Communication Over a Wireless Network With Random Connections

A network of nodes in which pairs communicate over a shared wireless medium is analyzed. We consider the maximum total aggregate traffic flow possible as given by the number of users multiplied by their data rate. The model in this paper differs substantially from the many existing approaches in that the channel connections in this network are entirely random: rather than being governed by geometry and a decay-versus-distance law, the strengths of the connections between nodes are drawn independently from a common distribution. Such a model is appropriate for environments where the first-order effect that governs the signal strength at a receiving node is a random event (such as the existence of an obstacle), rather than the distance from the transmitter. It is shown that the aggregate traffic flow as a function of the number of nodes n is a strong function of the channel distribution. In particular, for certain distributions the aggregate traffic flow is at least n/(log n)^d for some d≫0, which is significantly larger than the O(sqrt n) results obtained for many geometric models. The results provide guidelines for the connectivity that is needed for large aggregate traffic. The relation between the proposed model and existing distance-based models is shown in some cases

Computational methods for finding long simple cycles in complex networks

© 2017 Elsevier B.V. Detection of long simple cycles in real-world complex networks finds many applications in layout algorithms, information flow modelling, as well as in bioinformatics. In this paper, we propose two computational methods for finding long cycles in real-world networks. The first method is an exact approach based on our own integer linear programming formulation of the problem and a data mining pipeline. This pipeline ensures that the problem is solved as a sequence of integer linear programs. The second method is a multi-start local search heuristic, which combines an initial construction of a long cycle using depth-first search with four different perturbation operators. Our experimental results are presented for social network samples, graphs studied in the network science field, graphs from DIMACS series, and protein-protein interaction networks. These results show that our formulation leads to a significantly more efficient exact approach to solve the problem than a previous formulation. For 14 out of 22 networks, we have found the optimal solutions. The potential of heuristics in this problem is also demonstrated, especially in the context of large-scale problem instances

A Tight Algorithm for Strongly Connected Steiner Subgraph On Two Terminals With Demands

Given an edge-weighted directed graph $G=(V,E)$ on $n$ vertices and a set $T=\{t_1, t_2, \ldots, t_p\}$ of $p$ terminals, the objective of the \scss ($p$-SCSS) problem is to find an edge set $H\subseteq E$ of minimum weight such that $G[H]$ contains an $t_{i}\rightarrow t_j$ path for each $1\leq i\neq j\leq p$. In this paper, we investigate the computational complexity of a variant of $2$-SCSS where we have demands for the number of paths between each terminal pair. Formally, the \sharinggeneral problem is defined as follows: given an edge-weighted directed graph $G=(V,E)$ with weight function $\omega: E\rightarrow \mathbb{R}^{\geq 0}$, two terminal vertices $s, t$, and integers $k_1, k_2$ ; the objective is to find a set of $k_1$ paths $F_1, F_2, \ldots, F_{k_1}$ from $s\leadsto t$ and $k_2$ paths $B_1, B_2, \ldots, B_{k_2}$ from $t\leadsto s$ such that $\sum_{e\in E} \omega(e)\cdot \phi(e)$ is minimized, where $\phi(e)= \max \Big\{|\{i\in [k_1] : e\in F_i\}|\ ,\ |\{j\in [k_2] : e\in B_j\}|\Big\}$. For each $k\geq 1$, we show the following: The \sharing problem can be solved in $n^{O(k)}$ time. A matching lower bound for our algorithm: the \sharing problem does not have an $f(k)\cdot n^{o(k)}$ algorithm for any computable function $f$, unless the Exponential Time Hypothesis (ETH) fails. Our algorithm for \sharing relies on a structural result regarding an optimal solution followed by using the idea of a "token game" similar to that of Feldman and Ruhl. We show with an example that the structural result does not hold for the \sharinggeneral problem if $\min\{k_1, k_2\}\geq 2$. Therefore \sharing is the most general problem one can attempt to solve with our techniques.Comment: To appear in Algorithmica. An extended abstract appeared in IPEC '1