ﻻ يوجد ملخص باللغة العربية
In this paper we initiate the study of property testing in simultaneous and non-simultaneous multi-party communication complexity, focusing on testing triangle-freeness in graphs. We consider the $textit{coordinator}$ model, where we have $k$ players receiving private inputs, and a coordinator who receives no input; the coordinator can communicate with all the players, but the players cannot communicate with each other. In this model, we ask: if an input graph is divided between the players, with each player receiving some of the edges, how many bits do the players and the coordinator need to exchange to determine if the graph is triangle-free, or $textit{far}$ from triangle-free? For general communication protocols, we show that $tilde{O}(k(nd)^{1/4}+k^2)$ bits are sufficient to test triangle-freeness in graphs of size $n$ with average degree $d$ (the degree need not be known in advance). For $textit{simultaneous}$ protocols, where there is only one communication round, we give a protocol that uses $tilde{O}(k sqrt{n})$ bits when $d = O(sqrt{n})$ and $tilde{O}(k (nd)^{1/3})$ when $d = Omega(sqrt{n})$; here, again, the average degree $d$ does not need to be known in advance. We show that for average degree $d = O(1)$, our simultaneous protocol is asymptotically optimal up to logarithmic factors. For higher degrees, we are not able to give lower bounds on testing triangle-freeness, but we give evidence that the problem is hard by showing that finding an edge that participates in a triangle is hard, even when promised that at least a constant fraction of the edges must be removed in order to make the graph triangle-free.
For integers $n$ and $k$, the density Hales-Jewett number $c_{n,k}$ is defined as the maximal size of a subset of $[k]^n$ that contains no combinatorial line. We show that for $k ge 3$ the density Hales-Jewett number $c_{n,k}$ is equal to the maximal
In the subgraph-freeness problem, we are given a constant-size graph $H$, and wish to determine whether the network contains $H$ as a subgraph or not. The emph{property-testing} relaxation of the problem only requires us to distinguish graphs that ar
The classical communication complexity of testing closeness of discrete distributions has recently been studied by Andoni, Malkin and Nosatzki (ICALP19). In this problem, two players each receive $t$ samples from one distribution over $[n]$, and the
Linear programming is a powerful method in combinatorial optimization with many applications in theory and practice. For solving a linear program quickly it is desirable to have a formulation of small size for the given problem. A useful approach for
Hub Labeling (HL) is a data structure for distance oracles. Hierarchical HL (HHL) is a special type of HL, that received a lot of attention from a practical point of view. However, theoretical questions such as NP-hardness and approximation guarantee