OF THE THESIS Partial Order Reduction for Planning by You Xu Master of Science in Computer Science Washington University in St. Louis, 2009 Research Advisor: Professor Yixin Chen Partial Order Reduction (POR) is a technique that reduces the search space by recognizing interchangeable orders between actions and expanding only a subset of all possible orders during the search. It has been extensively studied in model checking and has proven to be an enabling technique for reducing the search space and costs. Several POR algorithms have been proposed in planning, including the Expansion Core (EC) and Stratified Planning (SP) algorithms. Being orthogonal to the development of accurate heuristic functions, these reduction methods show great potential to improve the planning efficiency from a new perspective. However, it is unclear how these POR methods relate to each other and whether there exist stronger reduction methods. In this thesis, we have proposed a unifying theory that provides a necessary and sufficient condition for two actions to be semi-commutative. We have also revealed that semi-commutativity is the central property that enables POR. We have also interpreted both EC and SP algorithms using this new theory. Further, we have ii proposed new, stronger POR algorithms based on the new theory. We have also applied these new algorithms to solve benchmark problems across various planning domains. Experimental results have shown significant search cost reduction.

MOTIVATION:No individual assembly algorithm addresses all the known limitations of assembling short-length sequences. Overall reduced sequence contig length is the major problem that challenges the usage of these assemblies. We describe an algorithm to take advantages of different assembly algorithms or sequencing platforms to improve the quality of next-generation sequence (NGS) assemblies.RESULTS:The algorithm is implemented as a graph accordance assembly (GAA) program. The algorithm constructs an accordance graph to capture the mapping information between the target and query assemblies. Based on the accordance graph, the contigs or scaffolds of the target assembly can be extended, merged or bridged together. Extra constraints, including gap sizes, mate pairs, scaffold order and orientation, are explored to enforce those accordance operations in the correct context. We applied GAA to various chicken NGS assemblies and the results demonstrate improved contiguity statistics and higher genome and gene coverage.AVAILABILITY:GAA is implemented in OO perl and is available here: http://sourceforge.net/projects/gaa-wugi/.CONTACT:lye@genome.wustl.edu

Traditional AI search methods search in a state space typically modelled as a directed graph. Prohibitively large sizes of state space graphs make complete or optimal search expensive. A key observation, as exemplified by the SAS+ formalism for planning, is that most commonly a state-space graph can be decomposed into subgraphs, linked by constraints. We propose a novel space reduction algorithm that exploits such structure. The result reveals that standard search algorithms may explore many redundant paths. Our method provides an automatic way to remove such redundancy. At each state, we expand only the subgraphs within a dependency closure satisfying certain sufficient conditions instead of all the subgraphs. Theoretically we prove that the proposed algorithm is completeness-preserving as well as optimality-preserving. We show that our reduction method can significantly reduce the search cost on a collection of planning domains.

Most planning problems have strong structures. They can be decomposed into subdomains with causal dependencies. The idea of exploiting the domain decomposition has motivated previous work such as hierarchical planning and factored planing. However, these algorithms require extensive backtracking and lead to few efficient general-purpose planners. On the other hand, heuristic search has been a successful approach to automated planning. The domain decomposition of planning problems, unfortunately, is not directly and fully exploited by heuristic search. We propose a novel and general framework to exploit domain decomposition. Based on a structure analysis on the SAS+ planning formalism, we stratify the sub-domains of a planning problem into dependency layers. By recognizing the stratification of a planning structure, we propose a space reduction method that expands only a subset of executable actions at each state. This reduction method can be combined with state-space search, allowing us to simultaneously employ the strength of domain decomposition and high-quality heuristics. We prove that the reduction preserves completeness and optimality of search and experimentally verify its effectiveness in space reduction.

计算具有较小度的生成树是算法与复杂性研究的一个基本问题,同时在网络设计等领域具有重要应用.给定具有n个顶点的有向无环图G=(V,E)和根顶点r∈ V,最小度生成树问题欲求一棵以r为根的生成树T,使得在G的所有以r为根的生成树中T的最大度最小.给出该问题的一种迭代的多项式时间近似算法.该算法所求树的度不超过△*+1,其中△*为某一最优树的度.算法的时间复杂度为O(n2logn),其中n为顶点数目.算法没有运用过多的枚举,其实际运行时间要快得多.

In this paper, we focus on the directed minimum degree spanning tree problem and the minimum time broadcast problem. Firstly, we propose a polynomial time algorithm for the minimum degree spanning tree problem in directed acyclic graphs. The algorithm starts with an arbitrary spanning tree, and iteratively reduces the number of vertices of maximum degree. We can prove that the algorithm must reduce a vertex of the maximum degree for each phase, and finally result in an optimal tree. The algorithm terminates in O(mnlogn) time, where m and n are the numbers of edges and vertices of the graph, respectively. Moreover, we apply the new algorithm to the minimum time broadcast problem. Two consequences for directed acyclic graphs are: (1) the problem under the vertex-disjoint paths mode can be approximated within a factor of O(logn/logOPT) of the optimum in O(mnlogn)-time; (2) the problem under the edge-disjoint paths mode can be approximated within a factor of O(Δ*/logΔ*) of the optimum in O(mnlogn)-time, where Δ* is the minimum k such that there is a spanning tree of the graph with maximum degree k.

A randomized approximation algorithm for the set cover problem was proposed. Given a collection S of subsets of E = {e1, e2, ⋯, en} with weights on the subsets, the goal of the weighted set cover problem is to cover all the elements of E by picking a subset with the minimum total cost from S. For each uncovered element of the set E, a subset which contains the element was chosen with a certain elaborately designed probability distribution. The algorithm will not terminate until all the elements have been covered. The proposed algorithm computes a cover whose expected cost is at most B·opt where opt is the cost of some optimal cover for the problem, where B = maxe∈E{|L(e)|} and L(e) = {s|e∈s, s∈S}. The running time of the algorithm is O(n) where n is the cardinality of E.

The minimum degree spanning tree problem has been studied extensively. In this paper, we present a polynomial time algorithm for the minimum degree spanning tree problem on directed acyclic graphs. The algorithm starts with an arbitrary spanning tree, and iteratively reduces the number of vertices of maximum degree. We can prove the algorithm must reduce a vertex of the maximum degree for each phase, and finally result in an optimal tree. The algorithm terminates in O(mnlogn) time, where m and n are the number of edges and vertices respectively.