Expansion in Lifts of Graphs
نویسندگان
چکیده
The central goal of this thesis is to better understand, and explicitly construct, expanding towers G1,G2, . . ., which are expander families with the additional constraint that Gn+1 is a lift of Gn . A lift G of H is a graph that locally looks like H , but may be globally di erent; lifts have been proposed as a more structured setting for elementary explicit constructions of expanders, and there have recently been promising results in this direction by Marcus, Spielman and Srivastava [MSS13], Bilu and Linial [BL06], and Rozenman, Shalev and Wigderson [RSW06]; besides that, expansion in lifts is related to the Unique Games Conjecture (e.g., Arora et al [AKK+08]). We develop the basic theory of spectral expanders and lifts in the generality of directed multigraphs, and give some examples of their applications. We then derive some group-theoretic structural properties of towers, and show that a large class of commonly used graph operations ‘respect’ lifts. These two insights allow us to give a di erent perspective on an existing construction [RSW06], show that standard iterative constructions of expanders can be adjusted to give expander towers almost ‘for free’, and give a new elementary construction, along the lines of Ben-Aroya and Ta-Shma [BATS11], of a fully-explicit expanding tower of almost optimal spectral expanders. 1As required by the Computer Science concentration thesis guidelines.
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تاریخ انتشار 2015