Monotonicity Testing and Shortest-Path Routing on the Cube

• Published in: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques pp. 462 - 475
• Main Authors: Briët, Jop, Chakraborty, Sourav, García-Soriano, David, Matsliah, Arie
• Format: Book Chapter
• Language: English
• Published: Berlin, Heidelberg Springer Berlin Heidelberg 2010
• Series: Lecture Notes in Computer Science
• ISBN: 9783642153686; 3642153682
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-642-15369-3_35

We study the problem of monotonicity testing over the hypercube. As previously observed in several works, a positive answer to a natural question about routing properties of the hypercube network would imply the existence of efficient monotonicity testers. In particular, if any set of source-sink pairs on the directed hypercube (with all sources and all sinks distinct) can be connected with edge-disjoint paths, then monotonicity of functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f:{\{0,1\}}^n \rightarrow {\mathcal R}$\end{document} can be tested with O(n/ε) queries, for any totally ordered range \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal R}$\end{document}. More generally, if at least a μ(n) fraction of the pairs can always be connected with edge-disjoint paths then the query complexity is O(n/(εμ(n)) ). We construct a family of instances of Ω(2n) pairs in n-dimensional hypercubes such that no more than roughly a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac{1}{\sqrt{n}}$\end{document} fraction of the pairs can be simultaneously connected with edge-disjoint paths. This answers an open question of Lehman and Ron [LR01], and suggests that the aforementioned appealing combinatorial approach for deriving query-complexity upper bounds from routing properties cannot yield, by itself, query-complexity bounds better than ≈ n3/2. Additionally, our construction can also be used to obtain a strong counterexample to Szymanski’s conjecture about routing on the hypercube. In particular, we show that for any δ> 0, the n-dimensional hypercube is not \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n^{\frac 12 -\delta}$\end{document}-realizable with shortest paths, while previously it was only known that hypercubes are not 1-realizable with shortest paths. We also prove a lower bound of Ω(n/ε) queries for one-sided non-adaptive testing of monotonicity over the n-dimensional hypercube, as well as additional bounds for specific classes of functions and testers.