Twin-width I: Tractable FO Model Checking

Inspired by a width invariant defined on permutations by Guillemot and Marx [SODA’14], we introduce the notion of twin-width on graphs and on matrices. Proper minor-closed classes, bounded rank-width graphs, map graphs, K t -free unit d -dimensional ball graphs, posets with antichains of bounded size, and proper subclasses of dimension-2 posets all have bounded twin-width. On all these classes (except map graphs without geometric embedding) we show how to compute in polynomial time a sequence of d -contractions , witness that the twin-width is at most d . We show that FO model checking, that is deciding if a given first-order formula ϕ evaluates to true for a given binary structure G on a domain D , is FPT in |ϕ| on classes of bounded twin-width, provided the witness is given. More precisely, being given a d -contraction sequence for G , our algorithm runs in time f ( d ,|ϕ |) · |D| where f is a computable but non-elementary function. We also prove that bounded twin-width is preserved under FO interpretations and transductions (allowing operations such as squaring or complementing a graph). This unifies and significantly extends the knowledge on fixed-parameter tractability of FO model checking on non-monotone classes, such as the FPT algorithm on bounded-width posets by Gajarský et al. [FOCS’15].

Babai’s conjecture for high-rank classical groups with random generators

Let $$G = {\text {SCl}}_n(q)$$ G = SCl n ( q ) be a quasisimple classical group with n large, and let $$x_1, \ldots , x_k \in G$$ x 1 , … , x k ∈ G be random, where $$k \ge q^C$$ k ≥ q C . We show that the diameter of the resulting Cayley graph is bounded by $$q^2 n^{O(1)}$$ q 2 n O ( 1 ) with probability $$1 - o(1)$$ 1 - o ( 1 ) . In the particular case $$G = {\text {SL}}_n(p)$$ G = SL n ( p ) with p a prime of bounded size, we show that the same holds for $$k = 3$$ k = 3 .

A Synopsis Based Approach for Itemset Frequency Estimation over Massive Multi-Transaction Stream

The streams where multiple transactions are associated with the same key are prevalent in practice, e.g., a customer has multiple shopping records arriving at different time. Itemset frequency estimation on such streams is very challenging since sampling based methods, such as the popularly used reservoir sampling, cannot be used. In this article, we propose a novel k -Minimum Value (KMV) synopsis based method to estimate the frequency of itemsets over multi-transaction streams. First, we extract the KMV synopses for each item from the stream. Then, we propose a novel estimator to estimate the frequency of an itemset over the KMV synopses. Comparing to the existing estimator, our method is not only more accurate and efficient to calculate but also follows the downward-closure property. These properties enable the incorporation of our new estimator with existing frequent itemset mining (FIM) algorithm (e.g., FP-Growth) to mine frequent itemsets over multi-transaction streams. To demonstrate this, we implement a KMV synopsis based FIM algorithm by integrating our estimator into existing FIM algorithms, and we prove it is capable of guaranteeing the accuracy of FIM with a bounded size of KMV synopsis. Experimental results on massive streams show our estimator can significantly improve on the accuracy for both estimating itemset frequency and FIM compared to the existing estimators.

On Polyhedral Realization with Isosceles Triangles

Answering a question posed by Joseph Malkevitch, we prove that there exists a polyhedral graph, with triangular faces, such that every realization of it as the graph of a convex polyhedron includes at least one face that is a scalene triangle. Our construction is based on Kleetopes, and shows that there exists an integer i such that all convex i-iterated Kleetopes have a scalene face. However, we also show that all Kleetopes of triangulated polyhedral graphs have non-convex non-self-crossing realizations in which all faces are isosceles. We answer another question of Malkevitch by observing that a spherical tiling of Dawson (Renaissance Banff, Bridges Conference, pp. 489–496, 2005) leads to a fourth infinite family of convex polyhedra in which all faces are congruent isosceles triangles, adding one to the three families previously known to Malkevitch. We prove that the graphs of convex polyhedra with congruent isosceles faces have bounded diameter and have dominating sets of bounded size.

Partitioning planar graphs with girth at least $ 9 $ into an edgeless graph and a graph with bounded size components

<abstract><p>In this paper, we study the problem of partitioning the vertex set of a planar graph with girth restriction into parts, also referred to as color classes, such that each part induces a graph with components of bounded order. An ($ \mathcal{I} $, $ \mathcal{O}_{k} $)-partition of a graph $ G $ is the partition of $ V(G) $ into two non-empty subsets $ V_{1} $ and $ V_{2} $, such that $ G[V_{1}] $ is an edgeless graph and $ G[V_{2}] $ is a graph with components of order at most $ k $. We prove that every planar graph with girth 9 and without intersecting $ 9 $-face admits an ($ \mathcal{I} $, $ \mathcal{O}_{6} $)-partition. This improves a result of Choi, Dross and Ochem (2020) which says every planar graph with girth at least $ 9 $ admits an ($ \mathcal{I} $, $ \mathcal{O}_{9} $)-partition.</p></abstract>

Random Spanning Forests and Hyperbolic Symmetry

We study (unrooted) random forests on a graph where the probability of a forest is multiplicatively weighted by a parameter $$\beta >0$$ β > 0 per edge. This is called the arboreal gas model, and the special case when $$\beta =1$$ β = 1 is the uniform forest model. The arboreal gas can equivalently be defined to be Bernoulli bond percolation with parameter $$p=\beta /(1+\beta )$$ p = β / ( 1 + β ) conditioned to be acyclic, or as the limit $$q\rightarrow 0$$ q → 0 with $$p=\beta q$$ p = β q of the random cluster model. It is known that on the complete graph $$K_{N}$$ K N with $$\beta =\alpha /N$$ β = α / N there is a phase transition similar to that of the Erdős–Rényi random graph: a giant tree percolates for $$\alpha > 1$$ α > 1 and all trees have bounded size for $$\alpha <1$$ α < 1 . In contrast to this, by exploiting an exact relationship between the arboreal gas and a supersymmetric sigma model with hyperbolic target space, we show that the forest constraint is significant in two dimensions: trees do not percolate on $${\mathbb {Z}}^2$$ Z 2 for any finite $$\beta >0$$ β > 0 . This result is a consequence of a Mermin–Wagner theorem associated to the hyperbolic symmetry of the sigma model. Our proof makes use of two main ingredients: techniques previously developed for hyperbolic sigma models related to linearly reinforced random walks and a version of the principle of dimensional reduction.

New number-theoretic cryptographic primitives

This paper introduces new prq-based one-way functions and companion signature schemes. The new signature schemes are interesting because they do not belong to the two common design blueprints, which are the inversion of a trapdoor permutation and the Fiat–Shamir transform. In the basic signature scheme, the signer generates multiple RSA-like moduli ni = pi2qi and keeps their factors secret. The signature is a bounded-size prime whose Jacobi symbols with respect to the ni’s match the message digest. The generalized signature schemes replace the Jacobi symbol with higher-power residue symbols. Given of their very unique design, the proposed signature schemes seem to be overlooked “missing species” in the corpus of known signature algorithms.

Rewriting the Description Logic ALCHIQ to Disjunctive Existential Rules

Especially in data-intensive settings, a promising reasoning approach for description logics (DLs) is to rewrite DL theories into sets of rules. Although many such approaches have been considered in the literature, there are still various relevant DLs for which no small rewriting (of polynomial size) is known. We therefore develop small rewritings for the DL \ALCHIQ -- featuring disjunction, number restrictions, and inverse roles -- to disjunctive Datalog. By admitting existential quantifiers in rule heads, we can improve this result to yield only rules of bounded size, a property that is common to all rewritings that were implemented in practice so far.

Approximate Pareto Set for Fair and Efficient Allocation: Few Agent Types or Few Resource Types

In fair division of indivisible goods, finding an allocation that satisfies fairness and efficiency simultaneously is highly desired but computationally hard. We solve this problem approximately in polynomial time by modeling it as a bi-criteria optimization problem that can be solved efficiently by determining an approximate Pareto set of bounded size. We focus on two criteria: max-min fairness and utilitarian efficiency, and study this problem for the setting when there are only a few item types or a few agent types. We show in both cases that one can construct an approximate Pareto set in time polynomial in the input size, either by designing a dynamic programming scheme, or a linear-programming algorithm. Our techniques strengthen known methods and can be potentially applied to other notions of fairness and efficiency as well.