Fault-Tolerant, but Paradoxical Path-Finding in Physical and Conceptual Systems

We report our initial investigations into reliability and path-finding based models and propose future areas of interest. Inspired by broken sidewalks during on-campus construction projects, we develop two models for navigating this "unreliable network." These are based on a concept of "accumulating risk" backward from the destination, and both operate on directed acyclic graphs with a probability of failure associated with each edge. The first serves to introduce and has faults addressed by the second, more conservative model. Next, we show a paradox when these models are used to construct polynomials on conceptual networks, such as design processes and software development life cycles. When the risk of a network increases uniformly, the most reliable path changes from wider and longer to shorter and narrower. If we let professional inexperience--such as with entry level cooks and software developers--represent probability of edge failure, does this change in path imply that the novice should follow instructions with fewer "back-up" plans, yet those with alternative routes should be followed by the expert?Comment: 8 page

An Efficient Algorithm for Computing Network Reliability in Small Treewidth

We consider the classic problem of Network Reliability. A network is given together with a source vertex, one or more target vertices, and probabilities assigned to each of the edges. Each edge appears in the network with its associated probability and the problem is to determine the probability of having at least one source-to-target path. This problem is known to be NP-hard. We present a linear-time fixed-parameter algorithm based on a parameter called treewidth, which is a measure of tree-likeness of graphs. Network Reliability was already known to be solvable in polynomial time for bounded treewidth, but there were no concrete algorithms and the known methods used complicated structures and were not easy to implement. We provide a significantly simpler and more intuitive algorithm that is much easier to implement. We also report on an implementation of our algorithm and establish the applicability of our approach by providing experimental results on the graphs of subway and transit systems of several major cities, such as London and Tokyo. To the best of our knowledge, this is the first exact algorithm for Network Reliability that can scale to handle real-world instances of the problem.Comment: 14 page

Tree decompositions of graphs: Saving memory in dynamic programming

AbstractWe propose a simple and effective heuristic to save memory in dynamic programming on tree decompositions when solving graph optimization problems. The introduced “anchor technique” is based on a tree-like set covering problem. We substantiate our findings by experimental results. Our strategy has negligible computational overhead concerning running time but achieves memory savings for nice tree decompositions and path decompositions between 60% and 98%

A framework for network reliability problems on graphs of bounded treewidth

Abstract. In this paper, we consider problems related to the network reliability problem, restricted to graphs of bounded treewidth. We look at undirected simple graphs with each vertex and edge a number in [0, 1] associated. These graphs model networks in which sites and links can fail, with a given probability, independently of whether other sites or links fail or not. The number in [0, 1] associated to each element is the probability that this element does not fail. In addition, there are distinguished sets of vertices: a set S of servers, and a set L of clients. This paper presents a dynamic programming framework for graphs of bounded treewidth for computing for a large number of different properties Y whether Y holds for the graph formed by the nodes and edges that did not fail. For instance, it is shown that one can compute in linear time the probability that all clients are connected to at least one server, assuming the treewidth of the input graph is bounded. The classical S-terminal reliability problem can be solved in linear time as well using this framework. The method is applicable to a large number of related questions. Depending on the particular problem, the algorithm obtained by the method uses linear, polynomial, or exponential time.