The weighted k-center problem in graphs is a classical facility location problem where we place k centers on the graph, which minimize the maximum weighted distance of a vertex to its nearest center. We study this problem when the underlying graph is a cactus with n vertices and present an O(n log ^2 n) time algorithm for the same. This time complexity improves upon the O(n^2) time algorithm by Ben-Moshe et al. [2], which is the current state-of-the-art.

We address the problem of placing k sinks on dynamic-flow path networks with n vertices so as to minimize their maximum evacuation completion time. We develop two different algorithms that, when all edges have the same capacity, run respectively in O(n+k2log2 n) and O(n log n) time. When the edge capacities can be different, i.e., are general, they run respectively in O(n log n+k2log4 n) and O(n log3 n) time. These algorithms improve upon the previously most efficient algorithms, which had time complexities O(kn) and O(knlog2n), respectively, for the uniform and general edge capacity models. The improvements are achieved by moving from a dynamic programming based approach to a parametric-search based one.

Given a set P of n points in the plane, the k -center problem is to find k congruent disks of minimum possible radius such that their union covers all the points in P . The 2-center problem is a special case of the k -center problem that has been extensively studied in the recent past [ 7 , 20 , 22 ]. In this paper, we consider a generalized version of the 2-center problem called proximity connected 2-center (PCTC) problem. In this problem, we are also given a parameter δ ≥ 0 and we have the additional constraint that the distance between the centers of the disks should be at most δ . Note that when δ = 0 , the PCTC problem is reduced to the 1-center(minimum enclosing disk) problem and when δ tends to infinity, it is reduced to the 2-center problem. The PCTC problem first appeared in the context of wireless networks in 1992 [ 12 ], but obtaining a nontrivial deterministic algorithm for the problem remained open. In this paper, we resolve this open problem by providing a deterministic O ( n 2 log n ) time algorithm for the problem.

We present a linear time algorithm for the weighted k-center problem on trees for fixed k. This partially settles the long-standing question about the lower bound on the time complexity of the problem. The current time complexity of the best-known algorithm for the problem with k as part of the input is O(nlog⁡n) by Wang et al. (2018) [20]. Whether an O(n) time algorithm exists for arbitrary k is still open.

The k -center problem is one of the most well-known problems in combinatorial optimization which has been extensively studied in the past. In this paper, we introduce a generalized version of the k -center problem called proximity connected k -center (PCkC) problem. In this problem, we are given a set of demand points in a metric space and a parameter δ > 0. We are going to locate k center points such that the maximum distance of a demand point to its nearest center is minimized and each pair of centers can communicate with each other either directly or via other centers assuming that each center can directly communicate with any other center within the range of δ of itself. Note that when δ is large enough, the problem turns to the k -center problem and when δ tends to zero, the problem turns to the 1-center problem. We consider the PCkC problem when the underlying space is a path and present a sub-quadratic time algorithm for both the unweighted and the weighted demand points cases.

We present dynamic flow algorithms to solve the k -sink problem whose aim is to locate k sinks (evacuation centers) in such a way that the evacuation time of the last evacuee is minimized. In the confluent model , the evacuees originating from or passing through a vertex must evacuate to the same sink, and most known results on the k -sink problem adopt the confluent model. When the edge capacities are uniform (resp. general), our algorithms for non-confluent flow in the path networks run in O ( n + k 2 log 2 n ) (resp. O ( n log n + k 2 log 5 n )) time, where n is the number of vertices. Our algorithms for cycle networks run in O ( k 2 n log 2 n ) (resp. O ( k 2 n log 5 n )) time, when the edge capacities are uniform (resp. general)

The database skyline query (or non-domination query) has a spatial form: Given a set P with n point sites, and a point set S of m locations of interest, a site p∈P is a skyline point if and only if for each q∈P∖{p}, there exists at least one location s∈S that is closer to p than to q. We reduce the problem of determining skyline points to the problem of finding sites that have non-empty cells in an additively weighted Voronoi diagram under a convex distance function. The weights of said Voronoi diagram are derived from the coordinates of the sites of P, while the convex distance function is derived from the set of locations S. In the two-dimensional plane, this reduction gives an O((n+m)log⁡(n+m))-time algorithm to find the skyline points.

We consider the problem of locating a set of k sinks on a path network with general edge capacities that minimizes the sum of the evacuation times of all evacuees. We first present an O ( k n log 4 ⁡ n ) time algorithm when the edge capacities are non-uniform, where n is the number of vertices. We then present an O ( k n log 3 ⁡ n ) time algorithm when the edge capacities are uniform. We also present an O ( n log ⁡ n ) time algorithm for the special case where k = 1 and the edge capacities are non-uniform.

In this paper, we study some fundamental facility location problems from the space-efficient perspective. We show that 1-center problem in Euclidean space and in tree networks can be efficiently solved in constant-workspace model. We use a virtual pairing tree during the pruning stage that allows the maintenance of pruned elements. We also show that a feasible region during the prune-and-search process can always be maintained using O(1) space. These results realize the solutions to the problems in constant work-space model: linear programming in 2-d, 3-d, 2-center in trees, and largest disk inside a convex polygon.

The bilinear assignment problem (BAP) is a generalization of the well-known quadratic assignment problem. In this paper, we study the problem from the computational analysis point of view. Several classes of neighborhood structures are introduced for the problem along with some theoretical analysis. These neighborhoods are then explored within a local search and variable neighborhood search frameworks with multistart to generate robust heuristic algorithms. In addition, we present several very fast construction heuristics. Our systematic experimental analysis disclosed some interesting properties of the BAP, different from those of comparable models. We have also introduced benchmark test instances that can be used for future experiments on exact and heuristic algorithms for the problem.

This paper considers the problem of evacuating people located at vertices to a “sink” in a cycle network. In the “minmax-regret” version of this problem, the exact number of evacuees at each vertex is unknown, but only an interval for a possible number is given. We show that a minmax-regret 1-sink in cycle networks with uniform edge capacities can be found in $$O(n^2)$$ time, where n is the number of vertices. No correct algorithm was known before for this problem.

We investigate three kinds of optimization problems regarding n points in the 2-dimensional plane that need to be enclosed by squares. (1) Find a given number of squares that enclose all the points, minimizing the size of the largest square used. (2) Problem (1) with the additional condition that the center of each enclosing square must lie on one of the two given axis-parallel lines, which are either parallel or perpendicular. (3) Enclose the maximum number of points, using a specified number of squares of a fixed size. We propose different techniques to solve the above problems in cases where squares are axis-parallel or of arbitrary orientation, disjoint or overlapping. All the algorithms we use run in time that is a low-order polynomial in n , and improve upon the previous algorithms, if any.

Evacuation in emergency situations can be modeled by a dynamic flow network. Two criteria have been used before: one is the evacuation completion time and the other is the aggregate evacuation time of individual evacuees. The aim of this paper is to optimize the aggregate evacuation time in the simplest case, where the network is a path and only one evacuation center (called a sink) is to be introduced. The evacuees are initially located at the vertices, but their precise numbers are unknown, and are given by upper and lower bounds. Under this assumption, we compute the sink location that minimizes the maximum "regret." We present an $O(n^2\log n)$ time algorithm to solve this problem, improving upon the previously fastest $O(n^3)$ time algorithm, where $n$ is the number of vertices.

In emergencies such as earthquakes, nuclear accidents, etc., we need an evacuation plan. We model a street, a building corridor, etc. by a path network, and consider the problem of locating a set of k sinks on a dynamic flow path network with n vertices, where people are located, that minimizes the sum of the evacuation times of all evacuees. Our minsum model is more difficult to deal with than the minmax model, because the cost function is not monotone along the path. We present an O(kn(2) log(2) n) time algorithm for solving this problem, which is the first polynomial time result. If the edge capacities are uniform, we give an O(kn log(3) n) time algorithm.

In this paper we study the {\it bilinear assignment problem} (BAP) with size parameters $m$ and $n$, $m\leq n$. BAP is a generalization of the well known quadratic assignment problem and the three dimensional assignment problem and hence NP-hard. We show that BAP cannot be approximated within a constant factor unless P=NP even if the associated quadratic cost matrix $Q$ is diagonal. Further, we show that BAP remains NP-hard if $m = O(\sqrt[r]{n})$, for some fixed $r$, but is solvable in polynomial time if $m = O(\sqrt{\log n})$. When the rank of $Q$ is fixed, BAP is observed to admit FPTAS and when this rank is one, it is solvable in polynomial time under some additional restrictions. We then provide a necessary and sufficient condition for BAP to be equivalent to two linear assignment problems. A closed form expression to compute the average of the objective function values of all solutions is presented, whereas the median of the solution values cannot be identified in polynomial time, unless P=NP. We then provide polynomial time heuristic algorithms that find a solution with objective function value no worse than that of $(m-1)!(n-1)!$ solutions. However, computing a solution whose objective function value is no worse than that of $m!n!-\lceil\frac{m}{\beta}\rceil !\lceil\frac{n}{\beta}\rceil !$ solutions is NP-hard for any fixed rational number $\beta>1$.

Given the contour of a simple polygon P as an ordered set V of n vertices including a start vertex v, we model the optimization problem of representing P with a smallest-size unordered set S={V∪V′} of vertices, where V′ denotes an additional set of pseudo-vertices chosen along the edges of P such that P is perceivable uniquely by applying a progressive nearest-neighbor traversal rule. A traversal that uses the nearest-neighbor rule on the set S is said to perceive the polygon P if the traversal on S from the same start vertex v∈S visits the vertices in P in the same order when the following rule is applied: Recursively choose the next nearest neighbor v′∈S of v and then delete the last visited vertex v until all the vertices in S is traversed. The set S of vertices by itself should be tangible by touch (tactile information) in the sense that it is able to convey the perception of the shape to a blind reader in the same way as it was described in its input. A desirable objective in this context is to find the smallest-cardinality set V′ such that P can be perceived uniquely from S={V∪V′} using the nearest-neighbor traversal rule. In this paper, we propose to choose a set V⁎ with a sufficiently large cardinality such that the unordered set S⁎={V∪V⁎} can be used to perceive P using the nearest-neighbor traversal rule. We also compute an upper bound on |V⁎| constructed by the proposed algorithm, in terms of certain geometric parameters of the polygon P.

We address the problem of locating k sinks on dynamic flow path networks with n vertices in such a way that the evacuation completion time to them is minimized. Our two algorithms run in O(nlog n + k^2log ^4 n) and O(nlog ^3 n) time, respectively. When all edges have the same capacity, we also present two algorithms which run in O(n + k^2log ^2n) time and O(nlog n) time, respectively. These algorithms together improve upon the previously most efficient algorithms, which have time complexities O(knlog ^2n) [1] and O(kn) [11], in the general and uniform edge capacity cases, respectively. The above results are achieved by organizing relevant data for subpaths in a strategic way during preprocessing, and the final results are obtained by extracting/merging them in an efficient manner.

This article presents a linear time algorithm to solve a variant of the minimum enclosing circle (MEC) problem. The inputs are a point set S of size n, and a point b in the plane called the free point. Our goal is to locate a circle center o∗ such that the maximum distance of all points in S to o∗ divided by the distance from o∗ to b is minimized. The original investigation by Qiu et al. [5] found an O(n log n) algorithm using the furthest point Voronoi diagram of the point set S. This problem can be formulated as a generalized linear programming problem when the domain for the optimal solution is restricted and therefore, can be solved in linear expected time [3]. We describe here a simple deterministic linear time algorithm based on Meggido’s prune-and-search solution to the standard problem [4]. We extend our technique to solve similar variants of the MEC problem where the free point is replaced with other geometric objects such as a free line, a free line segment, and a set of free points.

Given a set of n weighted points on the x-y plane, we want to find a step function consisting of k horizontal steps such that the maximum vertical weighted distance from any point to a step is minimized. We solve this problem in On time when k is a constant. Our approach relies on the prune-and-search technique, and can be adapted to design similar linear time algorithms to solve the line-constrained k-center problem and the size-k histogram construction problem as well.