It is proved that the problem of defining a bounded cost set of nodes S such that the influence spreading from S in G, within a given time bound, is as large as possible, and that the problem is NP-hard, even in simple networks like complete graphs and trees.
ABSTRACT We address the Positive Influence Dominating Set problem (PIDS), which involves identifying the smallest group of vertices in a given network that maximizes influence throughout the network. Formally, given a graph G = (V, E), representing the network, we look for a subset of vertices P ⊆ V such that each vertex v i ∈ V \ P has at least ρ deg G (v i ) neighbors in P, where deg G (v i ) is the degree of v i , and 0 < ρ < 1. The problem is known to be NP-hard in general graphs and remains NP-hard even when restricted to certain particular classes, such as chordal graphs. In this paper, we develop an exact algorithm based on the Branch and Bound approach for solving PIDS on chordal graphs. The effectiveness of the proposed algorithm is demonstrated through numerical experiments conducted on various randomly generated chordal graphs.
Y A Bekhti, M. Lalou, Méziane Aïder et al.· Pesquisa Operacional· 0 citations
We study the shortest-path problem on graphs with positive real-valued edge weights. Given a source vertex $s$ and a target vertex $t$, the goal is to calculate the length of the shortest path from $s$ to $t$. We are particularly interested in instances that can be solved in sublinear time. Recently, Haeupler, Hlad\'ik, Rozho\v{n}, Tarjan, and T\v{e}tek proved that (a version of) the bidirectional Dijkstra's algorithm is instance-optimal on positively weighted multigraphs, both directed and undirected, considering the number of vertices and edges queried by the algorithm. However, multigraphs are not the canonical setting for the shortest-path problem. The problem is typically formulated on simple graphs without loops and parallel edges. They therefore left as an open problem whether bidirectional Dijkstra remains instance-optimal on simple weighted graphs. We answer this question, but for simple graphs, the answer is more complex, depending on the setting. We show that bidirectional Dijkstra is still instance-optimal on simple undirected weighted graphs under the order-oblivious model, where incident edges are given in a random order. In contrast, under the order-dependent model, where incident edges have a given order, we show that bidirectional Dijkstra is not instance-optimal. For simple directed weighted graphs, we show that bidirectional Dijkstra is not instance-optimal under either the order-oblivious or the order-dependent model, being off by a factor of $\Theta(m/n)$ in both cases. We further show that no algorithm can have instance-optimality ratio $o(m/n)$ under the order-dependent model, or under the order-oblivious model when $m=O(n\sqrt{n})$. On the positive side, the above results imply that bidirectional Dijkstra is instance-optimal up to logarithmic factors on all sparse directed and undirected graphs satisfying $m/n=\log^{O(1)} n$.
Christian Bertram, Mads Vestergaard Jensen, Mikkel Thorup et al.· 0 citations
Designing networks that minimize distances and satisfy structural constraints is a fundamental task across transportation, communication, and biological systems. Here, we consider the problem of finding, for a given degree sequence, the network structure displaying the smallest possible average shortest-path length. While exact solutions are available in linear time for trees, such an optimization problem becomes computationally infeasible as soon as loops are allowed in the networks. We propose a fast algorithm to construct approximate solutions to such a degree-constrained distance-minimization problem. Accordingly, edges are first created between high-degree nodes; then, additional connections are placed following the rules of the standard configuration model. In spite of its simplicity, the algorithm displays outstanding performance as demonstrated in our systematic experiments on both synthetic and real degree sequences. Our method is particularly effective on synthetic degree sequences displaying medium levels of heterogeneity. When applied to degree sequences of real networks, the proposed algorithm is able to reduce the all-pair shortest path of real structures by 20%, on average. We perform a validation on small-sized networks, where we compare the shortest-path distance of the networks generated with our algorithm against those obtained via simulated annealing optimization. Although simulated annealing yields slightly better structures, our proposed algorithm provides nearly identical solutions at a substantially lower computational cost, making it a solid method in applications concerning large-scale systems.