=|E|. Especially it runs linear time if G is connected. This algorithm uses a subroutine for finding a subtree with the maximum weight among p-leaf trees of a given vertex-weighted tree. For the problem we give a greedy-based linear-time algorithm, which is an extension of the linear-time algorithm for finding a longest path of a given tree presented by E. W. Dijkstra around 1960. Moreover, we show some polynomial solvable cases, e.g., a given graph is a tree or (k-1)-edge-connected, and NP-hard cases, e.g., a vertex-cost function is given or G is a digraph.

Let G = (V,E) be an edge weighted graph with n vertices and m edges. For a given integer p with 1 < p < n, we call a set X V of p vertices a p-maximizer if X has a property that the edge-connectivity of the graph obtained by contracting X into a single vertex is no less than that of the graph obtained by contracting any other subset of p vertices. In this paper, we first show that there always exists an ordering v1,v2,...,vn of vertices in V such that, for each i = 2,3,...,n - 1, set {v1,v2,...,vi} is an i-maximizer. We give an O(mn + n2log n) time algorithm for finding such an ordering and then show an application to the source location problem.

In this paper, we consider a network design problem with hub-and-spoke structure. We propose a relaxation technique for the problem where the location of hub nodes is given and decides the allocation of non-hub nodes to one of the hub nodes. We linearize the non-convex quadratic objective function of the original problem, introducing Hitchcock transportation problems defined for each pair of non-hub nodes. We provide two linear relaxation problems, one based on the Hitchcock transportation problems and the other on the dual Hitchcock transportation problems. We show the tightness of the lower bounds obtained by our formulations by computational experiences.

In this paper we consider a compound problem of dynamic flows and sink location in a tree network. Given a dynamic flow network of tree structure with initial supplies at vertices, the problem is to find a vertex v as a sink in the network such that we can send all the initial suplies to v as quick as possible. This problem can be regarded as a dynamic flow version of the 1-center problem in a tree network. We present an O(n2) time algorithm for the sink location problem, where n is the number of vertices in the network.

The redundancy allocation problem for a series-parallel system is a well known as one of NP-hard combinatorial problems and it generally belongs to the class of nonlinear integer programming (nIP) problem. Many researchers have developed the various methods which can be roughly categorized into exact solution methods, approximate methods, and heuristic methods. Though each method has both advantages and disadvantage, the heuristic methods have been received much attention since other methods involve more computation effort and usually require larger computer memory. Genetic algorithm (GA) as one of heuristic optimization techniques is a robust evolutionary optimization search technique with very few restrictions concerning with the various design problems. However, GAs cannot guarantee the optimality and sometimes can suffer from the premature convergence situation of its solution, because it has some unknown parameters and it neither uses a priori knowledge nor exploits the local search information. To improve these problems in GA, this paper proposes an effective hybrid genetic algorithm based on, 1) fuzzy logic controller (FLC) to automatically regulate GA parameters and 2) incorporation of the iterative hill climbing method to perform local exploitation around the near optimum solution for solving redundancy allocation problem. The effectiveness of this proposed method is demonstrated by comparison results with other conventional methods on two different types of redundancy allocation problems.

We consider a single-vehicle scheduling problem on a tree, where each vertex has a job with a release time and a processing time and each edge has a travel time. There is a single vehicle which starts from a start vertex s and reaches a goal vertex g after finishing all jobs. In particular, s is called a home location if s = g. The objective of the problem is to find a depth-first routing on T so as to minimize the completion time. In this paper, we first show that the minimum completion times of the problem for all home locations s V can be simultaneously computed in O(n) time, once the problem with a specified home location s V has been solved, where n is the number of vertices. We also show that given a specified start vertex s, the minimum completion times for all goal vertices g can be computed in O(n) time.

For a given graph G=(V, E), edge capacities c(e) for edges e E, and flow-demands d(v) for nodes v V, a problem of finding the minimum size source-set S V such that the maximum flow-amount between S and v is greater than or equal to d(v) for every v V is called generalized plural cover problem (GPC). For this problem, O(np s(n,m)) time algorithm is presented, where n, m, and p is the number of nodes, edges, and different values of d(v), respectively, and s(n,m) is the time required to solve the maximum flow problem of a graph with n nodes and m edges. This algorithm also constructs a set of optimal solutions in the same time. This property is convenient for applying it to real problems, because other constraints can also be taken into account.

The lower-bounded p-collection problem is the problem where to locate p sinks in a flow network with lower bounds such that the value of a maximum flow is maximum. This paper discusses the cover problems corresponding to the lower bounded p-collection problem. We consider the complexity of the cover problem, and we show polynomial time algorithms for its subproblems in a network with tree structure.

For given integerp, a flow networkNwithn vertices, and sources inN, a problem of finding location ofp sinks inN which maximize the value of maximum flow from sources to sinks is calledp-collection problem. This problem is NP-hard even if a given network is a tree, but a pseudo-polynomial time algorithm is possible for such a network. This paper proposes a new pseudo-polynomial time algorithm for a tree-type network, which is simpler and more efficient than the previous algorithm, and has time complexity of O(p2n2Cc min {Cc,Cd}), whereCc andCd are the maximum edge capacity and the maximum vertex weight, respectively.

In this paper we extend the p-collection problem to a flow network with lower bounds, and call the extended problem the lower-bounded p-collection problem. First we discuss the complexity of this problem to show NP-hardness for a network with path structure. Next we present a linear time algorithm for the lower-bounded 1-collection problem in a network with tree structure, and a pseudo-polynomial time algorithm with dynamic programming type for the lower-bounded p-collection problem in a network with tree structure. Using the pseudo-polynomial time algorithm, we show an exponential algorithm, which is efficient in a connected network with few cycles, for the lower-bounded p-collection problem.

The p-collection problem is where to locate p sinks in a flow network such that the value of a maximum flow is maximum. In this paper we show complexity results of the p-collection problem. We prove that the decision problem corresponding to the p-collection problem is strongly NP-complete. Although location problems (the p-center problem and the p-median problem) in networks and flow networks with tree structure is solvable in polynomial time, we prove that the decision problem of the p-collection problem in networks with tree structure, is weakly NP-complete. And we show a polynomial time algorithm for the subproblem of the p-collection problem such that the degree sum of vertices with degree3 in a network, is bound to some constant K0.

This paper proposes an optimal free-sensors allocation problem (OFSAP) in safety monitoring systems. OFSAP is the problem of deciding the optimal allocation of several sensors, which we call free sensors, to plural objects. The solution of OFSAP gives the optimal allocation which minimizes expected losses caused by failed dangerous (FD)-failures and failed safe (FS)-failures; a FD-failure is to fail to generate an alarm for unsafe object and a FS-failure is to generate an alarm for safe object. We show an unexpected result that a safer object should be monitored by more sensors under certain conditions.