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Reduction and fixed points of Boolean networks and linear network coding solvability.

Gadouleau, Maximilien and Richard, Adrien and Fanchon, Eric (2016) 'Reduction and fixed points of Boolean networks and linear network coding solvability.', IEEE transactions on information theory., 62 (5). pp. 2504-2519.


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 = f0; 1g) 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.

Item Type:Article
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Publisher statement:© 2016 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.
Date accepted:05 March 2016
Date deposited:18 March 2016
Date of first online publication:21 March 2016
Date first made open access:18 March 2016

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