A characterization of the interval distance monotone graphs

AbstractA simple connected graph G is said to be interval distance monotone if the interval I(u,v) between any pair of vertices u and v in G induces a distance monotone graph. Aı¨der and Aouchiche [Distance monotonicity and a new characterization of hypercubes, Discrete Math. 245 (2002) 55–62] proposed the following conjecture: a graph G is interval distance monotone if and only if each of its intervals is either isomorphic to a path or to a cycle or to a hypercube. In this paper we verify the conjecture

On paths and cycles dominating hypercubes

AbstractThe aim of the present paper is to study the properties of the hypercube related to the concept of domination. We derive upper and lower bounds and prove an interpolation theorem for related invariants

Nearly Optimal Bounds for Sample-Based Testing and Learning of kk-Monotone Functions

We study monotonicity testing of functions $f \colon \{0,1\}^d \to \{0,1\}$ using sample-based algorithms, which are only allowed to observe the value of $f$ on points drawn independently from the uniform distribution. A classic result by Bshouty-Tamon (J. ACM 1996) proved that monotone functions can be learned with $\exp(O(\min\{\frac{1}{\varepsilon}\sqrt{d},d\}))$ samples and it is not hard to show that this bound extends to testing. Prior to our work the only lower bound for this problem was $\Omega(\sqrt{\exp(d)/\varepsilon})$ in the small $\varepsilon$ parameter regime, when $\varepsilon = O(d^{-3/2})$, due to Goldreich-Goldwasser-Lehman-Ron-Samorodnitsky (Combinatorica 2000). Thus, the sample complexity of monotonicity testing was wide open for $\varepsilon \gg d^{-3/2}$. We resolve this question, obtaining a tight lower bound of $\exp(\Omega(\min\{\frac{1}{\varepsilon}\sqrt{d},d\}))$ for all $\varepsilon$ at most a sufficiently small constant. In fact, we prove a much more general result, showing that the sample complexity of $k$-monotonicity testing and learning for functions $f \colon \{0,1\}^d \to [r]$ is $\exp(\Theta(\min\{\frac{rk}{\varepsilon}\sqrt{d},d\}))$. For testing with one-sided error we show that the sample complexity is $\exp(\Theta(d))$. Beyond the hypercube, we prove nearly tight bounds (up to polylog factors of $d,k,r,1/\varepsilon$ in the exponent) of $\exp(\widetilde{\Theta}(\min\{\frac{rk}{\varepsilon}\sqrt{d},d\}))$ on the sample complexity of testing and learning measurable $k$-monotone functions $f \colon \mathbb{R}^d \to [r]$ under product distributions. Our upper bound improves upon the previous bound of $\exp(\widetilde{O}(\min\{\frac{k}{\varepsilon^2}\sqrt{d},d\}))$ by Harms-Yoshida (ICALP 2022) for Boolean functions ($r=2$)

Property testing for distributions on partially ordered sets

Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2007.Includes bibliographical references (p. 24).We survey the results of Rubinfeld, Batu et al. ([2], [3]) on testing distributions for monotonicity, and testing distributions known to be monotone for uniformity. We extend some of their results to new partial orders, and provide evidence for some new conjectural lower bounds. Our results apply to various partial orders: bipartite graphs, lines,, trees, grids, and hypercubes.by Punyashloka Biswal.M.Eng

A THEORY OF QUALITATIVE SIMILARITY

The central result of this paper establishes an isomorphism between two types of mathematical structures: ""ternary preorders"" and ""convex topologies."" The former are characterized by reflexivity, symmetry and transitivity conditions, and can be interpreted geometrically as ordered betweenness relations; the latter are defined as intersection-closed families of sets satisfying an ""abstract convexity"" property. A large range of examples is given. As corollaries of the main result we obtain a version of Birkhoff''s representation theorem for finite distributive lattices, and a qualitative version of the representation of ultrametric distances by indexed taxonomic hierarchies.