Možná vylepšení pro algoritmy řešící problém Min-Power Symmetric Connectivity

Abstract

V teto praci se zabyvame problemem minimalizace celkove spotreby energie bezdratove site, ktery je formalne znamy jako Min-Power Symmetric Connectivity. Modelem bezdratove site je hranove ohodnoceny neorientovany graf. Cilem je najit souvisly podgraf, ktery pokryva vsechny vrcholy, s minimalnim souctem cen vsech vrcholu. Cena vrcholu je dana nejvetsi vahou hrany incidentni s danym vrcholem. Je znamo, ze se jedna o NP-tezky problem. Toto tvrzeni rozsirime a ukazeme, ze problem je NP-tezky pro konkretni tridu uplnych grafu. Dale predstavime dva postupy umoznujici efektivnejsi reseni Min-Power Symmetric Connectivity problemu. Jedna z metod se zameruje na hledani tzv. povinneho podgrafu. Ten jsme schopni efektivne nalezt za pomoci dolnich mezi cen vrcholu. Predstavime metodu, jak tyto dolni meze cen vrcholu zvysit. Dale se zabyvame moznou redukci poctu hran. Ukazeme metodu identifikujici nadbytecne hrany grafu, cimz snizime vypocetni slozitost ulohy.In this work, we deal with the problem of minimising the total power consumption of a wireless network, which is formally known as Min-Power Symmetric Connectivity. The wireless network is represented by an edge-weighted undirected graph. The goal is to find a connected spanning subgraph with the minimum sum of the costs across all vertices. The cost of a vertex is given by the largest weight of the edge incident to the given vertex. It is known that the problem is NP-hard. We extend this result and show that the problem is NP-hard for a particular class of complete graphs. Next, we present two routines for solving Min-Power Symmetric Connectivity more efficiently. One of the methods focuses on finding the so-called obligatory subgraph. We are able to find it efficiently using lower bounds on the vertex cost. We introduce a method for incrementing these vertex lower bounds. Subsequently, we focus on the possible reduction in the number of edges. We present a method for identifying redundant edges in a graph, thereby reducing the computational complexity of the task.