Ordering constraints on trees

We survey recent results about ordering constraints on trees and discuss their applications. Our main interest lies in the family of recursive path orderings which enjoy the properties of being total, well-founded and compatible with the tree constructors. The paper includes some new results, in particular the undecidability of the theory of lexicographic path orderings in case of a non-unary signature

Termination of rewriting strategies: a generic approach

We propose a generic termination proof method for rewriting under strategies, based on an explicit induction on the termination property. Rewriting trees on ground terms are modeled by proof trees, generated by alternatively applying narrowing and abstracting steps. The induction principle is applied through the abstraction mechanism, where terms are replaced by variables representing any of their normal forms. The induction ordering is not given a priori, but defined with ordering constraints, incrementally set during the proof. Abstraction constraints can be used to control the narrowing mechanism, well known to easily diverge. The generic method is then instantiated for the innermost, outermost and local strategies.Comment: 49 page

Learning optimization models in the presence of unknown relations

In a sequential auction with multiple bidding agents, it is highly challenging to determine the ordering of the items to sell in order to maximize the revenue due to the fact that the autonomy and private information of the agents heavily influence the outcome of the auction. The main contribution of this paper is two-fold. First, we demonstrate how to apply machine learning techniques to solve the optimal ordering problem in sequential auctions. We learn regression models from historical auctions, which are subsequently used to predict the expected value of orderings for new auctions. Given the learned models, we propose two types of optimization methods: a black-box best-first search approach, and a novel white-box approach that maps learned models to integer linear programs (ILP) which can then be solved by any ILP-solver. Although the studied auction design problem is hard, our proposed optimization methods obtain good orderings with high revenues. Our second main contribution is the insight that the internal structure of regression models can be efficiently evaluated inside an ILP solver for optimization purposes. To this end, we provide efficient encodings of regression trees and linear regression models as ILP constraints. This new way of using learned models for optimization is promising. As the experimental results show, it significantly outperforms the black-box best-first search in nearly all settings.Comment: 37 pages. Working pape

A New Perspective on Clustered Planarity as a Combinatorial Embedding Problem

The clustered planarity problem (c-planarity) asks whether a hierarchically clustered graph admits a planar drawing such that the clusters can be nicely represented by regions. We introduce the cd-tree data structure and give a new characterization of c-planarity. It leads to efficient algorithms for c-planarity testing in the following cases. (i) Every cluster and every co-cluster (complement of a cluster) has at most two connected components. (ii) Every cluster has at most five outgoing edges. Moreover, the cd-tree reveals interesting connections between c-planarity and planarity with constraints on the order of edges around vertices. On one hand, this gives rise to a bunch of new open problems related to c-planarity, on the other hand it provides a new perspective on previous results.Comment: 17 pages, 2 figure

Improved Bounds on the Phase Transition for the Hard-Core Model in 2-Dimensions

For the hard-core lattice gas model defined on independent sets weighted by an activity $\lambda$, we study the critical activity $\lambda_c(\mathbb{Z}^2)$ for the uniqueness/non-uniqueness threshold on the 2-dimensional integer lattice $\mathbb{Z}^2$. The conjectured value of the critical activity is approximately $3.796$. Until recently, the best lower bound followed from algorithmic results of Weitz (2006). Weitz presented an FPTAS for approximating the partition function for graphs of constant maximum degree $\Delta$ when $\lambda<\lambda_c(\mathbb{T}_\Delta)$ where $\mathbb{T}_\Delta$ is the infinite, regular tree of degree $\Delta$. His result established a certain decay of correlations property called strong spatial mixing (SSM) on $\mathbb{Z}^2$ by proving that SSM holds on its self-avoiding walk tree $T_{\mathrm{saw}}^\sigma(\mathbb{Z}^2)$ where $\sigma=(\sigma_v)_{v\in \mathbb{Z}^2}$ and $\sigma_v$ is an ordering on the neighbors of vertex $v$. As a consequence he obtained that $\lambda_c(\mathbb{Z}^2)\geq\lambda_c( \mathbb{T}_4) = 1.675$. Restrepo et al. (2011) improved Weitz's approach for the particular case of $\mathbb{Z}^2$ and obtained that $\lambda_c(\mathbb{Z}^2)>2.388$. In this paper, we establish an upper bound for this approach, by showing that, for all $\sigma$, SSM does not hold on $T_{\mathrm{saw}}^\sigma(\mathbb{Z}^2)$ when $\lambda>3.4$. We also present a refinement of the approach of Restrepo et al. which improves the lower bound to $\lambda_c(\mathbb{Z}^2)>2.48$.Comment: 19 pages, 1 figure. Polished proofs and examples compared to earlier versio