Efficient testing of large 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.

We study the space complexity of refuting unsatisfiable random k-CNFs in the resolution proof system. We prove that for any large enough /spl Delta/, with high probability a random k-CNF over n variables and /spl Delta/n clauses requires resolution clause space of /spl Omega/(n/spl middot//spl Delta//sup -1+/spl epsiv//k-2-/spl epsiv//), for any 0 >/spl radic/n. This bound is nearly tight. Specifically, we show that with high probability, a random 3-CNF with /spl Delta/n clauses requires tree-like refutation size of exp(/spl Omega/(n//spl Delta//sup 1+/spl epsiv//1-/spl epsiv//)), for any 0</spl epsiv/<1/2. Our space lower bound is the consequence of three main contributions. 1. We introduce a 2-player matching game on bipartite graphs G to prove that there are no perfect matchings in G. 2. We reduce lower bounds for the clause space of a formula F in resolution to lower bounds for the complexity of the game played on the bipartite graph G(F) associated with F. 3. We prove that the complexity of the game is large whenever G is an expander graph. Finally, a simple probabilistic analysis shows that for a random formula F, with high probability G(F) is an expander. We also extend our result to the case of G-PHP, a generalization of the pigeonhole principle based on bipartite graphs G. We prove that the clause space for G-PHP can be reduced to the game complexity on G.

A graph property is monotone if it is closed under removal of vertices and edges. In this paper we consider the following edge-deletion problem; given a monotone property P and a graph G, compute the smallest number of edge deletions that are needed in order to turn G into a graph satisfying P. We denote this quantity by E/sub P/'(G). The first result of this paper states that the edge-deletion problem can be efficiently approximated for any monotone property. 1) For any /spl epsiv/ > 0 and any monotone property P, there is a deterministic algorithm, which given a graph G of size n, approximates E/sub P/'(G) in time O(n/sup 2/) to within an additive error of /spl epsiv/n/sup 2/. Given the above, a natural question is for which monotone properties one can obtain better additive approximations of E/sub P/'. Our second main result essentially resolves this problem by giving a precise characterization of the monotone graph properties for which such approximations exist; 1. If there is a bipartite graph that does not satisfy P, then there is a /spl delta/ > 0 for which it is possible to approximate E/sub P/' to within an additive error of n/sup 2-/spl delta// in polynomial time. 2) On the other hand, if all bipartite graphs satisfy P, then for any /spl delta/ > 0 it is NP-hard to approximate E/sub P/' to within an additive error of n/sup 2-/spl delta//. While the proof of (1) is simple, the proof of (2) requires several new ideas and involves tools from extremal graph theory together with spectral techniques. This approach may be useful for obtaining other hardness of approximation results. Interestingly, prior to this work it was not even known that computing E/sub P/' precisely for the properties in (2) is NP-hard. We thus answer (in a strong form) a question of Yannakakis [1981], who asked in 1981 if it is possible to find a large and natural family of graph properties for which computing E/sub P/' is NP-hard.

We introduce a simple erasure recovery algorithm for codes derived from cascades of sparse bipartite graphs and analyze the algorithm by analyzing a corresponding discrete-time random process. As a result, we obtain a simple criterion involving the fractions of nodes of different degrees on both sides of the graph which is necessary and sufficient for the decoding process to finish successfully with high probability. By carefully designing these graphs we can construct for any given rate R and any given real number /spl epsiv/ a family of linear codes of rate R which can be encoded in time proportional to ln(1//spl epsiv/) times their block length n. Furthermore, a codeword can be recovered with high probability from a portion of its entries of length (1+/spl epsiv/)Rn or more. The recovery algorithm also runs in time proportional to n ln(1//spl epsiv/). Our algorithms have been implemented and work well in practice; various implementation issues are discussed.

An important theme of recent research in Ramsey theory has been establishing pseudorandomness properties of Ramsey graphs. An N N -vertex graph is called C C -Ramsey if it has no homogeneous set of size C log ⁡ N C\log N . A theorem of Bukh and Sudakov, solving a conjecture of Erdős, Faudree, and Sós, shows that any C C -Ramsey N N -vertex graph contains an induced subgraph with Ω C ( N 1 / 2 ) \Omega _C(N^{1/2}) distinct degrees. We improve this to Ω C ( N 2 / 3 ) \Omega _C(N^{2/3}) , which is tight up to the constant factor. We also show that any N N -vertex graph with N &gt; ( k − 1 ) ( n − 1 ) N &gt; (k-1)(n-1) and n ≥ n 0 ( k ) = Ω ( k 9 ) n\geq n_0(k) = \Omega (k^9) either contains a homogeneous set of order n n or an induced subgraph with k k distinct degrees. The lower bound on N N here is sharp, as shown by an appropriate Turán graph, and confirms a conjecture of Narayanan and Tomon.

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 present a randomized parallel algorithm for the Voronoi diagram of line segments on coarse grained parallel machines, which, for any input of n line segments on P processors (n=/spl Omega/(P/sup 3+/spl epsiv//), for any /spl epsiv/>0) requires (with high probability) O([n log n/P]) local operations, O(n/P) global-operations (or messages) per processor, O(1) levels of global data dependency (or O(1) communication phases). This generic parallel algorithm can be transformed to the CRCW PRAM model in O(log n) time with O(n) processors and thus provides a different solution from, and conceptually much simpler than, that of (Rajasekaran, 1994).

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.

Extends B. Alpern & F.B. Schneider's linear time characterization of safety and liveness properties to branching time, where properties are sets of trees. We define two closure operators that give rise to the following four extremal types of properties: universally safe, existentially safe, universally live and existentially live. The distinction between universal and existential properties captures the difference between the CTL (computation tree logic) path quantifiers /spl forall/ (for all paths) and /spl exist/ (there is a path). We show that every branching time property is the intersection of an existentially safe property and an existentially live property, a universally safe property and a universally live property, and an existentially safe property and a universally live property. We also examine how our closure operators behave on linear-time properties. We then focus on sets of finitely branching trees and show that our closure operators agree on linear-time safety properties. Furthermore, if a set of trees is given implicitly as a Rabin tree automaton /spl Bscr/, we show that it is possible to compute the Rabin automata corresponding to the closures of the language of /spl Bscr/. This allows us to effectively compute /spl Bscr//sub safe/ and /spl Bscr//sub live/ such that the language of /spl Bscr/ is the intersection of the languages of /spl Bscr//sub safe/ and /spl Bscr//sub live/. As above, /spl Bscr//sub safe/ and /spl Bscr//sub live/ can be chosen so that their languages are existentially safe and existentially live, universally safe and universally live, or existentially safe and universally live.

We consider the problem of estimating the size of a maximum matching when the edges are revealed in a streaming fashion. When the input graph is planar, we present a simple and elegant streaming algorithm that, with high probability, estimates the size of a maximum matching within a constant factor using Õ( n 2/3 ) space, where n is the number of vertices. The approach generalizes to the family of graphs that have bounded arboricity, which include graphs with an excluded constant-size minor. To the best of our knowledge, this is the first result for estimating the size of a maximum matching in the adversarial-order streaming model (as opposed to the random-order streaming model) in o ( n ) space. We circumvent the barriers inherent in the adversarial-order model by exploiting several structural properties of planar graphs, and more generally, graphs with bounded arboricity. We further reduce the required memory size to Õ(√ n ) for three restricted settings: (i) when the input graph is a forest; (ii) when we have 2-passes and the input graph has bounded arboricity; and (iii) when the edges arrive in random order and the input graph has bounded arboricity. Finally, we design a reduction from the Boolean Hidden Matching Problem to show that there is no randomized streaming algorithm that estimates the size of the maximum matching to within a factor better than 3/2 and uses only o ( n 1/2 ) bits of space. Using the same reduction, we show that there is no deterministic algorithm that computes this kind of estimate in o ( n ) bits of space. The lower bounds hold even for graphs that are collections of paths of constant length.

Neighborhood unions and hamiltonian properties in graphs

We discuss a new algorithmic type of problem in random graphs studying the minimum number of queries one has to ask about adjacency between pairs of vertices of a random graph in order to find a subgraph which possesses some target property with high probability. In this paper we focus on finding long paths in when for some fixed constant . This random graph is known to have typically linearly long paths. To have edges with high probability in one clearly needs to query at least pairs of vertices. Can we find a path of length economically, i.e., by querying roughly that many pairs? We argue that this is not possible and one needs to query significantly more pairs. We prove that any randomised algorithm which finds a path of length with at least constant probability in with must query at least pairs of vertices. This is tight up to the factor. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 71–85, 2017

Hierarchical Glocal Attention Pooling for Graph Classification

The Maximal Independent Set (MIS) formulation tackles the graph coloring problem (GCP) as the partitioning of vertices of a graph into a minimum number of maximal independent sets as each MIS can be assigned a unique color. Mehrotra and Trick [5] solved the MIS formulation with an exact IP approach, but they were restricted to solving smaller or easier instances. For harder instances, it might be impossible to get the optimal solution within a reasonable computation time. We develop a heuristic algorithm, hoping that we can solve these problems in more reasonable time. However, though heuristics can find a near-optimal solution extremely fast compared to the exact approaches, there is still significant variations in performance that can only be explained by the fact that certain structures or properties in graphs may be better suited to some heuristics more than others. Selecting the best algorithm on average across all instances does not help us pick the best one for a particular instance. The need to understand how the best heuristic for a particular class of instance depends on these graph properties is an important issue. In this research, we use data mining to select the best solution strategies for classes of graph coloring instances.

Properties of graphs that can be characterized by the spectrum of the adjacency matrix of the graph have been studied systematically recently. Motivated by the complexity of these properties, we show that there are such properties for which testing whether a graph has that property can be NP-hard (or belong to other computational complexity classes consisting of even harder problems). In addition, we discuss a possible spectral characterization of some well-known NP-hard properties. In particular, for every integer k≥6 we construct a pair of k-regular cospectral graphs, where one graph is Hamiltonian and the other one not.