The Complexity of Surjective Homomorphism Problems -- a Survey

We survey known results about the complexity of surjective homomorphism problems, studied in the context of related problems in the literature such as list homomorphism, retraction and compaction. In comparison with these problems, surjective homomorphism problems seem to be harder to classify and we examine especially three concrete problems that have arisen from the literature, two of which remain of open complexity

Binary Decision Diagrams: from Tree Compaction to Sampling

Any Boolean function corresponds with a complete full binary decision tree. This tree can in turn be represented in a maximally compact form as a direct acyclic graph where common subtrees are factored and shared, keeping only one copy of each unique subtree. This yields the celebrated and widely used structure called reduced ordered binary decision diagram (ROBDD). We propose to revisit the classical compaction process to give a new way of enumerating ROBDDs of a given size without considering fully expanded trees and the compaction step. Our method also provides an unranking procedure for the set of ROBDDs. As a by-product we get a random uniform and exhaustive sampler for ROBDDs for a given number of variables and size

A hard-sphere model on generalized Bethe lattices: Statics

We analyze the phase diagram of a model of hard spheres of chemical radius one, which is defined over a generalized Bethe lattice containing short loops. We find a liquid, two different crystalline, a glassy and an unusual crystalline glassy phase. Special attention is also paid to the close-packing limit in the glassy phase. All analytical results are cross-checked by numerical Monte-Carlo simulations.Comment: 24 pages, revised versio

Computational Complexity of Graph Partition under Vertex-Compaction to an Irreflexive Hexagon

In this paper, we solve a long-standing graph partition problem under vertex-compaction that has been of interest since about 1999. The graph partition problem that we consider in this paper is to decide whether or not it is possible to partition the vertices of a graph into six distinct non-empty sets A, B, C, D, E, and F, such that the vertices in each set are independent, i.e., there is no edge within any set, and an edge is possible but not necessary only between the pairs of sets A and B, B and C, C and D, D and E, E and F, and F and A, and there is no edge between any other pair of sets. We study the problem as the vertex-compaction problem for an irreflexive hexagon (6-cycle). Determining the computational complexity of this problem has been a long-standing problem of interest since about 1999, especially after the results of open problems obtained by the author on a related compaction problem appeared in 1999. We show in this paper that the vertex-compaction problem for an irreflexive hexagon is NP-complete. Our proof can be extended for larger even irreflexive cycles, showing that the vertex-compaction problem for an irreflexive even k-cycle is NP-complete, for all even k geq 6

Drawing Activity Diagrams

Activity diagrams experience an increasing importance in the design and description of software systems. Unfortunately, previous approaches for automatic layout support fail or are just insufficient to capture the complexity of the related requirements. We propose a new approach tailored to the needs of activity diagrams which combines the advantages of two fundamental layout concepts called "Sugiyama's approach" and "topology-shape-metrics approach", originally developed for layered layouts of directed graphs and for orthogonal layout of undirected graphs respectively

Graphulo Implementation of Server-Side Sparse Matrix Multiply in the Accumulo Database

The Apache Accumulo database excels at distributed storage and indexing and is ideally suited for storing graph data. Many big data analytics compute on graph data and persist their results back to the database. These graph calculations are often best performed inside the database server. The GraphBLAS standard provides a compact and efficient basis for a wide range of graph applications through a small number of sparse matrix operations. In this article, we implement GraphBLAS sparse matrix multiplication server-side by leveraging Accumulo's native, high-performance iterators. We compare the mathematics and performance of inner and outer product implementations, and show how an outer product implementation achieves optimal performance near Accumulo's peak write rate. We offer our work as a core component to the Graphulo library that will deliver matrix math primitives for graph analytics within Accumulo.Comment: To be presented at IEEE HPEC 2015: http://www.ieee-hpec.org

Glassy dynamics in granular compaction: sand on random graphs

We discuss the use of a ferromagnetic spin model on a random graph to model granular compaction. A multi-spin interaction is used to capture the competition between local and global satisfaction of constraints characteristic for geometric frustration. We define an athermal dynamics designed to model repeated taps of a given strength. Amplitude cycling and the effect of permanently constraining a subset of the spins at a given amplitude is discussed. Finally we check the validity of Edwards' hypothesis for the athermal tapping dynamics.Comment: 13 pages Revtex, minor changes, to appear in PR

On random graphs and the statistical mechanics of granular matter

The dynamics of spins on a random graph with ferromagnetic three-spin interactions is used to model the compaction of granular matter under a series of taps. Taps are modelled as the random flipping of a small fraction of the spins followed by a quench at zero temperature. We find that the density approached during a logarithmically slow compaction - the random-close-packing density - corresponds to a dynamical phase transition. We discuss the the role of cascades of successive spin-flips in this model and link them with density-noise power fluctuations observed in recent experiments.Comment: minor changes, to appear in EP

Strengthening Model Checking Techniques with Inductive Invariants

This paper describes optimized techniques to efficiently compute and reap benefits from inductive invariants within SAT-based model checking. We address sequential circuit verification, and we consider both equivalences and implications between pairs of nodes in the logic networks. First, we present a very efficient dynamic procedure, based on equivalence classes and incremental SAT, specifically oriented to reduce the set of checked invariants. Then, we show how to effectively integrate the computation of inductive invariants within state-of-the-art SAT-based model checking procedures. Experiments (on more than 600 designs) show the robustness of our approach on verification instances on which stand-alone techniques fai