On The Communication Complexity of Property Testing in Graphs

Constant-time algorithms are powerful tools, since they run by reading only a constant-sized part of each input. Property testing is the most popular research framework for constant-time algorithms. In property testing, an algorithm determines whether a given instance satisfies some predetermined property or is far from satisfying the property with high probability by reading a constant-sized part of the input. A property is said to be testable if there is a constant-time testing algorithm for the property. This chapter covers property testing on graphs and games. The fields of graph algorithms and property testing are two of the main streams of research on discrete algorithms and computational complexity. In the section on graphs in this chapter, we present some important results, particularly on the characterization of testable graph properties. At the end of the section, we show results that we published in 2020 on a complete characterization (necessary and sufficient condition) of testable monotone or hereditary properties in the bounded-degree digraphs. In the section on games, we present results that we published in 2019 showing that the generalized chess, Shogi (Japanese chess), and Xiangqi (Chinese chess) are all testable. We believe that this is the first results for testable EXPTIME-complete problems.

The robust and complex real-time applications and dramatically increased sensor capabilities may play a vital role in enhancing the lifespan of WSNs. On the other hand majority of WSNs operate on battery powered infrastructure, therefore in order to enhance the lifespan maximization a robust and highly efficient protocols are required to be developed that can effectively minimize the battery utilization and the overall computational as well as communication complexity also could be minimized. Optimizations required to be adopted at the Routing Layer, MAC Layer and the Radio Layer of the wireless sensor node. In this paper in order to achieve a better network performance an scheme of elephant swarm optimization has been implemented which enables optimization of routing algorithm, adaptive radio link optimization and balanced TDMA MAC scheduling. The proposed Elephant Based Swarm Optimization scheme is analyzed and compared with the popular LEACH and PSO Protocols and results proves that the EBSO algorithm is the best among PSO and LEACH schemes. Keywords: Active node ratio, Cross-layer design, Elephant Based Swarm Optimization (EBSO), LEACH, Network lifespan, Particle Swarm Optimization (PSO).

We consider the question of whether adaptivity can improve the complexity of property testing algorithms in the dense graphs model. It is known that there can be at most a quadratic gap between adaptive and non-adaptive testers in this model, but it was not known whether any gap indeed exists. In this work we reveal such a gap.Specifically, we focus on the well studied property of bipartiteness. Bogdanov and Trevisan (IEEE Symposium on Computational Complexity, 2004) proved a lower bound of Ω(1/ε 2) on the query complexity of non-adaptive testing algorithms for bipartiteness. This lower bound holds for graphs with maximum degree O(εn). Our main result is an adaptive testing algorithm for bipartiteness of graphs with maximum degree O(εn) whose query complexity is \(\tilde{O}(1/\epsilon^{3/2})\). A slightly modified version of our algorithm can be used to test the combined property of being bipartite and having maximum degree O(εn). Thus we demonstrate that adaptive testers are stronger than non-adaptive testers in the dense graphs model.We note that the upper bound we obtain is tight up-to polylogarithmic factors, in view of the Ω(1/ε 3/2) lower bound of Bogdanov and Trevisan for adaptive testers. In addition we show that \(\tilde{O}(1/\epsilon^{3/2})\) queries also suffice when (almost) all vertices have degree \(\Omega(\sqrt \epsilon \cdot n)\). In this case adaptivity is not necessary.KeywordsQuery ComplexityTesting AlgorithmAdaptive TesterDense GraphNeighbor QueryThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We study the power of four query models in the context of property testing in general graphs (i.e., with arbitrary edge densities), where our main case study is the problem of testing k-colorability. Two query types, which have been studied extensively in the past, are pair queries and neighbor queries. The former corresponds to asking whether there is an edge between any particular pair of vertices, and the latter to asking for the i’th neighbor of a particular vertex. We show that while for pair queries, testing k-colorability requires a number of queries that is a monotone decreasing function in the average degree d, the query complexity in the case of neighbor queries remains roughly the same for every density and for large values of k. We also consider a combined model that allows both types of queries, and we propose a new, stronger, query model, related to the field of Group Testing. We give one-sided error upper and lower bounds for all the models, where the bounds are nearly tight for three of the models. In some of the cases our lower bounds extend to two-sided error algorithms.The problem of testing k-colorability was previously studied in the contexts of dense and sparse graphs, and in our proofs we unify approaches from those cases, and also provide some new tools and techniques that may be of independent interest.Keywordspair queriesneighbor queriesgroup queriesk-colorability

This dissertation is concerned with the application of notions and methods from the field of information theory to the field of communication complexity. It consists of two main parts. In the first part of the dissertation, we prove lower bounds on the randomized two-party communication complexity of functions that arise from read-once boolean formulae. A read-once boolean formula is a formula in propositional logic with the property that every variable appears exactly once. Such a formula can be represented by a tree, where the leaves correspond to variables, and the internal nodes are labeled by binary connectives. Under certain assumptions, this representation is unique. Thus, one can define the depth of a formula as the depth of the tree that represents it. The complexity of the evaluation of general read-once formulae has attracted interest mainly in the decision tree model. In the communication complexity model many interesting results deal with specific read-once formulae, such as disjointness and tribes. In this dissertation we use information theory methods to prove lower bounds that hold for any read-once formula. Our lower bounds are of the form n(f )/c^d(f ) , where n(f ) is the number of variables and d(f ) is the depth of the formula, and they are optimal up to the constant in the base of the denominator. In the second part of the dissertation, we explore the applicability of the information-theoretic method in the number-on-the-forehead model. The work of Bar-Yossef, Jayram, Kumar & Sivakumar [BYJKS04] revealed a beautiful connection between Hellinger distance and two-party randomized communication protocols. Inspired by their work and motivated by the open questions in the number-on-the-forehead model, we introduce the notion of Hellinger volume. We show that it lower bounds the information cost of multi-party protocols. We provide a small toolbox that allows one to manipulate several Hellinger volume terms and also to lower bound a Hellinger volume when the distributions involved satisfy certain conditions. In doing so, we prove a new upper bound on the difference between the arithmetic mean and the geometric mean in terms of relative entropy. Finally, we show how to apply the new tools to obtain a lower bound on the informational complexity of the ANDk function.

Communication Complexity and Data Structures

A few years ago, Blais, Brody, and Matulef: Comput. Complex. 21(2), 311–358 (2012) presented a methodology for proving lower bounds for property testing problems by reducing them from problems in communication complexity. Recently, Bhrushundi, Chakraborty, and Kulkarni (2014) showed that some reductions of this type can be deconstructed to two separate reductions, from communication complexity to randomized parity decision trees and from the latter to property testing. This work follows up on these ideas. We introduce a model called linear-access algorithms, which is a generalization of randomized parity decision trees, and show several methods to reduce communication complexity problems to problems for linear-access algorithms and problems for linear-access algorithms to property testing problems. This approach yields a new interpretation for several well-known reductions, since we present these reductions as a composition of two steps with fundamentally different functionalities. Furthermore, we demonstrate the potential of proving lower bounds on property testing problems by reducing them directly from problems for linear-access algorithms. In particular, we provide an alternative and simple proof for a known lower bound of Ω(k) queries on testing “k-linearity”; that is, the property of k-sparse linear functions over $\mathbb {F}_{2}$ . This alternative proof relies on a theorem by Linial and Samorodnitsky: Combinatorica 22(4), 497–522 (2002). We then extend this result to a new lower bound of Ω(s) queries for testing s-sparse degree-d polynomials over $\mathbb {F}_{2}$ , for any $d\in \mathbb {N}$ . In addition we provide a simple proof for the hardness of testing some families of linear subcodes.

This article documents two programmatic comments regarding testing graph properties, which I made during the Dagstuhl workshop on Sublinear-Time Algorithms (July 2005). The first comment advocates paying more attention to the dependence of the tester’s complexity on the proximity parameter. The second comment advocates paying more attention to the question of testing general graphs (rather than dense or bounded-degree ones). In addition, this article includes a suggestion to view and discuss property testing within the framework of promise problems.KeywordsProperty testinggraph properties

One of the most fundamental questions in graph property testing is to characterize the combinatorial structure of properties that are testable with a constant number of queries. We work towards an answer to this question for the bounded-degree graph model introduced in [Goldreich, Ron, 2002], where the input graphs have maximum degree bounded by a constant $d$. In this model, it is known (among other results) that every \emph{hyperfinite} property is constant-query testable [Newman, Sohler, 2013], where, informally, a graph property is hyperfinite, if for every $\delta >0$ every graph in the property can be partitioned into small connected components by removing $\delta n$ edges. In this paper we show that hyperfiniteness plays a role in \emph{every} testable property, i.e. we show that every testable property is either finite (which trivially implies hyperfiniteness and testability) or contains an infinite hyperfinite subproperty. A simple consequence of our result is that no infinite graph property that only consists of expander graphs is constant-query testable. Based on the above findings, one could ask if every infinite testable non-hyperfinite property might contain an infinite family of expander (or near-expander) graphs. We show that this is not true. Motivated by our counter-example we develop a theorem that shows that we can partition the set of vertices of every bounded degree graph into a constant number of subsets and a separator set, such that the separator set is small and the distribution of $k$-disks on every subset of a partition class, is roughly the same as that of the partition class if the subset has small expansion.

We call F : {0, 1}n × {0, 1}n → {0, 1} a symmetric XOR function if for a functionS : {0, 1, ..., n} → {0, 1}, F(x, y) = S(|x⊕y|), for any x, y ∈ {0, 1}n, where |x⊕y| is theHamming weight of the bit...

The Role of Communication Complexity in Distributed Computing

We develop a new technique for proving lower bounds in property testing, by showing a strong connection between testing and communication complexity. We give a simple scheme for reducing communication problems to testing problems, thus allowing us to use known lower bounds in communication complexity to prove lower bounds in testing. This scheme is general and implies a number of new testing bounds, as well as simpler proofs of several known bounds. For the problem of testing whether a boolean function is k-linear (a parity function on k variables), we achieve a lower bound of Ω(k) queries, even for adaptive algorithms with two-sided error, thus confirming a conjecture of Goldreich (2010). The same argument behind this lower bound also implies a new proof of known lower bounds for testing related classes such as k-juntas. For some classes, such as the class of monotone functions and the class of s-sparse GF(2) polynomials, we significantly strengthen the best known bounds.

We develop a new technique for proving lower bounds in property testing, by showing a strong connection between testing and communication complexity. We give a simple scheme for reducing communication problems to testing problems, thus allowing us to use known lower bounds in communication complexity to prove lower bounds in testing. This scheme is general and implies a number of new testing bounds, as well as simpler proofs of several known bounds. For the problem of testing whether a Boolean function is k-linear (a parity function on k variables), we achieve a lower bound of Ω(k) queries, even for adaptive algorithms with two-sided error, thus confirming a conjecture of Goldreich (2010a). The same argument behind this lower bound also implies a new proof of known lower bounds for testing related classes such as k-juntas. For some classes, such as the class of monotone functions and the class of s-sparse GF(2) polynomials, we significantly strengthen the best known bounds.

A function f: Fn2 → {0, 1} is odd-cycle-free if there are no x1,..., xk e Fn2 with k an odd integer such that f(x1) =... = f(xk) = 1 and x1 +... + xk = 0. We show that one can distinguish odd-cycle-free functions from those e-far from being odd-cycle-free by making poly(1/e) queries to an evaluation oracle. We give two proofs of this result, each shedding light on a different connection between testability of properties of Boolean functions and of dense graphs.The first problem we study is directly reducing testing linear-invariant properties of Boolean functions to testing associated graph properties. We show a black-box reduction from testing odd-cycle-freeness to testing bipartiteness of graphs. Such reductions have been shown previously (Kral-Serra-Vena, Israel J. Math 2011; Shapira, STOC 2009) for monotone linear-invariant properties defined by forbidding solutions to a finite number of equations. But for odd-cycle-freeness whose description involves an infinite number of forbidden equations, a reduction to graph property testing was not previously known. If one could show such a reduction more generally for any linear-invariant property closed under restrictions to subspaces, then it would likely lead to a characterization of the one-sided testable linear-invariant properties, an open problem raised by Sudan.The second issue we study is whether there is an efficient canonical tester for linear-invariant properties of Boolean functions. A canonical tester for linear-invariant properties operates by picking a random linear subspace and then checking if the restriction of the input function to the subspace satisfies a fixed property or not. The question is whether for every linear-invariant property, there is a canonical tester for which there is only a polynomial blowup from the optimal query complexity. We answer the question affirmatively for odd-cycle-freeness. The general question still remains open.

This article reflects on different approaches to multimodal communication in text linguistics. For some time text linguistics deals with multimodal texts, however, the communicative and functional complexity of language-image-texts could give a reason to establish a new discipline – image linguistics, which is based on texts linguistics and develops already existing text theory and analytical methods. Nevertheless, within this article I propose a compromise and vote for developing of multimodal text linguistics.