Well Quasi Orders in Subclasses of Bounded Treewidth Graphs and Their Algorithmic Applications

We show that three subclasses of bounded treewidth graphs are well quasi ordered by refinements of the minor order. Specifically, we prove that graphs with bounded vertex cover are well quasi ordered by the induced subgraph order, graphs with bounded feedback vertex set are well quasi ordered by the topological-minor order, and graphs with bounded circumference are well quasi ordered by the induced minor order. Our results give algorithms for recognizing any graph family in these classes which is closed under the corresponding minor order refinement.     

Approximation Algorithms for Treewidth

This paper presents algorithms whose input is an undirected graph, and whose output is a tree decomposition of width that approximates the optimal, the treewidth of that graph. The algorithms differ in their computation time and their approximation guarantees. The first algorithm works in polynomial-time and finds a factor-O(log OPT) approximation, where OPT is the treewidth of the graph. This is the first polynomial-time algorithm that approximates the optimal by a factor that does not depend on n, the number of nodes in the input graph. As a result, we get an algorithm for finding pathwidth within a factor of O(log OPT⋅log n) from the optimal. We also present algorithms that approximate the treewidth of a graph by constant factors of 3.66, 4, and 4.5, respectively and take time that is exponential in the treewidth. These are more efficient than previously known algorithms by an exponential factor, and are of practical interest. Finding triangulations of minimum treewidth for graphs is central to many problems in computer science. Real-world problems in artificial intelligence, VLSI design and databases are efficiently solvable if we have an efficient approximation algorithm for them. Many of those applications rely on weighted graphs. We extend our results to weighted graphs and weighted treewidth, showing similar approximation results for this more general notion. We report on experimental results confirming the effectiveness of our algorithms for large graphs associated with real-world problems.     

Reduction Algorithms for Graphs of Small Treewidth

This paper presents a number of new ideas and results on graph reduction applied to graphs of bounded treewidth. S. Arnborg, B. Courcelle, A. Proskurowski, and D. Seese (J. Assoc. Comput. Mach.40, 1134–1164 (1993)) have shown that many decision problems on graphs can be solved in linear time on graphs of bounded treewidth, using a finite set of reduction rules. These algorithms can be used to solve problems on graphs of bounded treewidth without the need to obtain a tree decomposition of the input graph first. We show that the reduction method can be extended to solve the construction variants of many decision problems on graphs of bounded treewidth, including all problems definable in monadic second order logic. We also show that a variant of these reduction algorithms can be used to solve (constructive) optimization problems in O(n) time. For example, optimization and construction variants of I S and H C N can be solved in this way on graphs of small treewidth. Additionally, we show that the results of H. L. Bodlaender and T. Hagerup (SIAM J. Comput.27, 1725–1746 (1998)) can be applied to our reduction algorithms, which results in parallel reduction algorithms that use O(n) operations and O(log n log* n) time on an EREW PRAM, or O(log n) time on a CRCW PRAM.     

Safe Reduction Rules for Weighted Treewidth

Several sets of reductions rules are known for preprocessing a graph when computing its treewidth. In this paper we give reduction rules for a weighted variant of treewidth, motivated by the analysis of algorithms for probabilistic networks. We present two general reduction rules that are safe for weighted treewidth. They generalise many of the existing reduction rules for treewidth. Experimental results show that these reduction rules can significantly reduce the problem size for several instances of real-life probabilistic networks.     

Dynamic Algorithms for Graphs of Bounded Treewidth

The formalism of monadic second-order (MS) logic has been very successful in unifying a large number of algorithms for graphs of bounded treewidth. We extend the elegant framework of MS logic from static problems to dynamic problems, in which queries about MS properties of a graph of bounded treewidth are interspersed with updates of vertex and edge labels. This allows us to unify and occasionally strengthen a number of scattered previous results obtained in an ad hoc manner and to enable solutions to a wide range of additional problems to be derived automatically. As an auxiliary result of independent interest, we dynamize a data structure of Chazelle for answering queries about products of labels along paths in a tree with edges labeled by elements of a semigroup.     

Balancing Bounded Treewidth Circuits

We use algorithmic tools for graphs of small treewidth to address questions in complexity theory. For our main construction, we prove that multiplicatively disjoint arithmetic circuits of size n ^O(1) and treewidth k can be simulated by bounded fan-in arithmetic formulas of depth O(k ²logn). From this we derive an analogous statement for syntactically multilinear arithmetic circuits, which strengthens the central theorem of M. Mahajan and B.V.R. Rao (Proc. 33rd International Symposium on Mathematical Foundations of Computer Science, vol. 5162, pp. 455–466, 2008). We show our main construction has the following three applications:

• Bounded width arithmetic circuits of size n ^O(1) can be balanced to depth O(logn), provided chains of iterated multiplication in the circuit are of length O(1).
• Boolean bounded fan-in circuits of size n ^O(1) and treewidth k can be simulated by bounded fan-in formulas of depth O(k ²logn). This strengthens in the non-uniform setting the known inclusion that SC⁰?NC¹.
• We demonstrate treewidth restricted cases of Directed-Reachability and Circuit Value Problem that can be solved in LogDCFL.

We also give a construction showing, for both arithmetic and Boolean circuits, that any circuit of size n ^O(1) and treewidth O(log ⁱ n) can be simulated by a circuit of width O(log ⁱ⁺¹ n) and size n ^c , where c=O(1), if i=0, and c=O(loglogn) otherwise.     

I/O-Efficient Algorithms for Graphs of Bounded Treewidth

We present an algorithm that takes I/Os (sort(N)=Θ((N/(DB))log  _M/B (N/B)) is the number of I/Os it takes to sort N data items) to compute a tree decomposition of width at most k, for any graph G of treewidth at most k and size N, where k is a constant. Given such a tree decomposition, we use a dynamic programming framework to solve a wide variety of problems on G in I/Os, including the single-source shortest path problem and a number of problems that are NP-hard on general graphs. The tree decomposition can also be used to obtain an optimal separator decomposition of G. We use such a decomposition to perform depth-first search in G in  I/Os. As important tools that are used in the tree decomposition algorithm, we introduce flippable DAGs and present an algorithm that computes a perfect elimination ordering of a k-tree in I/Os. The second contribution of our paper, which is of independent interest, is a general and simple framework for obtaining I/O-efficient algorithms for a number of graph problems that can be solved using greedy algorithms in internal memory. We apply this framework in order to obtain an improved algorithm for finding a maximal matching and the first deterministic I/O-efficient algorithm for finding a maximal independent set of an arbitrary graph. Both algorithms take I/Os. The maximal matching algorithm is used in the tree decomposition algorithm. An abstract of this paper was presented at the 12th Annual ACM-SIAM Symposium on Discrete Algorithms, Proceedings, pp. 89–90, 2001. Research of A. Maheshwari supported by NSERC. Part of this work was done while the second author was a Ph.D. student at the School of Computer Science of Carleton University.     

Treewidth Lower Bounds with Brambles

In this paper we present a new technique for computing lower bounds for graph treewidth. Our technique is based on the fact that the treewidth of a graph G is the maximum order of a bramble of G minus one. We give two algorithms: one for general graphs, and one for planar graphs. The algorithm for planar graphs is shown to give a lower bound for both the treewidth and branchwidth that is at most a constant factor away from the optimum. For both algorithms, we report on extensive computational experiments that show that the algorithms often give excellent lower bounds, in particular when applied to (close to) planar graphs. This work was partially supported by the Netherlands Organisation for Scientific Research NWO (project Treewidth and Combinatorial Optimisation) and partially by the DFG research group “Algorithms, Structure, Randomness” (Grant number GR 883/9-3, GR 883/9-4).     

Treewidth computations II. Lower bounds

For several applications, it is important to be able to compute the treewidth of a given graph and to find tree decompositions of small width reasonably fast. Good lower bounds on the treewidth of a graph can, amongst others, help to speed up branch and bound algorithms that compute the treewidth of a graph exactly. A high lower bound for a specific graph instance can tell that a dynamic programming approach for solving a problem is infeasible for this instance. This paper gives an overview of several recent methods that give lower bounds on the treewidth of graphs.     

Treewidth computations I. Upper bounds

For more and more applications, it is important to be able to compute the treewidth of a given graph and to find tree decompositions of small width reasonably fast.This paper gives an overview of several upper bound heuristics that have been proposed and tested for the problem of determining the treewidth of a graph and finding tree decompositions. Each of the heuristics produces tree decompositions whose width may be larger than the optimal width. However, experiments show that in many cases, the heuristics give tree decompositions whose width is close to the exact treewidth of the input graphs.     

Algorithmic methodologies for FPGA-based vision

This paper proposes a strategy for the design of computer vision on Field Programmable Gate Arrays (FPGAs). We show that there are certain advantages to the approach of designing an algorithm to specifically suit hardware rather than attempting to replicate the behavior of software implementations in hardware. We justify this approach through the analysis of two case studies. In the first case study, we present FPGA implementations of two corner detectors. We make a number of observations which point to the advantages of an FPGA-tailored algorithm. In the second case study, we investigate the feasibility of this approach by designing a proof-of-concept face detection algorithm that was designed specifically for an FPGA. We show that this design allows for high detection speed on a low-cost FPGA device, although the same algorithm would not be considered in software. Finally, we conclude that FPGAs offer special opportunities for specialized algorithms that are infeasible in software.     

基于视觉传感器SLAM问题的算法构建

自主移动机器人通过自身携带的传感器来感知周围环境是实现其智能导航的前提。由于视觉传感器只能检测路标的方位角,不能提供距离信息,当利用视觉传感器完成机器人的同步定位及地图创建（SLAM）时,由于路标距离信息的缺失,会带来新路标的初始化及旧路标关联等难题。为识别新路标方位,采用滞后的三角型测量法来估计新路标的距离,从而解决了新路标初始化及旧路标关联等问题,并最终建立了一套适用于视觉传感器的SLAM算法及在Matlab上建立的仿真模型。仿真实验结果表明,算法满足概率估计的一致性和收敛性条件,是一种有效的仅检测路标方位角的同步定位及地图创建算法,证明方法的有效性。