We survey recent trends in practical algorithms for balanced graph partitioning, point to applications and discuss future research directions.

/pdf/recent-advances-in-graph-partitioning-4eois71ejs.pdf

Recent Advances in Graph Partitioning

This work presents a contribution to the solution of the average agreement problem on a network with quantized links. Starting from the well-known linear diffusion algorithm, we propose a simple and effective adaptation that is able to preserve the average of states and to drive the system near to the consensus value, when a uniform quantization is applied to communication between agents. The properties of this algorithm are investigated both by a worst-case analysis and by a probabilistic analysis, and are shown to depend on the spectral properties of the evolution matrix. A special attention is devoted to the issue of the dependence of the performance on the number of agents, and several examples are given. Copyright © 2008 John Wiley & Sons, Ltd.

Average consensus on networks with quantized communication

The increasing role of quantified Boolean logic in many applications calls for practically efficient decision procedures. One of the most promising paradigms is the semantic tree format implemented in the style of the DPLL procedure. In this paper, so-called learningtec hniques like intelligent backtrackingand caching of lemmas which proved useful in the pure propositional case are generalised to the quantified Boolean case and the occuring differences are discussed. Due to the strong restriction of the variable selection in semantic tree procedures for quantified Boolean formulas, learning methods are more important than in the propositional case, as we demonstrate. Furthermore, in addition to the caching of lemmas, significant advances can be achieved by techniques based on the caching of models, too. The theoretical effect of these improvements is illustrated by a comparison of the search spaces on pathological examples. We also describe the basic features of the system Semprop, which is an efficient implementation of (some of) the developed techniques, and give the results of an experimental evaluation of the system on a number of practical examples.

/pdf/lemma-and-model-caching-in-decision-procedures-for-1qo09ilhvi.pdf

Lemma and Model Caching in Decision Procedures for Quantified Boolean Formulas

Deciding the satisfiability of a Quantified Boolean Formula (QBF) is an important research issue in Artificial Intelligence. Many reasoning tasks involving planning [1], abduction, reasoning about knowledge, non monotonic reasoning [2], can be directly mapped into the problem of deciding the satisfiability of a QBF. In this paper we present QuBE, a system for deciding QBFs satisfiability.

/pdf/qube-a-system-for-deciding-quantified-boolean-formulas-2ol9vlj91c.pdf

QUBE: A System for Deciding Quantified Boolean Formulas Satisfiability

We propose a new declarative planning language, called K, which is based on principles and methods of logic programming. In this language, transitions between states of knowledge can be described, rather than transitions between completely described states of the world, which makes the language well suited for planning under incomplete knowledge. Furthermore, our formalism enables the use of default principles in the planning process by supporting negation as failure. Nonetheless, K also supports the representation of transitions between states of the world (i.e., states of complete knowledge) as a special case, which shows that the language is very flexible. As we demonstrate on particular examples, the use of knowledge states may allow for a natural and compact problem representation. We then provide a thorough analysis of the computational complexity of K, and consider different planning problems, including standard planning and secure planning (also known as conformant planning) problems. We show that these problems have different complexities under various restrictions, ranging from NP to NEXPTIME in the propositional case. Our results form the theoretical basis for the DLVk system, which implements the language K on top of the DLV logic programming system.

A logic programming approach to knowledge-state planning: Semantics and complexity

In this paper, we present PQSOLVE, a distributed theorem-prover for Quantified Boolean Formulae First, we introduce our sequential algorithm QSOLVE, which uses new heuristics and improves the use of known heuristics to prune the search tree As a result, QSOLVE is more efficient than the QSAT-solvers previously known We have parallelized QSOLVE The resulting distributed QSAT-solver PQSOLVE uses parallel search techniques, which we have developed for distributed game tree search PQSOLVE runs efficiently on distributed systems, i e parallel systems without any shared memory We briefly present experiments that show a speedup of about 114 on 128 processors To the best of our knowledge we are the first to introduce an efficient parallel QSAT-solver

/pdf/a-distributed-algorithm-to-evaluate-quantified-boolean-3k9s1h7xqn.pdf

A Distributed Algorithm to Evaluate Quantified Boolean Formulae

The NP-hard graph partitioning problem is an important subtask in load balancing and many other applications. It requires the division of a graph's vertex set into P equally sized subsets such that some objective function is optimized. State-of-the-art libraries addressing this problem show several deficiencies: they are hard to parallelize, focus on small edge-cuts instead of few boundary vertices, and often produce disconnected partitions. This work introduces our novel graph partitioning and repartitioning heuristic Bubble-FOS/C. In contrast to other libraries, Bubble-FOS/C does not try to minimize the edge-cut explicitly, but focuses instead implicitly on good partition shapes. The shapes are optimized by diffusion processes that are embedded into an iterative framework. This approach incorporates a high degree of parallelism. Besides describing the evolution process that led to the new diffusion scheme FOS/C used by Bubble-FOS/C, we reveal some of FOS/C's properties and propose a number of enhancements for a fast and reliable implementation. Our experiments, in which we compare sequential and parallel Bubble-FOS/C implementations to the state-of-the-art libraries Metis and Jostle, reveal that our new heuristic is slower, but generates high-quality solutions that are often superior.

Graph partitioning and disturbed diffusion

We propose a load balancing heuristic for parallel adaptive finite element method (FEM) simulations. In contrast to most existing approaches, the heuristic focuses on good partition shapes rather than on minimizing the classical edge-cut metric. By applying algebraic multigrid (AMG), we are able to speed up the two most time consuming calculations of the approach while maintaining its large amount of natural parallelism.

/pdf/accelerating-shape-optimizing-load-balancing-for-parallel-10fv0tchhp.pdf

Accelerating shape optimizing load balancing for parallel FEM simulations by algebraic multigrid

Using space-filling curves to partition unstructured finite element meshes is a widely applied strategy when it comes to distributing load among several computation nodes. Compared to more elaborated graph partitioning packages, this geometric approach is relatively easy to implement and very fast. However, results are not expected to be as good as those of the latter. In this paper we present results of our experiments comparing the quality of partitionings computed with different types of space-filling curves to those generated with the graph partitioning package Metis.

Partitioning finite element meshes using space-filling curves

The task of balancing dynamically generated work load occurs in a wide range of parallel and distributed applications. Diffusion based schemes, which belong to the class of nearest neighbor load balancing algorithms, are a popular way to address this problem. Originally created to equalize the amount of arbitrarily divisible load among the nodes of a static and homogeneous network, they have been generalized to heterogeneous topologies. Additionally, some simple diffusion algorithms have been adapted to work in dynamic networks as well. However, if the load is not divisible arbitrarily but consists of indivisible unit size tokens, diffusion schemes are not able to balance the load properly. In this paper we consider the problem of balancing indivisible unit size tokens on heterogeneous systems. By modifying a randomized strategy invented for homogeneous systems, we can achieve an asymptotically minimal expected overload in l1, l2 and l1 norm while only slightly increasing the run-time by a logarithmic factor. Our experiments show that this additional factor is usually not required in applications.

/pdf/distributing-unit-size-workload-packages-in-heterogeneous-31qp63s2gf.pdf

Stefan Schamberger

Papers

A Distributed Algorithm to Evaluate Quantified Boolean Formulae

Graph partitioning and disturbed diffusion

Accelerating shape optimizing load balancing for parallel FEM simulations by algebraic multigrid

Partitioning finite element meshes using space-filling curves

Distributing Unit Size Workload Packages in Heterogeneous Networks