Reachability problems in interval-constrained and cardinality-constrained graphs

Abstract

In this paper, we present two new graph structures that can be used to model information routing in heterogeneous communication networks. These are cardinality-constrained graphs (CCGs) and interval-constrained graphs (ICGs). In both cardinality- and interval-constrained graphs, the nodes in the graph are labeled. In a CCG, each label is associated with an upper bound. A path in a CCG is considered valid, if the number of nodes on the path with any particular label is at most the upper bound associated with that label. In an ICG, each label is associated with both an upper bound and a lower bound. A path in an ICG is considered valid, if the number of nodes on the path with any particular label is between the upper and lower bounds associated with that label. In this paper, we examine the computational complexities of finding valid paths in both CCGs and ICGs. In particular, we show that these problems are NP-hard. Additionally, we propose novel algorithms for finding paths in CCGs and a subset of ICGs. We also demonstrate the effectiveness of our algorithms on CCGs and ICGs of various sizes. The past decade has seen a surge in the degree of hardware-heterogeneous device interconnectivity, Additionally, there has been a proliferation of resource-constrained wireless sensor networks. Thus, the problems of finding valid paths in CCGs and ICGs are of interest. Nodes in wireless sensor networks can be faulty, vulnerable, energy-constrained, or untrusted. These nodes may also have differing degrees of performance or power efficiency. This can pose a significant challenge in routing the flow of information between nodes. The heterogeneity of nodes in such networks is captured through the use of CCGs and ICGs.