On the Query Complexity of Testing Orientations for Being Eulerian

Locally decodable codes (LDCs) allow any single encoded message symbol to be retrieved from a codeword with good probability by reading only a tiny number of codeword symbols, even if the codeword is partially corrupted. LDCs have surprisingly many applications in computer science and mathematics (we refer to [Yekhanin, 2012; Lovett, 2007] for extensive surveys). But despite their ubiquity, they are poorly understood. Of particular interest is the tradeoff between the codeword length N as a function of message length k when the query complexity - the number of probed codeword symbols - and alphabet size are constant. The Hadamard code is a 2-query LDC of length N = 2^O(k) and this length is optimal in the 2-query regime [Lovett, 2007]. For q ≥ 3, near-exponential gaps persist between the best-known upper and lower bounds. The family of Reed-Muller codes, which generalize the Hadamard code, were for a long time the best-known examples, giving q-query LDCs of length exp(O(k^{1/(q-1)})), until breakthrough constructions of matching vector LDCs of Yekhanin and Efremenko [Yekhanin, 2008; Efremenko, 2012].
\nIn contrast with other combinatorial objects such as expander graphs, the probabilistic method has so far not been successfully used to beat the best explicit LDC constructions. In [Lovett, 2007], a probabilistic framework was given that could in principle yield best-possible LDCs, albeit non-constructively. A special instance of this framework connects LDCs with a probabilistic version of Szemerédi’s theorem. The setup for this is as follows: For a finite abelian group G of size N = |G|, let D ⊆ G be a random subset where each element is present with probability ρ independently of all others. For k ≥ 3 and ε ∈ (0,1), let E be the event that every subset A ⊆ G of size |A| ≥ ε |G| contains a proper k-term arithmetic progression with common difference in D. For fixed ε > 0 and sufficiently large N, it is an open problem to determine the smallest value of ρ - denoted ρ_k - such that Pr[E] ≥ 1/2. In [Lovett, 2007] it is shown that there exist k-query LDCs of message length Ω(ρ_k N) and codeword length O(N). As such, Szemerédi’s theorem with random differences, in particular lower bounds on ρ_k, can be used to show the existence of LDCs. Conversely, this connection indirectly implies the best-known upper bounds on ρ_k for all k ≥ 3 [Lovett, 2007; Lovett, 2007]. However, a conjecture from [Lovett, 2007] states that over ℤ_N we have ρ_k ≤ O_k(N^{-1}log N) for all k, which would be best-possible. Truth of this conjecture would imply that over this group, Szemerédi’s theorem with random differences cannot give LDCs better than the Hadamard code. For finite fields, Altman [Lovett, 2007] showed that this is false. In particular, over 𝔽_pⁿ for p odd, he proved that ρ₃ ≥ Ω(p^{-n} n²); generally, ρ_k ≥ Ω(p^{-n} n^{k-1}) holds when p ≥ k+1 [Lovett, 2007]. In turn, these bounds are conjectured to be optimal for the finite-field setting, which would imply that over finite fields, Szemerédi’s theorem with random differences cannot give LDCs better than Reed-Muller codes. 
\nThe finite-field conjecture is motivated mainly by the possibility that so-called dual functions can be approximated well by polynomial phases, functions of the form e^{2π i P(x)/p} where P is a multivariate polynomial over 𝔽_p. We show that this is false. Using Yekhanin’s matching-vector-code construction, we give dual functions of order k over 𝔽_pⁿ that cannot be approximated in L_∞-distance by polynomial phases of degree k-1. This answers in the negative a natural finite-field analog of a problem of Frantzikinakis over ℕ [Lovett, 2007].

The goal of property testing is to quickly distinguish between objects which satisfy a property and objects that are ϵ-far from satisfying the property. There are now several general results in this area which show that natural properties of combinatorial objects can be tested with constant query complexity, depending only on ϵ and the property, and not on the size of the object being tested. The upper bound on the query complexity coming from the proof techniques are often enormous and impractical. It remains a major open problem if better bounds hold.Hoppen, Kohayakawa, Moreira, and Sampaio conjectured and Klimosova and Kral' proved that hereditary permutation properties are strongly testable, i.e., can be tested with respect to Kendall's tau distance. The query complexity bound coming from this proof is huge. Even for testing a single forbidden subpermutation it is of Ackermann-type in 1/ϵ. We give a new proof which gives a polynomial bound in 1/ϵ for testing a single forbidden subpermutation.Maybe surprisingly, for testing with respect to the rectangular distance, we prove there is a universal (not depending on the property), polynomial in 1/ϵ query complexity bound for two-sided testing hereditary properties of sufficiently large permutations. We further give a nearly linear bound with respect to a closely related metric which also depends on the smallest forbidden subpermutation for the property. Finally, we show that several different permutation metrics of interest are related to the rectangular distance, yielding similar results for testing with respect to these metrics.

Focusing on property testing tasks that have query complexity that is independent of the size of the tested object (i.e., depends on the proximity parameter only), we prove the existence of a rich hierarchy of the corresponding complexity classes. That is, for essentially any function $$q : (0, 1] \rightarrow \mathbb{N}$$ , we prove the existence of properties for which $$\epsilon$$ -testing has query complexity $$\Theta(q(\Theta(\epsilon)))$$ . Such results are proved in three standard domains that are often considered in property testing: generic functions, adjacency predicates describing (dense) graphs, and incidence functions describing bounded-degree graphs. These results complement hierarchy theorems of Goldreich, Krivelevich, Newman, and Rozenberg (Computational Complexity, 2012), which refer to the dependence of the query complexity on the size of the tested object, and focus on the case that the proximity parameter is set to some small positive constant. We actually combine both flavors and get tight results on the query complexity of testing when allowing the query complexity to depend on both the size of the object and the proximity parameter.

Consider property testing on bounded degree graphs and let $\\varepsilon>0$\ndenote the proximity parameter. A remarkable theorem of Newman-Sohler (SICOMP\n2013) asserts that all properties of planar graphs (more generally hyperfinite)\nare testable with query complexity only depending on $\\varepsilon$. Recent\nadvances in testing minor-freeness have proven that all additive and monotone\nproperties of planar graphs can be tested in $poly(\\varepsilon^{-1})$ queries.\nSome properties falling outside this class, such as Hamiltonicity, also have a\nsimilar complexity for planar graphs. Motivated by these results, we ask: can\nall properties of planar graphs can be tested in $poly(\\varepsilon^{-1})$\nqueries? Is there a uniform query complexity upper bound for all planar\nproperties, and what is the "hardest" such property to test? We discover a\nsurprisingly clean and optimal answer. Any property of bounded degree planar\ngraphs can be tested in $\\exp(O(\\varepsilon^{-2}))$ queries. Moreover, there is\na matching lower bound, up to constant factors in the exponent. The natural\nproperty of testing isomorphism to a fixed graph needs\n$\\exp(\\Omega(\\varepsilon^{-2}))$ queries, thereby showing that (up to\npolynomial dependencies) isomorphism to an explicit fixed graph is the hardest\nproperty of planar graphs. The upper bound is a straightforward adapation of\nthe Newman-Sohler analysis that tracks dependencies on $\\varepsilon$ carefully.\nThe main technical contribution is the lower bound construction, which is\nachieved by a special family of planar graphs that are all mutually far from\neach other. We can also apply our techniques to get analogous results for\nbounded treewidth graphs. We prove that all properties of bounded treewidth\ngraphs can be tested in $\\exp(O(\\varepsilon^{-1}\\log \\varepsilon^{-1}))$\nqueries. Moreover, testing isomorphism to a fixed forest requires\n$\\exp(\\Omega(\\varepsilon^{-1}))$ queries.\n

This article initiates the study of testing properties of directed graphs. In particular, the article considers the most basic property of directed graphs--acyclicity. Because the choice of representation affects the choice of algorithm, the two main representations of graphs are studied. For the adjacency-matrix representation, most appropriate for dense graphs, a testing algorithm is developed that requires query and time complexity of O(1/e2), where e is a distance parameter independent of the size of the graph. The algorithm, which can probe the adjacency matrix of the graph, accepts every graph that is acyclic, and rejects, with probability at least 2/3, every graph whose adjacency matrix should be modified in at least e fraction of its entries so that it becomes acyclic. For the incidence list representation, most appropriate for sparse graphs, an Ω(|V|1/3) lower bound is proved on the number of queries and the time required for testing, where V is the set of vertices in the graph. Along with acyclicity, this article considers the property of strong connectivity. Contrasting upper and lower bounds are proved for the incidence list representation. In particular, if the testing algorithm can query on both incoming and outgoing edges at each vertex, then it is possible to test strong connectivity in O(1/e) time and query complexity. On the other hand, if the testing algorithm only has access to outgoing edges, then Ω(√N) queries are required to test for strong connectivity.

In recent years, a variety of randomized constructions of sketching matrices have been devised, that have been used in fast algorithms for numerical linear algebra problems, such as least squares regression, low-rank approximation, and the approximation of leverage scores. A key property of sketching matrices is that of subspace embedding. In this paper, we study sketching matrices that are obtained from bipartite graphs that are sparse, i.e., have left degree s that is small. In particular, we explore two popular classes of sparse graphs, namely, expander graphs and magical graphs. For a given subspace ${\mathcal{U}} \subseteq {{\mathbb{R}}^n}$ of dimension k, we show that the magical graph with left degree s = 2 yields a (1 ± ϵ) ℓ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> -subspace embedding for ${\mathcal{U}}$, if the number of right vertices (the sketch size) $m = {\mathcal{O}}\left( {{k^2}/{\varepsilon ^2}} \right)$. The expander graph with $s = {\mathcal{O}}(\log k/\varepsilon )$ yields a subspace embedding for $m = {\mathcal{O}}\left( {k\log k/{\varepsilon ^2}} \right)$. We also discuss the construction of sparse sketching matrices with reduced randomness using expanders based on error-correcting codes. Empirical results on various synthetic and real datasets show that these sparse graph sketching matrices work very well in practice.

We consider the problem of distribution-free testing of the class of monotone monomials and the class of monomials over n variables.0 While there are very efficient algorithms for testing a variety of functions classes when the underlying distribution is uniform, designing distribution-free algorithms (which must work under any arbitrary and unknown distribution), tends to be a more challenging task. When the underlying distribution is uniform, Parnas et al. (SIAM Journal on Discrete Math, 2002) give an algorithm for testing (monotone) monomials whose query complexity does not depends on n, and whose dependence on the distance parameter is (inverse) linear. In contrast, Glasner and Servedio (in Proceedings of RANDOM, 2007) prove that every distribution-free testing algorithm for monotone monomials as well as for general monomials must have query complexity Ω(n1/5) (for a constant distance parameter e). In this paper we present distribution-free testing algorithms for these classes where the query complexity of the algorithms is O(n1/2/e). We note that as opposed to previous results for distribution-free testing, our algorithms do not build on the algorithms that work under the uniform distribution. Rather, we define and exploit certain structural properties of monomials (and functions that differ from them in a nonnegligible manner), which were not used in previous work on property testing.

The number of spanning trees (or complexity) of a graph can be used to measure the reliabilities of network and circuit design. Although many NP hard problems are related to the complexity of graphs, there are explicit formulas for the complexity of some specific graph classes. This paper mainly studies the explicit expression of the complexity of graphs obtained by graph operations. Through classical graph operations, we obtain some operation graphs generated by cycle and complete graph, and get the closed formulas for the complexity in these operation graphs. Compared with Daoud’s results, we obtain some interesting algebraic properties about the complexity of operation graphs.

We continue the study of the trade‐off between the length of probabilistically checkable proofs (PCPs) and their query complexity, establishing the following main results (which refer to proofs of satisfiability of circuits of size n): 1. We present PCPs of length $\exp(o(\log\log n)^2)\cdot n$ that can be verified by making $o(\log\log n)$ Boolean queries. 2. For every \epsilon>0, we present PCPs of length $\exp(\log^\epsilon n)\cdot n$ that can be verified by making a constant number of Boolean queries. In both cases, false assertions are rejected with constant probability (which may be set to be arbitrarily close to 1). The multiplicative overhead on the length of the proof, introduced by transforming a proof into a probabilistically checkable one, is just quasi polylogarithmic in the first case (of query complexity $o(\log\log n)$), and is $2^{(\log n)^\epsilon}$, for any $\epsilon > 0$, in the second case (of constant query complexity). Our techniques include the introduction of a new variant of PCPs that we call “robust PCPs of proximity.” These new PCPs facilitate proof composition, which is a central ingredient in the construction of PCP systems. (A related notion and its composition properties were discovered independently by Dinur and Reingold.) Our main technical contribution is a construction of a “length‐efficient” robust PCP of proximity. While the new construction uses many of the standard techniques used in PCP constructions, it does differ from previous constructions in fundamental ways, and in particular does not use the “parallelization” step of Arora et al. [J. ACM, 45 (1998), pp. 501–555]. The alternative approach may be of independent interest. We also obtain analogous quantitative results for locally testable codes. In addition, we introduce a relaxed notion of locally decodable codes and present such codes mapping k information bits to codewords of length $k^{1+\epsilon}$ for any $\epsilon>0$.

We witness an unprecedented proliferation of knowledge graphs that record millions of entities and their relationships. While knowledge graphs are structure-flexible and content-rich, they are difficult to use. The challenge lies in the gap between their overwhelming complexity and the limited database knowledge of non-professional users. If writing structured queries over “simple” tables is difficult, complex graphs are only harder to query. As an initial step toward improving the usability of knowledge graphs, we propose to query such data by example entity tuples, without requiring users to form complex graph queries. Our system, Graph Query By Example ( $\mathsf {GQBE}$ ), automatically discovers a weighted hidden maximum query graph based on input query tuples, to capture a user’s query intent. It then efficiently finds and ranks the top approximate matching answer graphs and answer tuples. We conducted experiments and user studies on the large Freebase and DBpedia datasets and observed appealing accuracy and efficiency. Our system provides a complementary approach to the existing keyword-based methods, facilitating user-friendly graph querying. To the best of our knowledge, there was no such proposal in the past in the context of graphs.

We present new quantum algorithms for Triangle Finding improving its best previously known quantum query complexities for both dense and sparse instances. For dense graphs on n vertices, we get a query complexity of O(n^(5/4)) without any of the extra logarithmic factors present in the previous algorithm of Le Gall [FOCS'14]. For sparse graphs with m >= n^(5/4) edges, we get a query complexity of O(n^(11/12) m^(1/6) sqrt(log n)), which is better than the one obtained by Le Gall and Nakajima [ISAAC'15] when m >= n^(3/2). We also obtain an algorithm with query complexity O(n^(5/6) (m log n)^(1/6) + d_2 sqrt(n)) where d_2 is the variance of the degree distribution. Our algorithms are designed and analyzed in a new model of learning graphs that we call extended learning graphs. In addition, we present a framework in order to easily combine and analyze them. As a consequence we get much simpler algorithms and analyses than previous algorithms of Le Gall based on the MNRS quantum walk framework [SICOMP'11].

In this paper we consider the problem of testing whether a graph has bounded arboricity. The family of graphs with bounded arboricity includes, among others, bounded-degree graphs, all minor-closed graph classes (e.g. planar graphs, graphs with bounded treewidth) and randomly generated preferential attachment graphs. Graphs with bounded arboricity have been studied extensively in the past, in particular since for many problems they allow for much more efficient algorithms and/or better approximation ratios. We present a tolerant tester in the sparse-graphs model. The sparse-graphs model allows access to degree queries and neighbor queries, and the distance is defined with respect to the actual number of edges. More specifically, our algorithm distinguishes between graphs that are ϵ-close to having arboricity α and graphs that c · ϵ-far from having arboricity 3α, where c is an absolute small constant. The query complexity and running time of the algorithm are1 [EQUATION] where n denotes the number of vertices and m denotes the number of edges. In terms of the dependence on n and m this bound is optimal up to poly-logarithmic factors since [EQUATION] queries are necessary (and the arboricity of a graph is always [EQUATION]. We leave it as an open question whether the dependence on 1/ϵ can be improved from quasi-polynomial to polynomial. Our techniques include an efficient local simulation for approximating the outcome of a global (almost) forest-decomposition algorithm as well as a tailored procedure of edge sampling.

We overview algorithmic results for classes of sparse graphs emphasizing new developments in this area. We focus on recently introduced classes of graphs with bounded expansion and nowhere-dense graphs and relate algorithmic meta-theorems for these classes of graphs to their analogues for proper minor-closed classes of graphs, classes of graphs with bounded tree-width, locally bounded tree-width and locally excluding a minor.

Shape-based metrics measure how faithfully a drawing D of a large graph G shows the structure of graph, by comparing the similarity between G and a proximity graph S computed from D. Although these metrics can successfully evaluate drawings of large graphs, they are limited to relatively sparse graphs, since existing metrics use planar proximity graphs GG (Gabriel Graph) and RNG (Relative Neighbourhood Graph). This paper presents new shape-based faithfulness metrics for evaluating drawings of large and complex graphs, using high-order prox-imity graphs k-GG and k-RNG. Extensive experiments demonstrate that our new shape-based metrics using degree-based proximity graphs dGG and dRNG can more accurately measure the faithful-ness of drawings of large and complex graphs, with a significant improvement of over 100% better, on average, than the existing shape-based metrics using GG and RNG. Moreover, we present a new shape change faithfulness metric DSC for evaluating drawings of dynamic graphs, by measuring how proportional the geometric shape change in the drawings of dynamic graphs is to the ground truth change in dynamic graphs. Validation using deformation experiments support that DSC can accurately measure shape change faithfulness in dynamic graph drawing. Furthermore, we present extensive comparison experiments of ten popular graph layouts using our new shape-based metrics dGG, dRNG and DSC, to recommend which layouts can give a better shape-faithful graph drawing for large and complex graphs.

We give an efficient perfect sampling algorithm for weighted, connected induced subgraphs (or graphlets ) of rooted, bounded degree graphs. Our algorithm utilizes a vertex-percolation process with a carefully chosen rejection filter and works under a percolation subcriticality condition. We show that this condition is optimal in the sense that the task of (approximately) sampling weighted rooted graphlets becomes impossible in finite expected time for infinite graphs and intractable for finite graphs when the condition does not hold. We apply our sampling algorithm as a subroutine to give near linear-time perfect sampling algorithms for polymer models and weighted non-rooted graphlets in finite graphs, two widely studied yet very different problems. This new perfect sampling algorithm for polymer models gives improved sampling algorithms for spin systems at low temperatures on expander graphs and unbalanced bipartite graphs, among other applications.