Searching a network for intruders is an interesting and difficult problem. Edge-searching is one such search model, in which intruders may exist anywhere along an edge. Since finding the minimum number of searchers necessary to search a graph is NP–complete, it is natural to look for bounds on the search number. We show lower bounds on the search number using minimum degree, girth, chromatic number, and colouring number.
Random geometric graphs (RGG) contain vertices whose points are uniformly distributed in a given plane and an edge between two distinct nodes exists when their distance is less than a given positive value. RGGs are appropriate for modeling multi-hop wireless networks consisting of n mobile devices with transmission radius r unit length that are independently and uniformly distributed randomly in an area. This work presents a novel paradigm to compute the subgraph probability in RGGs. In contrast to previous asymptotic bounds or approximation, the closed-form formulas we derived herein are fairly accurate and of practical value. The proposed paradigm can be used to make quantitative analyzes on the fundamental properties of multi-hop wireless networks.
We introduce a novel variant of the well known d-dimensional bin (or vector) packing problem. Given a sequence of non-negative d-dimensional vectors, the goal is to pack these into as few bins as possible, of the smallest possible size. In the classical problem, the bin size vector is given and the sequence can be partitioned arbitrarily. We study a variation where the vectors have to be packed in the order in which they arrive, and the bin size vector can be chosen once in the beginning. This setting gives rise to two combinatorial problems: one in which we want to minimize the number of bins used for a given total bin size, and one in which we want to minimize the total bin size for a given number of bins. We prove that both problems are NP-hard, and propose an LP based bicriteria (1ε,11−ε)-approximation algorithm. We give a 2-approximation algorithm for the version with a bounded number of bins. Furthermore, we investigate properties of natural greedy algorithms, and present an easy to implement heuristic, which is fast and performs well in practice. Experiments with the heuristic and an ILP formulation yield promising results on real world data.
The max-edge-coloring problem (MECP) is finding an edge colorings E 1, E 2, E 3, ..., E z of a weighted graph G = (V, E) to minimize ∑^z_i-1 max{w(e_k)|e_k∈ E_i} , where w(e k ) is the weight of e k . In the work, we discuss the complexity issues on the MECP and its variants. Specifically, we design a 2-approximmation algorithm for the max-edge-coloring problem on biplanar graphs, which is bipartite and has a biplanar drawing. Next, we show the splitting chromatic max-edge-coloring problem, a variant of MECP, is NP-complete even when the input graph is restricted to biplanar graphs. Finally, we also show that these two problems have applications in scheduling data redistribution on parallel computer systems.
For the problem of aligning two-dimensional NMR spectra of biological samples to determine if metabolite standards in a database can be excluded as possible constituents, we develop heuristic search algorithms that offer tremendous time savings when compared to manual methods. Xi et al [15] consider this problem and use statistical methods to reduce the search space and enumerate it. In this paper we consider the case when the statistical model is not available due to lack of data. We describe a simulated annealing algorithm and an algorithm that hybridizes simulated annealing with a shift neighborhood and a variant of reactive tabu search with a large neighborhood. Computational experiments based on data from physical experiments demonstrates that the hybrid is more effective than its constituents for runs with limited CPU time but that simulated annealing and the hybrid are roughly equal for longer runs.
