2014/12/17 by Maximilien Gadouleau, Gadouleau, Maximilien, Adrien Richard +4 · 1 citation
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #Cooperative Communication and Network Coding #DNA and Biological Computing #Discrete Mathematics (cs.DM) #Dynamical Systems (math.DS) #FOS: Computer and information sciences #FOS: Mathematics #Gene Regulatory Network Analysis #Information Theory (cs.IT) #cs.DM #cs.IT #math.DS #math.IT
paper · pdf · doi:10.48550/arxiv.1412.5310
arxiv created 2014/12/17 · openalex publication_date 2014/12/17 · arxiv updated 2014/12/18 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
Linear network coding transmits data through networks by letting the intermediate nodes combine the messages they receive and forward the combinations towards their destinations. The solvability problem asks whether the demands of all the destinations can be simultaneously satisfied by using linear network coding. The guessing number approach converts this problem to determining the number of fixed points of coding functions f:An→ An over a finite alphabet A (usually referred to as Boolean networks if A = \0,1\) with a given interaction graph, that describes which local functions depend on which variables. In this paper, we generalise the so-called reduction of coding functions in order to eliminate variables. We then determine the maximum number of fixed points of a fully reduced coding function, whose interaction graph has a loop on every vertex. Since the reduction preserves the number of fixed points, we then apply these ideas and results to obtain four main results on the linear network coding solvability problem. First, we prove that non-decreasing coding functions cannot solve any more instances than routing already does. Second, we show that triangle-free undirected graphs are linearly solvable if and only if they are solvable by routing. This is the first classification result for the linear network coding solvability problem. Third, we exhibit a new class of non-linearly solvable graphs. Fourth, we determine large classes of strictly linearly solvable graphs.