Identifying Codes and Covering Problems

Abstract

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> The identifying code problem for a given graph involves finding a minimum set of vertices whose neighborhoods uniquely overlap at any given graph vertex. Initially introduced in 1998, this problem has demonstrated its fundamental nature through a wide variety of applications, such as fault diagnosis, location detection, and environmental monitoring, in addition to deep connections to information theory, superimposed and covering codes, and tilings. This work establishes efficient reductions between the identifying code problem and the well-known set-covering problem, resulting in a tight hardness of approximation result and novel, provably tight polynomial-time approximations. The main results are also extended to <emphasis><formula formulatype="inline"><tex>$r$</tex> </formula></emphasis><emphasis emphasistype="boldital">-robust</emphasis> identifying codes and analogous <emphasis emphasistype="boldital">set </emphasis><emphasis><formula formulatype="inline"><tex>$(2r+1)$</tex></formula></emphasis><emphasis emphasistype="boldital">-multicover</emphasis> problems. Finally, empirical support is provided for the effectiveness of the proposed approximations, including good constructions for well-known topologies such as infinite two-dimensional grids. </para>