An upper bound for Cubicity in terms of Boxicity

AbstractAn axis-parallel b-dimensional box is a Cartesian product R1×R2×⋯×Rb where each Ri (for 1≤i≤b) is a closed interval of the form [ai,bi] on the real line. The boxicity of any graph G, box(G) is the minimum positive integer b such that G can be represented as the intersection graph of axis-parallel b-dimensional boxes. A b-dimensional cube is a Cartesian product R1×R2×⋯×Rb, where each Ri (for 1≤i≤b) is a closed interval of the form [ai,ai+1] on the real line. When the boxes are restricted to be axis-parallel cubes in b-dimension, the minimum dimension b required to represent the graph is called the cubicity of the graph (denoted by cub(G)). In this paper we prove that cub(G)≤⌈log2n⌉box(G), where n is the number of vertices in the graph. We also show that this upper bound is tight.Some immediate consequences of the above result are listed below: 1.Planar graphs have cubicity at most 3⌈log2n⌉.2.Outer planar graphs have cubicity at most 2⌈log2n⌉.3.Any graph of treewidth tw has cubicity at most (tw+2)⌈log2n⌉. Thus, chordal graphs have cubicity at most (ω+1)⌈log2n⌉ and circular arc graphs have cubicity at most (2ω+1)⌈log2n⌉, where ω is the clique number.The above upper bounds are tight, but for small constant factors

Boxicity and Cubicity of Asteroidal Triple free graphs

An axis parallel $d$-dimensional box is the Cartesian product $R_1 \times R_2 \times ... \times R_d$ where each $R_i$ is a closed interval on the real line. The {\it boxicity} of a graph $G$, denoted as \boxi(G), is the minimum integer $d$ such that $G$ can be represented as the intersection graph of a collection of $d$-dimensional boxes. An axis parallel unit cube in $d$-dimensional space or a $d$-cube is defined as the Cartesian product $R_1 \times R_2 \times ... \times R_d$ where each $R_i$ is a closed interval on the real line of the form $[a_i,a_i + 1]$. The {\it cubicity} of $G$, denoted as \cub(G), is the minimum integer $d$ such that $G$ can be represented as the intersection graph of a collection of $d$-cubes. Let $S(m)$ denote a star graph on $m+1$ nodes. We define {\it claw number} of a graph $G$ as the largest positive integer $k$ such that $S(k)$ is an induced subgraph of $G$ and denote it as \claw. Let $G$ be an AT-free graph with chromatic number $\chi(G)$ and claw number \claw. In this paper we will show that \boxi(G) \leq \chi(G) and this bound is tight. We also show that \cub(G) \leq \boxi(G)(\ceil{\log_2 \claw} +2) $\leq$ \chi(G)(\ceil{\log_2 \claw} +2). If $G$ is an AT-free graph having girth at least 5 then \boxi(G) \leq 2 and therefore \cub(G) \leq 2\ceil{\log_2 \claw} +4.Comment: 15 pages: We are replacing our earlier paper regarding boxicity of permutation graphs with a superior result. Here we consider the boxicity of AT-free graphs, which is a super class of permutation graph

Revisiting Interval Graphs for Network Science

The vertices of an interval graph represent intervals over a real line where overlapping intervals denote that their corresponding vertices are adjacent. This implies that the vertices are measurable by a metric and there exists a linear structure in the system. The generalization is an embedding of a graph onto a multi-dimensional Euclidean space and it was used by scientists to study the multi-relational complexity of ecology. However the research went out of fashion in the 1980s and was not revisited when Network Science recently expressed interests with multi-relational networks known as multiplexes. This paper studies interval graphs from the perspective of Network Science

Robustness Verification of Tree-based Models

We study the robustness verification problem for tree-based models, including decision trees, random forests (RFs) and gradient boosted decision trees (GBDTs). Formal robustness verification of decision tree ensembles involves finding the exact minimal adversarial perturbation or a guaranteed lower bound of it. Existing approaches find the minimal adversarial perturbation by a mixed integer linear programming (MILP) problem, which takes exponential time so is impractical for large ensembles. Although this verification problem is NP-complete in general, we give a more precise complexity characterization. We show that there is a simple linear time algorithm for verifying a single tree, and for tree ensembles, the verification problem can be cast as a max-clique problem on a multi-partite graph with bounded boxicity. For low dimensional problems when boxicity can be viewed as constant, this reformulation leads to a polynomial time algorithm. For general problems, by exploiting the boxicity of the graph, we develop an efficient multi-level verification algorithm that can give tight lower bounds on the robustness of decision tree ensembles, while allowing iterative improvement and any-time termination. OnRF/GBDT models trained on 10 datasets, our algorithm is hundreds of times faster than the previous approach that requires solving MILPs, and is able to give tight robustness verification bounds on large GBDTs with hundreds of deep trees.Comment: Hongge Chen and Huan Zhang contributed equall

BoxE: A Box Embedding Model for Knowledge Base Completion

Knowledge base completion (KBC) aims to automatically infer missing facts by exploiting information already present in a knowledge base (KB). A promising approach for KBC is to embed knowledge into latent spaces and make predictions from learned embeddings. However, existing embedding models are subject to at least one of the following limitations: (1) theoretical inexpressivity, (2) lack of support for prominent inference patterns (e.g., hierarchies), (3) lack of support for KBC over higher-arity relations, and (4) lack of support for incorporating logical rules. Here, we propose a spatio-translational embedding model, called BoxE, that simultaneously addresses all these limitations. BoxE embeds entities as points, and relations as a set of hyper-rectangles (or boxes), which spatially characterize basic logical properties. This seemingly simple abstraction yields a fully expressive model offering a natural encoding for many desired logical properties. BoxE can both capture and inject rules from rich classes of rule languages, going well beyond individual inference patterns. By design, BoxE naturally applies to higher-arity KBs. We conduct a detailed experimental analysis, and show that BoxE achieves state-of-the-art performance, both on benchmark knowledge graphs and on more general KBs, and we empirically show the power of integrating logical rules.Comment: Proceedings of the Thirty-Fourth Annual Conference on Advances in Neural Information Processing Systems (NeurIPS 2020). Code and data available at: http://www.github.com/ralphabb/Box