![]() ![]() An argument can be made that this hardness result is more meaningful than the trivial polynomial time promise algorithm. We show perhaps the surprising result that robustly finding a maximum independent set in a well-covered graph (i.e., a graph in which every maximal independent set is of the same size) is NP-hard. There exist problems that have a polynomial time promise solution, while being NP-hard if required to be robust. This is to be contrasted with the “promise” version of solving problems on restricted domains, in which there is a guarantee that the input is in the class, and an algorithm to “solve” the problem need not function correctly or even terminate if this guarantee is not met. Under this definition, an algorithm is required to be “robust,” i.e., it must produce correct output regardless of whether the input actually belongs to the restricted domain or not. ![]() We introduce a new definition of efficient algorithms for restricted domains. We then demonstrate in the third part how comparability graphs and interval graphs can be used for solving specific applied problems such as the seriation problem in archeology or certain scheduling problems over partially ordered sets. Again, we represent algorithmic methods for interval graph recognition and for solving combinatorial optimization problems on these graphs. The second part deals with the related class of interval graphs, which are exactly the incomparability graphs of interval orders. Topics dealt with are algorithmic methods and the necessary theoretical background for comparability graph recognition, for constructing all partial orders with the same comparability graphs, for decomposing comparability graphs and partial orders, for determining comparability invariants such as order dimension or jump number by decomposition, and for solving combinatorial optimization problems on comparability graphs. The first part of the paper gives a survey of this second aspect of comparability graphs. They constitute an important interface between graphs and partial orders both for theoretical investigations on their structural properties, and the development of efficient algorithmic methods for otherwise NP-hard combinatorial (optimization) problems on partial orders and their comparability graphs. The report also presents a hierarchy of dynamic networks based on dynamic graph properties, thereby offering a combinatorial alternative to the well-known mobility models typically used in simulations.Ĭomparability graphs are undirected graphs that represent the comparability relation of partial orders. We extend many graph theoretical concepts towards a dynamic variant and show how these new variants impact the solution of classical distributed problems. These tools include a dynamic graph formalism, various computational models, and communication models for distributed networks. In this report, we identify a collection of recent theoretical tools whose purpose is to model, describe, and leverage dynamic networks in a formal way. As a result, it is hard and sometimes impossible to guarantee, mathematically, that a given algorithm will reach its objectives once deployed in real conditions. Unfortunately, few theoretical tools to date have enabled the study of dynamic networks in a formal and rigorous way. This trend exists both in everyday life (e.g., smartphones, vehicles, and commercial satellites) and in a military context (e.g., dismounted soldiers or swarms of UAVs). By a case study using the IEEE 802.11 protocol as the underlying CSMA protocol, the proposed scheme pleads itself as a more efficient alternative to the RTS/CTS based collision avoidance scheme for large and dense multi-hop ad hoc networks with stationary nodes, such as wireless mesh and sensor networks.The number of telecommunication networks deployed in a dynamic environment is quickly growing. We provide a general framework for the SLICON scheme and compare its performance to the conventional RTS/CTS-based collision avoidance scheme. The contention among neighbors can be handled much more efficiently by a basic CSMA protocol as if operating in a single-hop network. In this work, we propose a novel hybrid channel access scheme that spatially limits the contention in the network such that 2-hop neighbors access the channel contention-free among each other whereas only the immediate neighbors may contend among each other. With rapid developments in the community mesh networks and wireless sensor networks research, the need for more efficient channel access techniques for multi-hop wireless networks has become eminent. ![]()
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