Planar multifacility location problems with tree structure and finite dominating sets

In this paper we deal with the planar location problem with forbidden regions. We consider the median objective with block norms and show that this problem is APX-hard, even when considering the Manhattan metric as distance function and polyhedral forbidden areas. As direct consequence, the problem cannot be approximated in polynomial time within a factor of 1.0019, unless $$P=NP$$ . In addition, we give a dominating set that contains at least one optimal solution. Based on this result an approximation algorithm is derived. For special instances it is possible to improve the algorithm. These instances include problems with bounded forbidden areas and a special structure as interrelation between the new facilities. For uniform weights, this algorithm becomes an FPTAS.

This paper considers two planar facility location problems while employing the Manhattan travel metric. We first consider the p-median problem in the presence of arbitrarily shaped barriers and convex forbidden regions. For this problem we establish that the search for an optimal solution can be restricted to a finite set of easily identifiable points. Next, we consider the stochastic queue median problem in the presence of arbitrarily shaped barriers. A procedure to obtain a global optimum solution for this problem is established. The results of the paper are illustrated via numerical examples. Finally, we comment on a connection between network location problems and planar location problems which use the Manhattan travel metric.

Facility location selection is a vital decision for companies that affects both cost and delivery time over the years. However, determination of the facility location is a NP-hard problem. A hybrid algorithm that combines revised weighted fuzzy c-means with Nelder Mead (RWFCM-NM) performs well when compared with well-known algorithms for the facility location problem (FLP) with deterministic customer demands and positions. The motivation of the study is both analyzing performance of the RWFCM-NM algorithm with probabilistic customer demands and positions and proposing a new approach for this problem. This paper develops two new algorithms for FLP when customer demands and positions are probabilistic. The proposed algorithms are a probabilistic fuzzy c-means algorithm and Nelder-Mead (Probabilistic FCM-NM), a probabilistic revised weighted fuzzy c-means algorithm and Nelder Mead (Probabilistic RWFCM-NM) for the un-capacitated planar multi-facility location problem when customer positions and customer demands are probabilistic with predetermined service level. Proposed algorithms performances were tested with 13 data sets and comparisons were made with four well known algorithms. According to the experimental results, probabilistic RWFCM-NM algorithm demonstrates superiority on compared algorithms in terms of total transportation costs.

This paper considers planar location problems with rectilinear distance and barriers where the objective function is any convex, nondecreasing function of distance. Such problems have a non-convex feasible region and a nonconvex objective function. Based on an equivalent problem with modified barriers, derived in a companion paper [3], the non convex feasible set is partitioned into a network and rectangular cells. The rectangular cells are further partitioned into a polynomial number of convex subcells, called convex domains, on which the distance function, and hence the objective function, is convex. Then the problem is solved over the network and convex domains for an optimal solution. Bounds are given that reduce the number of convex domains to be examined. The number of convex domains is bounded above by a polynomial in the size of the problem.

Equitable location of facilities in a region with probabilistic barriers to travel

A globally convergent algorithm for facility location on a sphere

On the bottleneck tree alignment problems

Given a set W of k sequences (strings) and a tree structure T with k leaves, each of which is labeled with a unique sequence in W, a tree alignment is to label a sequence to each internal node of T. The weight of an edge of the tree alignment is the distance, such as Hamming distance, Levenshtein (edit) distance or reversal distance, between the two sequences labeled to the two ends of the edge. The bottleneck tree alignment problem is to find a tree alignment such that the weight of the largest edge is minimized. A lifted tree alignment is a tree alignment, where each internal node v is labeled one of the sequences that was labeled to the children of v. The bottleneck lifted tree alignment problem is to find a lifted tree alignment such that the weight of the largest edge is minimized. In this paper, we show that the bottleneck tree alignment problem is NP-complete even when the tree structure is the binary tree and the weight function is metric. For special cases, we present an exact algorithm to solve the bottleneck lifted tree alignment problem in polynomial time. If the weight function is ultrametric, we show that any lifted tree alignment is an optimal bottleneck tree alignment.KeywordsEdit distancebottleneck tree alignmentmetricultrametricNP-complete

Given the demand between each origin-destination pair on a network, the planar hub location problem is to locate the multiple hubs anywhere on the plane and to assign the traffic to them so as to minimize the total travelling cost. The trips between any two points can be nonstop (no hubs used) or started by visiting any of the hubs. The travel cost between hubs is discounted with a factor. It is assumed that each point can be served by multiple hubs.

This paper studies the problem of deciding whether the present iteration point of some algorithm applied to a planar single- facility min-max location problem, with distances measured by either an/:norm or a polyhedral gauge, is optimal or not. It turns out that this problem is equivalent to the decision problem of whether 0 belongs to the convex hull of either a finite number of points in the plane or a finite number of different/q-circles_ R 2. Although both membership problems are theoretically solvable in polynomial time, the last problem is more difficult to solve in practice than the first one. Moreover, the second problem is solvable only in the weak sense, i.e., up to a predeter- mined accuracy. Unfortunately, these polynomial-time algorithms are not practical. Although this is a negative result, it is possible to construct an efficient and extremely simple linear-time algorithm to solve the first problem. Moreover, this paper describes an implementable procedure to reduce the second decision problem to the first with any desired precision. Finally, in the last section, some computational results for these algorithms are reported.

We consider the bin packing problem with d different item sizes s i and item multiplicities a i , where all numbers are given in binary encoding. This problem formulation is also known as the one-dimensional cutting stock problem . In this work, we provide an algorithm that, for constant d , solves bin packing in polynomial time. This was an open problem for all d\ge 3 . In fact, for constant d our algorithm solves the following problem in polynomial time: Given two d -dimensional polytopes P and Q , find the smallest number of integer points in P whose sum lies in Q . Our approach also applies to high multiplicity scheduling problems in which the number of copies of each job type is given in binary encoding and each type comes with certain parameters such as release dates, processing times, and deadlines. We show that a variety of high multiplicity scheduling problems can be solved in polynomial time if the number of job types is constant.

Understanding spatial correlation is vital in many fields including epidemiology and social science. Lee, Meeks and Pettersson (Stat. Comput. 2021) recently demonstrated that improved inference for areal unit count data can be achieved by carrying out modifications to a graph representing spatial correlations; specifically, they delete edges of the planar graph derived from border-sharing between geographic regions in order to maximise a specific objective function. In this paper we address the computational complexity of the associated graph optimisation problem. We demonstrate that this problem cannot be solved in polynomial time unless P = NP; we further show intractability for two simpler variants of the problem. We follow these results with two parameterised algorithms that exactly solve the problem in polynomial time in restricted settings. The first of these utilises dynamic programming on a tree decomposition, and runs in polynomial time if both the treewidth and maximum degree are bounded. The second algorithm is restricted to problem instances with maximum degree three, as may arise from triangulations of planar surfaces, but is an FPT algorithm when the maximum number of edges that can be removed is taken as the parameter.

We present a quantum polynomial time attack against the GMMSSZ branching program obfuscator of Garg et al. (TCC’16), when instantiated with the GGH13 multilinear map of Garg et al. (EUROCRYPT’13). This candidate obfuscator was proved secure in the weak multilinear map model introduced by Miles et al. (CRYPTO’16). Our attack uses the short principal ideal solver of Cramer et al. (EUROCRYPT’16), to recover a secret element of the GGH13 multilinear map in quantum polynomial time. We then use this secret element to mount a (classical) polynomial time mixed-input attack against the GMMSSZ obfuscator. The main result of this article can hence be seen as a classical reduction from the security of the GMMSSZ obfuscator to the short principal ideal problem (the quantum setting is then only used to solve this problem in polynomial time). As an additional contribution, we explain how the same ideas can be adapted to mount a quantum polynomial time attack against the DGGMM obfuscator of Döttling et al. (ePrint 2016), which was also proved secure in the weak multilinear map model.

Computation Offloading, sending computational tasks to more resourceful servers, is becoming a widely-used approach to save limited resources on mobile devices like battery life, storage, processor, etc. Given an application that is partitioned into multiple tasks, the offloading decisions can be made on each of them. However, considering the delay constraint and the extra costs on data transmission and remote computation, it is not trivial to make optimized decisions. Existing works have formulated offloading decision problems as either graph-partitioning or binary integer programming problems. The first approach can solve the problem in polynomial time but is not applicable to delay constraints. The second approach relies on an integer programming solver without a polynomial time guarantee. We provide an algorithm, DTP (Deterministic delay constrained Task Partitioning), to solve the offloading decision problem with delay constraints. DTP gives near-optimal solution and runs in polynomial time in the number of tasks. Going beyond prior work on linear delay constraints that apply only to serial tasks, we generalize the delay constraints to settings where the dependency between tasks can be described by a tree. Furthermore, we provide another algorithm, PTP (Probabilistic delay constrained Task Partitioning), which gives stronger QoS guarantees. Simulation results show that our algorithms are accurate and robust, and scale well with the number of tasks.

Partially-specified systems and specifications are used in formal methods such as stepwise design and query checking. Existing methods consider a setting in which the systems and their correctness are Boolean. In recent years there has been growing interest and need for quantitative formal methods, where systems may be weighted and specifications may be multi valued. Weighted automata, which map input words to a numerical value, play a key role in quantitative reasoning. Technically, every transition in a weighted automaton $\mathcal{A}$ has a cost, and the value $\mathcal{A}$ assigns to a finite word w is the sum of the costs on the transitions participating in the most expensive accepting run of $\mathcal{A}$ on w. We study parameterized weighted containment: given three weighted automata $\mathcal{A}, \mathcal{B}$, and $\mathcal{C}$, with $\mathcal{B}$ being partial, the goal is to find an assignment to the missing costs in $\mathcal{B}$ so that we end up with $\mathcal{B}'$ for which $\mathcal{A} \leq \mathcal{B}' \leq \mathcal{C}$, where ≤ is the weighted counterpart of containment. We also consider a one-sided version of the problem, where only $\mathcal{A}$ or only $\mathcal{C}$ are given in addition to $\mathcal{B}$, and the goal is to find a minimal assignment with which $\mathcal{A} \leq B'$ or, respectively, a maximal one with which $\mathcal{B}' \leq \mathcal{C}$. We argue that both problems are useful in stepwise design of weighted systems as well as approximated minimization of weighted automata. We show that when the automata are deterministic, we can solve the problems in polynomial time. Our solution is based on the observation that the set of legal assignments to k missing costs forms a k-dimensional polytope. The technical challenge is to find an assignment in polynomial time even though the polytope is defined by means of exponentially many inequalities. We do so by using a powerful mathematical tool that enables us to develop a divide-and-conquer algorithm based on a separation oracle for polytopes. For nondeterministic automata, the weighted setting is much more complex, and in fact even non-parameterized containment is undecidable. We are still able to study variants of the problems, where containment is replaced by simulation.