Finding the most vital node of a shortest path

AbstractIn an undirected, 2-node connected graph G=(V,E) with positive real edge lengths, the distance between any two nodes r and s is the length of a shortest path between r and s in G. The removal of a node and its incident edges from G may increase the distance from r to s. A most vital node of a given shortest path from r to s is a node (other than r and s) whose removal from G results in the largest increase of the distance from r to s. In the past, the problem of finding a most vital node of a given shortest path has been studied because of its implications in network management, where it is important to know in advance which component failure will affect network efficiency the most. In this paper, we show that this problem can be solved in O(m+nlogn) time and O(m) space, where m and n denote the number of edges and the number of nodes in G

Max flow vitality in general and stst-planar graphs

The \emph{vitality} of an arc/node of a graph with respect to the maximum flow between two fixed nodes $s$ and $t$ is defined as the reduction of the maximum flow caused by the removal of that arc/node. In this paper we address the issue of determining the vitality of arcs and/or nodes for the maximum flow problem. We show how to compute the vitality of all arcs in a general undirected graph by solving only $2(n-1)$ max flow instances and, In $st$-planar graphs (directed or undirected) we show how to compute the vitality of all arcs and all nodes in $O(n)$ worst-case time. Moreover, after determining the vitality of arcs and/or nodes, and given a planar embedding of the graph, we can determine the vitality of a `contiguous' set of arcs/nodes in time proportional to the size of the set.Comment: 12 pages, 3 figure

Robust optimization with incremental recourse

In this paper, we consider an adaptive approach to address optimization problems with uncertain cost parameters. Here, the decision maker selects an initial decision, observes the realization of the uncertain cost parameters, and then is permitted to modify the initial decision. We treat the uncertainty using the framework of robust optimization in which uncertain parameters lie within a given set. The decision maker optimizes so as to develop the best cost guarantee in terms of the worst-case analysis. The recourse decision is ``incremental"; that is, the decision maker is permitted to change the initial solution by a small fixed amount. We refer to the resulting problem as the robust incremental problem. We study robust incremental variants of several optimization problems. We show that the robust incremental counterpart of a linear program is itself a linear program if the uncertainty set is polyhedral. Hence, it is solvable in polynomial time. We establish the NP-hardness for robust incremental linear programming for the case of a discrete uncertainty set. We show that the robust incremental shortest path problem is NP-complete when costs are chosen from a polyhedral uncertainty set, even in the case that only one new arc may be added to the initial path. We also address the complexity of several special cases of the robust incremental shortest path problem and the robust incremental minimum spanning tree problem

Physiology-Aware Rural Ambulance Routing

In emergency patient transport from rural medical facility to center tertiary hospital, real-time monitoring of the patient in the ambulance by a physician expert at the tertiary center is crucial. While telemetry healthcare services using mobile networks may enable remote real-time monitoring of transported patients, physiologic measures and tracking are at least as important and requires the existence of high-fidelity communication coverage. However, the wireless networks along the roads especially in rural areas can range from 4G to low-speed 2G, some parts with communication breakage. From a patient care perspective, transport during critical illness can make route selection patient state dependent. Prompt decisions with the relative advantage of a longer more secure bandwidth route versus a shorter, more rapid transport route but with less secure bandwidth must be made. The trade-off between route selection and the quality of wireless communication is an important optimization problem which unfortunately has remained unaddressed by prior work. In this paper, we propose a novel physiology-aware route scheduling approach for emergency ambulance transport of rural patients with acute, high risk diseases in need of continuous remote monitoring. We mathematically model the problem into an NP-hard graph theory problem, and approximate a solution based on a trade-off between communication coverage and shortest path. We profile communication along two major routes in a large rural hospital settings in Illinois, and use the traces to manifest the concept. Further, we design our algorithms and run preliminary experiments for scalability analysis. We believe that our scheduling techniques can become a compelling aid that enables an always-connected remote monitoring system in emergency patient transfer scenarios aimed to prevent morbidity and mortality with early diagnosis treatment.Comment: 6 pages, The Fifth IEEE International Conference on Healthcare Informatics (ICHI 2017), Park City, Utah, 201