Iterative-deepening search with on-line tree size prediction

The memory requirements of best-first graph search algorithms such as A* often prevent them from solving large problems. The best-known approach for coping with this issue is iterative deepening, which performs a series of bounded depth-first searches. Unfortunately, iterative deepening only performs well when successive cost bounds visit a geometrically increasing number of nodes. While it happens to work acceptably for the classic sliding tile puzzle, IDA* fails for many other domains. In this paper, we present an algorithm that adaptively chooses appropriate cost bounds on-line during search. During each iteration, it learns a model of the search tree that helps it to predict the bound to use next. Our search tree model has three main benefits over previous approaches: (1) it will work in domains with real-valued heuristic estimates, (2) it can be trained on-line, and (3) it is able to make more accurate predictions with only a small number of training examples. We demonstrate the power of our improved model by using it to control an iterative-deepening A* search on-line. While our technique has more overhead than previous methods for controlling iterative-deepening A*, it can give more robust performance by using its experience to accurately double the amount of search effort between iterations.     

Path recovery in frontier search for multiobjective shortest path problems

Frontier search is a best-first graph search technique that allows significant memory savings over previous best-first algorithms. The fundamental idea is to remove from memory already explored nodes, keeping only open nodes in the search frontier. However, once the goal node is reached, additional techniques are needed to recover the solution path. This paper describes and analyzes a path recovery procedure for frontier search applied to multiobjective shortest path problems. Differences with the scalar case are outlined, and performance is evaluated over a random problem set.     

AND/OR search spaces for graphical models

The paper introduces an AND/OR search space perspective for graphical models that include probabilistic networks (directed or undirected) and constraint networks. In contrast to the traditional (OR) search space view, the AND/OR search tree displays some of the independencies present in the graphical model explicitly and may sometimes reduce the search space exponentially. Indeed, most algorithmic advances in search-based constraint processing and probabilistic inference can be viewed as searching an AND/OR search tree or graph. Familiar parameters such as the depth of a spanning tree, treewidth and pathwidth are shown to play a key role in characterizing the effect of AND/OR search graphs vs. the traditional OR search graphs. We compare memory intensive AND/OR graph search with inference methods, and place various existing algorithms within the AND/OR search space.     

Breadth-first heuristic search

Recent work shows that the memory requirements of A* and related graph-search algorithms can be reduced substantially by only storing nodes that are on or near the search frontier, using special techniques to prevent node regeneration, and recovering the solution path by a divide-and-conquer technique. When this approach is used to solve graph-search problems with unit edge costs, we show that a breadth-first search strategy can be more memory-efficient than a best-first strategy. We also show that a breadth-first strategy allows a technique for preventing node regeneration that is easier to implement and can be applied more widely. The breadth-first heuristic search algorithms introduced in this paper include a memory-efficient implementation of breadth-first branch-and-bound search and a breadth-first iterative-deepening A* algorithm that is based on it. Computational results show that they outperform other systematic search algorithms in solving a range of challenging graph-search problems.     

Finding smallest k-Compact tree set for keyword queries on graphs using mapreduce

Keyword search is integrated in many applications on account of the convenience to convey users’ query intention. Most existing works in keyword search on graphs modeled the query results as individual minimal connected trees or connected graphs that contain the keywords. We observe that significant overlap may exist among those query results, which would affect the result diversification. Besides, most solutions required accessing graph data and pre-built indexes in memory, which is not suitable to process big dataset. In this paper, we define the smallest k-compact tree set as the keyword query result, where no shared graph node exists between any two compact trees. We then develop a progressive A* based scalable solution using MapReduce to compute the smallest k-compact tree set, where the computation process could be stopped once the generated compact tree set is sufficient to compute the keyword query result. We conduct experiments to show the efficiency of our proposed algorithm.     

A Flexible Algorithm for Generating All the Spanning Trees in Undirected Graphs

In this paper we propose an algorithm for generating all the spanning trees in undirected graphs. The algorithm requires O (n+m+ τ n) time where the given graph has n vertices, m edges, and τ spanning trees. For outputting all the spanning trees explicitly, this time complexity is optimal. Our algorithm follows a special rooted tree structure on the skeleton graph of the spanning tree polytope. The rule by which the rooted tree structure is traversed is irrelevant to the time complexity. In this sense, our algorithm is flexible. If we employ the depth-first search rule, we can save the memory requirement to O (n+m). A breadth-first implementation requires as much as O (m+ τ n) space, but when a parallel computer is available, this might have an advantage. When a given graph is weighted, the best-first search rule provides a ranking algorithm for the minimum spanning tree problem. The ranking algorithm requires O (n+ m + τ n) time and O (m+ τ n) space when we have a minimum spanning tree. Received January 21, 1995; revised February 19, 1996.     

Scheduling task graphs optimally with A*

Scheduling tasks onto the processors of a parallel system is a crucial part of program parallelisation. Due to the NP-hard nature of the task scheduling problem, scheduling algorithms are based on heuristics that try to produce good rather than optimal schedules. Nevertheless, in certain situations it is desirable to have optimal schedules, for example for time-critical systems or to evaluate scheduling heuristics. This paper investigates the task scheduling problem using the A* search algorithm which is a best-first state space search. The adaptation of the A* search algorithm for the task scheduling problem is referred to as the A* scheduling algorithm. The A* scheduling algorithm can produce optimal schedules in reasonable time for small to medium sized task graphs with several tens of nodes. In comparison to a previous approach, the here presented A* scheduling algorithm has a significantly reduced search space due to a much improved consistent and admissible cost function f(s) and additional pruning techniques. Experimental results show that the cost function and the various pruning techniques are very effective for the workload. Last but not least, the results show that the proposed A* scheduling algorithm significantly outperforms the previous approach.     

Partial-order reduction and trail improvement in directed model checking

In this paper we present work on trail improvement and partial-order reduction in the context of directed explicit-state model checking. Directed explicit-state model checking employs directed heuristic search algorithms such as A* or best-first search to improve the error-detection capabilities of explicit-state model checking. We first present the use of directed explicit-state model checking to improve the length of already established error trails. Second, we show that partial-order reduction, which aims at reducing the size of the state space by exploiting the commutativity of concurrent transitions in asynchronous systems, can coexist well with directed explicit-state model checking. Finally, we illustrate how to mitigate the excessive length of error trails produced by partial-order reduction in explicit-state model checking. In this context we also propose a combination of heuristic search and partial-order reduction to improve the length to already provided counterexamples.     

Efficient branch-and-bound algorithms on a two-level memory system

Branch-and-bound algorithms in a system with a two-level memory hierarchy were evaluated. An efficient implementation depends on the disparities in the numbers of subproblems expanded between the depth-first and best-first searches as well as the relative speeds of the main and secondary memories. A best-first search should be used when it expands a much smaller number of subproblems than that of a depth-first search, and the secondary memory is relatively slow. In contrast, a depth-first search should be used when the number of expanded subproblems is close to that of a best-first search. The choice is not as clear for cases in between these cases are studied. Two strategies are proposed and analyzed: a specialized virtual-memory system that matches the architectural design with the characteristics of the existing algorithm, and a modified branch-and-bound algorithm that can be tuned to the characteristic of the problem and the architecture. The latter strategy illustrates that designing a better algorithm is sometimes more effective that tuning the architecture alone     

A comparison of multiobjective depth-first algorithms

Many real world problems involve several, usually conflicting, objectives. Multiobjective analysis deals with these problems locating trade-offs between different optimal solutions. Regarding graph search problems, several algorithms based on best-first and depth-first approaches have been proposed to return the set of all Pareto optimal solutions. This article presents a detailed comparison between two representatives of multiobjective depth-first algorithms, PIDMOA* and MO-DF-BnB. Both of them extend previous single-objective search algorithms with linear-space requirements to the multiobjective case. Experimental analyses on their time performance over tree-shaped search spaces are presented. The results clarify the fitness of both algorithms to parameters like the number or depth of goal nodes.     

A decision tree approach to graph and subgraph isomorphism detection

A new approach to the problem of graph and subgraph isomorphism detection from an input graph to a database of model graphs is proposed in this paper. It is based on a preprocessing step in which the model graphs are used to create a decision tree. At run time, subgraph isomorphisms are detected by means of decision tree traversal. If we neglect the time needed for preprocessing, the computational complexity of the new graph algorithm is only polynomial in the number of input graph vertices. In particular, it is independent of the number of model graphs and the number of edges in any of the graphs. However, the decision tree is of exponential size. Several pruning techniques which aim at reducing the size of the decision tree are presented. A computational complexity analysis of the new method is given and its behavior is studied in a number of practical experiments with randomly generated graphs.     

State-set branching: Leveraging BDDs for heuristic search

In this article, we present a framework called state-set branching that combines symbolic search based on reduced ordered Binary Decision Diagrams (BDDs) with best-first search, such as A* and greedy best-first search. The framework relies on an extension of these algorithms from expanding a single state in each iteration to expanding a set of states. We prove that it is generally sound and optimal for two A* implementations and show how a new BDD technique called branching partitioning can be used to efficiently expand sets of states. The framework is general. It applies to any heuristic function, evaluation function, and transition cost function defined over a finite domain. Moreover, branching partitioning applies to both disjunctive and conjunctive transition relation partitioning. An extensive experimental evaluation of the two A* implementations proves state-set branching to be a powerful framework. The algorithms outperform the ordinary A* algorithm in almost all domains. In addition, they can improve the complexity of A* exponentially and often dominate both A* and blind BDD-based search by several orders of magnitude. Moreover, they have substantially better performance than BDDA*, the currently most efficient BDD-based implementation of A*.     

Enhanced iterative-deepening search

Iterative-deepening searches mimic a breadth-first node expansion with a series of depth-first searches that operate with successively extended search horizons. They have been proposed as a simple way to reduce the space complexity of best-first searches like A* from exponential to linear in the search depth. But there is more to iterative-deepening than just a reduction of storage space. As the authors show, the search efficiency can be greatly improved by exploiting previously gained node information. The information management techniques considered here owe much to their counterparts from the domain of two-player games, namely the use of fast-execution memory functions to guide the search. The authors' methods not only save node expansions, but are also faster and easier to implement than previous proposals     

Parallel Branch-and-Bound Formulations for AND/OR Tree Search

This paper discusses two general schemes for performing branch-and-bound (B&B) search in parallel. These schemes are applicable in principle to most of the problems which can be solved by B&B. The schemes are implemented for SSS*, a versatile algorithm having applications in game tree search, structural pattern analysis, and AND/OR graph search. The performance of parallel SSS* is studied in the context of AND/OR tree and game tree search. The paper concludes with comments on potential applications of these parallel implementations of SSS* in structural pattern analysis and game playing.     

基于搜索树的平面图支配集算法

许多来自工业应用的优化问题都是NP难问题。确定参数可解FPT作为处理这类问题的另外一种思路,在最近的10多年中受到了广泛的关注。支配集问题是图论中最重要的NP完全的组合优化问题之一,即使对于FPT体系而言,一般图中的支配集问题属于W[2]完全的,意味着不可能设计出复杂度为f(k)no(1)的算法。在本文中,我们考虑在给定的平面图G=(V,E)中参数化支配集问题,给定参数k,看是否存在大小为k的顶点集合支配图中的其他顶点,当把问题限定在平面图上,这个问题属于确定参数可解。本文给出了基于两组归约规则的搜索树算法,通过使用规约技术化简实例,构造搜索树,得到了复杂度为O(8kn)的算法,同时通过相关实验结果显示了归约规则对算法的作用。     

EvoGeneSys,a new evolutionary approach to graph generation

Graphs are widely used to represent complex and structured information of interest in various fields of science and engineering. When using graph representations, problems of special interest often imply searching. For example, searching for the prototypes representing a dataset of graphs or for the graph that optimizes a set of parameters. In any case, it is necessary that the problem solution be expressed in terms of graphs. Therefore, defining effective methods for automatically generating single graphs, or sets of graphs, representing problem solutions, is a key issue. A new evolutionary computation-based approach specifically devised for generating graphs is presented. The method is based on a special data structure, called multilist, which allows the encoding of any type of graph, directed or undirected, with or without attributes. Graph encoding by multilists makes it possible to define effective crossover and mutation operators, overcoming the problems normally encountered when implementing genetic operators on graphs. Further advantages of the proposed approach are that it does not require any problem specific knowledge and it is able to search for graphs whose number of nodes is not known a priori. Three sets of experiments were performed to test the proposed approach and the solutions found were compared with those obtained by other approaches proposed in the literature.     

Single machine weighted earliness–tardiness penalty problem with a common due date

In this paper, we have considered a class of single machine job scheduling problems where the objective is to minimize the weighted sum of earliness–tardiness penalties of jobs. The weights are job-independent but they depend on whether a job is early or tardy. The restricted version of the problem where the common due date is smaller than a critical value, is known to be NP-complete. While dynamic programming formulation runs out of memory for large problem instances, depth-first branch-and-bound formulation runs slow for large problems since it uses a tree search space. In this paper, we have suggested an algorithm to optimally solve large instances of the restricted version of the problem. The algorithm uses a graph search space. Unlike dynamic programming, the algorithm can output optimal solutions even when available memory is limited. It has been found to run faster than dynamic programming and depth-first branch-and-bound formulations and can solve much larger instances of the problem in reasonable time. New upper and lower bounds have been proposed and used. Experimental findings are given in detail.Scope and purposeA class of single machine problems arising out of scheduling jobs in JIT environment has been considered in this paper. The objective is to minimize the total weighted earliness–tardiness penalties of jobs. In this paper, we have presented a new algorithm and conducted extensive empirical runs to show that the new algorithm performs much better than the existing approaches in solving large instances of the problem.     

Constraint models for graceful graphs

We present three constraint models of the problem of finding a graceful labelling of a graph, or proving that the graph is not graceful. An experimental comparison of the models applied to different classes of graph is given. The first model seems a natural way to represent the problem, but explores a much larger search tree than the other models. The second model does much less search, by making the most constrained decisions first, but is slow because the constraints are time-consuming to propagate. The third model combines the best features of the others, doing little more search than the second model while being much the fastest of the three. The comparison of the three models provides a useful case-study of modelling problems as constraint satisfaction problems. In addition, we show that constraint programming can be a useful tool for the study of graceful graphs; the models presented here have contributed many new results.     

Cut-and-solve: An iterative search strategy for combinatorial optimization problems

Branch-and-bound and branch-and-cut use search trees to identify optimal solutions to combinatorial optimization problems. In this paper, we introduce an iterative search strategy which we refer to as cut-and-solve and prove optimality and termination for this method. This search is different from traditional tree search as there is no branching. At each node in the search path, a relaxed problem and a sparse problem are solved and a constraint is added to the relaxed problem. The sparse problems provide incumbent solutions. When the constraining of the relaxed problem becomes tight enough, its solution value becomes no better than the incumbent solution value. At this point, the incumbent solution is declared to be optimal. This strategy is easily adapted to be an anytime algorithm as an incumbent solution is found at the root node and continuously updated during the search.Cut-and-solve enjoys two favorable properties. Since there is no branching, there are no “wrong” subtrees in which the search may get lost. Furthermore, its memory requirement is negligible. For these reasons, it has potential for problems that are difficult to solve using depth-first or best-first search tree methods.In this paper, we demonstrate the cut-and-solve strategy by implementing a generic version of it for the Asymmetric Traveling Salesman Problem (ATSP). Our unoptimized implementation outperformed state-of-the-art solvers for five out of seven real-world problem classes of the ATSP. For four of these classes, cut-and-solve was able to solve larger (sometimes substantially larger) problems. Our code is available at our websites.     

Using graph parsing for automatic graph drawing

This paper presents a procedure for automatically drawing directed graphs. Our system, Clan-based Graph Drawing Tool (CG), uses a unique clan-based graph decomposition to determine intrinsic substructures (clans) in the graph and to produce a parse tree. The tree is given attributes that specify the node layout. CG then uses tree properties with the addition of “routing nodes” to route the edges. The objective of the system is to provide, automatically, an aesthetically pleasing visual layout for arbitrary directed graphs. The prototype has shown the strengths of this approach. The innovative strategy of clan-based graph decomposition is the first digraph drawing technique to analyze locality in the graph in two dimensions. The typical approach to drawing digraphs uses a single dimension, level, to arrange the nodes