Extensive facility location problems on networks with equity measures

AbstractThis paper deals with the problem of locating path-shaped facilities of unrestricted length on networks. We consider as objective functions measures conceptually related to the variability of the distribution of the distances from the demand points to a facility. We study the following problems: locating a path which minimizes the range, that is, the difference between the maximum and the minimum distance from the vertices of the network to a facility, and locating a path which minimizes a convex combination of the maximum and the minimum distance from the vertices of the network to a facility, also known in decision theory as the Hurwicz criterion. We show that these problems are NP-hard on general networks. For the discrete versions of these problems on trees, we provide a linear time algorithm for each objective function, and we show how our analysis can be extended also to the continuous case

Designing Overlapping Networks for Publish-Subscribe Systems

From the publish-subscribe systems of the early days of the Internet to the recent emergence of Web 3.0 and IoT (Internet of Things), new problems arise in the design of networks centered at producers and consumers of constantly evolving information. In a typical problem, each terminal is a source or sink of information and builds a physical network in the form of a tree or an overlay network in the form of a star rooted at itself. Every pair of pub-sub terminals that need to be coordinated (e.g. the source and sink of an important piece of control information) define an edge in a bipartite demand graph; the solution must ensure that the corresponding networks rooted at the endpoints of each demand edge overlap at some node. This simple overlap constraint, and the requirement that each network is a tree or a star, leads to a variety of new questions on the design of overlapping networks. In this paper, for the general demand case of the problem, we show that a natural LP formulation has a non-constant integrality gap; on the positive side, we present a logarithmic approximation for the general demand case. When the demand graph is complete, however, we design approximation algorithms with small constant performance ratios, irrespective of whether the pub networks and sub networks are required to be trees or stars

Location problems with multiple criteria

This chapter analyzes multicriteria continuous, network, and discrete location problems. In the continuous framework, we provide a complete description of the set of weak Pareto, Pareto, and strict Pareto locations for a general Q-criteria location problem based on the characterization of three criteria problems. In the network case, the set of Pareto locations is characterized for general networks as well as for tree networks using the concavity and convexity properties of the distance function on the edges. In the discrete setting, the entire set of Pareto locations is characterized using rational generating functions of integer points in polytopes. Moreover, we describe algorithms to obtain the solutions sets (the different Pareto locations) using the above characterizations. We also include a detailed complexity analysis. A number of references has been cited throughout the chapter to avoid the inclusion of unnecessary technical details and also to be useful for a deeper analysis

Reconfigurations of Combinatorial Problems: Graph Colouring and Hamiltonian Cycle

We explore algorithmic aspects of two known combinatorial problems, Graph Colouring and Hamiltonian Cycle, by examining properties of their solution space. One can model the set of solutions of a combinatorial problem $P$ by the solution graph $R(P)$, where vertices are solutions of $P$ and there is an edge between two vertices, when the two corresponding solutions satisfy an adjacency reconfiguration rule. For example, we can define the reconfiguration rule for graph colouring to be that two solutions are adjacent when they differ in colour in exactly one vertex. The exploration of the properties of the solution graph $R(P)$ can give rise to interesting questions. The connectivity of $R(P)$ is the most prominent question in this research area. This is reasonable, since the main motivation for modelling combinatorial solutions as a graph is to be able to transform one into the other in a stepwise fashion, by following paths between solutions in the graph. Connectivity questions can be made binary, that is expressed as decision problems which accept a 'yes' or 'no' answer. For example, given two specific solutions, is there a path between them? Is the graph of solutions $R(P)$ connected? In this thesis, we first show that the diameter of the solution graph $R_{l}(G)$ of vertex $l$-colourings of k-colourable chordal and chordal bipartite graphs $G$ is $O(n^2)$, where $l > k$ and n is the number of vertices of $G$. Then, we formulate a decision problem on the connectivity of the graph colouring solution graph, where we allow extra colours to be used in order to enforce a path between two colourings with no path between them. We give some results for general instances and we also explore what kind of graphs pose a challenge to determine the complexity of the problem for general instances. Finally, we give a linear algorithm which decides whether there is a path between two solutions of the Hamiltonian Cycle Problem for graphs of maximum degree five, and thus providing insights towards the complexity classification of the decision problem

Orthogonal Graph Drawing with Inflexible Edges

We consider the problem of creating plane orthogonal drawings of 4-planar graphs (planar graphs with maximum degree 4) with constraints on the number of bends per edge. More precisely, we have a flexibility function assigning to each edge $e$ a natural number $\mathrm{flex}(e)$, its flexibility. The problem FlexDraw asks whether there exists an orthogonal drawing such that each edge $e$ has at most $\mathrm{flex}(e)$ bends. It is known that FlexDraw is NP-hard if $\mathrm{flex}(e) = 0$ for every edge $e$. On the other hand, FlexDraw can be solved efficiently if $\mathrm{flex}(e) \ge 1$ and is trivial if $\mathrm{flex}(e) \ge 2$ for every edge $e$. To close the gap between the NP-hardness for $\mathrm{flex}(e) = 0$ and the efficient algorithm for $\mathrm{flex}(e) \ge 1$, we investigate the computational complexity of FlexDraw in case only few edges are inflexible (i.e., have flexibility~$0$). We show that for any $\varepsilon > 0$ FlexDraw is NP-complete for instances with $O(n^\varepsilon)$ inflexible edges with pairwise distance $\Omega(n^{1-\varepsilon})$ (including the case where they induce a matching). On the other hand, we give an FPT-algorithm with running time $O(2^k\cdot n \cdot T_{\mathrm{flow}}(n))$, where $T_{\mathrm{flow}}(n)$ is the time necessary to compute a maximum flow in a planar flow network with multiple sources and sinks, and $k$ is the number of inflexible edges having at least one endpoint of degree 4.Comment: 23 pages, 5 figure

Discrete Convex Functions on Graphs and Their Algorithmic Applications

The present article is an exposition of a theory of discrete convex functions on certain graph structures, developed by the author in recent years. This theory is a spin-off of discrete convex analysis by Murota, and is motivated by combinatorial dualities in multiflow problems and the complexity classification of facility location problems on graphs. We outline the theory and algorithmic applications in combinatorial optimization problems

Approximation Algorithms for the Capacitated Domination Problem

We consider the {\em Capacitated Domination} problem, which models a service-requirement assignment scenario and is also a generalization of the well-known {\em Dominating Set} problem. In this problem, given a graph with three parameters defined on each vertex, namely cost, capacity, and demand, we want to find an assignment of demands to vertices of least cost such that the demand of each vertex is satisfied subject to the capacity constraint of each vertex providing the service. In terms of polynomial time approximations, we present logarithmic approximation algorithms with respect to different demand assignment models for this problem on general graphs, which also establishes the corresponding approximation results to the well-known approximations of the traditional {\em Dominating Set} problem. Together with our previous work, this closes the problem of generally approximating the optimal solution. On the other hand, from the perspective of parameterization, we prove that this problem is {\it W[1]}-hard when parameterized by a structure of the graph called treewidth. Based on this hardness result, we present exact fixed-parameter tractable algorithms when parameterized by treewidth and maximum capacity of the vertices. This algorithm is further extended to obtain pseudo-polynomial time approximation schemes for planar graphs

The Agricultural Spraying Vehicle Routing Problem With Splittable Edge Demands

In horticulture, spraying applications occur multiple times throughout any crop year. This paper presents a splittable agricultural chemical sprayed vehicle routing problem and formulates it as a mixed integer linear program. The main difference from the classical capacitated arc routing problem (CARP) is that our problem allows us to split the demand on a single demand edge amongst robotics sprayers. We are using theoretical insights about the optimal solution structure to improve the formulation and provide two different formulations of the splittable capacitated arc routing problem (SCARP), a basic spray formulation and a large edge demands formulation for large edge demands problems. This study presents solution methods consisting of lazy constraints, symmetry elimination constraints, and a heuristic repair method. Computational experiments on a set of valuable data based on the properties of real-world agricultural orchard fields reveal that the proposed methods can solve the SCARP with different properties. We also report computational results on classical benchmark sets from previous CARP literature. The tested results indicated that the SCARP model can provide cheaper solutions in some instances when compared with the classical CARP literature. Besides, the heuristic repair method significantly improves the quality of the solution by decreasing the upper bound when solving large-scale problems.Comment: 25 pages, 8 figure

Location routing problems on trees

This paper addresses combined location/routing problems defined on trees, where not all vertices have to be necessarily visited. A mathematical programming formulation is presented, which has the integrality property. The formulation models a directed forest where each connected component hosts at least one open facility, which becomes the root of the component. The problems considered can also be optimally solved with ad-hoc solution algorithms. Greedy type optimal algorithms are presented for the cases when all vertices have to be visited and facilities have no set-up costs. Facilities set-up costs can be handled with low-order interchanges, whose optimality check is a re-statement of the complementary slackness conditions of the proposed formulation. The general problems where not all vertices have to be necessarily visited can also be optimally solved with low-order optimal algorithms based on recursions.Peer ReviewedPostprint (author's final draft