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### Solving problems on recursively constructed graphs

Richard B. Borie, R. Gary Parker, Craig A. Tovey
2008 ACM Computing Surveys
Fast algorithms can be created for many graph problems when instances are confined to classes of graphs that are recursively constructed.  ...  that legitimize their identity as recursively constructed graphs.  ...  Solving Problems on Recursively Constructed Graphs · 21 s A B C D E F H I J K A A B A E B E B B A B A B E E E C C D C H D H D D C D C D H H H E A B E F F I F J I J I H C D H I F I F I J J J J F I J K C  ...

### Using modular decomposition technique to solve the maximum clique problem [article]

Irina Utkina
2017 arXiv   pre-print
Our algorithm takes the modular decomposition tree from the paper of Tedder et. al. and finds solution recursively. Also, we propose algorithms to construct graphs with modules.  ...  In this article we use the modular decomposition technique for exact solving the weighted maximum clique problem.  ...  of MC problem and then solve it using some general algorithm for weighted maximum clique problem.  ...

### AN IMPROVED ALGORITHM FOR FINDING TREE DECOMPOSITIONS OF SMALL WIDTH

LJUBOMIR PERKOVIĆ, BRUCE REED
2000 International Journal of Foundations of Computer Science
A consequence is that the fundamental disjoint rooted paths problem can now be solved in O(n 2 ) time. This is the primary motivation for this paper.  ...  We present a modi cation of Bodlaender's linear time algorithm that, for constant k, determines whether an input graph G = (V; E) has treewidth k and, if so, constructs a tree decomposition of G of width  ...  Second, we have streamlined the construction of G (the graph the algorithm recurses on) so the algorithm is simpler and the graph G is smaller.  ...

### Page 4309 of Mathematical Reviews Vol. , Issue 88h [page]

1988 Mathematical Reviews
Construction of recursive programs: Principles of construction; Example 1. Draw- ing a figure within a figure; Example 2. Binary search; Example 3. Sorting by partitioning (‘quicksort’); Example 4.  ...  One algorithm is a vectorized matrix-vector multiplication, and the other is a new, vectorized, overlapping-block iterative method for solving finite element equations.”  ...

### Algorithmic Meta-Theorems for Monotone Submodular Maximization [article]

Masakazu Ishihata, Takanori Maehara, Tomas Rigaux
2018 arXiv   pre-print
We consider a monotone submodular maximization problem whose constraint is described by a logic formula on a graph.  ...  Formally, we prove the following three 'algorithmic metatheorems.' (1) If the constraint is specified by a monadic second-order logic on a graph of bounded treewidth, the problem is solved in n^O(1) time  ...  Acknowledgment We thank Antoine Amarilli for confirming the structuredness of their construction of DNNF in [1] .  ...

### Binary search and recursive graph problems

William I. Gasarch, Katia S. Guimarães
1997 Theoretical Computer Science
A graph G = (V, E) is recursive if every node of G has a finite number of neighbors, and both V and E are recursive (i.e., decidable).  ...  We examine the complexity of identifying the number of connected components of an infinite recursive graph, and other related problems, both when an upper bound to that value is given a priori or not.  ...  We show that ]log(c + l)l queries is a tight bound on the number of queries necessary to solve the problem, even if a more powerful oracle is used.  ...

### An efficient programming model for memory-intensive recursive algorithms using parallel disks

Vlad Slavici, Daniel Kunkle, Gene Cooperman, Stephen Linton
2012 Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation - ISSAC '12
We demonstrate the ideas on a largest computation of its kind: the determinization via subset construction and minimization of very large nondeterministic finite set automata (NFA).  ...  Research in this area over the past 10 years has focused on per-problem solutions, without providing much insight into the connection between legacy algorithms and out-of-core algorithms.  ...  This type of recursion corresponds to a callgraph in which a sub-problem can be solved by aggregating the results of n sub-sub-problems.  ...

### On the complexity of finding the chromatic number of a recursive graph II: the unbounded case

Richard Beigel, William I. Gasarch
1989 Annals of Pure and Applied Logic
Most of our results are also true for highly recursive graphs (graphs where one can determine the neighbors of a given node recursively), though there are some interesting differences when queries to K  ...  A recursive graph is a graph whose edge set and vertex set are both recursive.  ...  In this section we show that if f(n) queries suffice to solve the unbounded search problem, then f(n) queries can be used to find the chromatic number of a recursive graph; and conversely that if d(n)  ...

### On energy optimized averaging in wireless sensor networks

Ioannis Ch. Paschalidis, Binbin Li
2009 Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference
This problem has been shown to be NP-complete and can only be solved for small instances. We devise a semi-definite relaxation and establish bounds on the optimal cost.  ...  We also develop a series of graph-based algorithms that yield energy efficient bidirectional spanning trees and establish associated bounds on the optimal cost.  ...  • Recursive SDP from MILP • Relaxation • Sampling and Rounding , • The Framework of Recursive SDP Algorithm •Bounds Distributed Graph-Based Algorithms • Augmented Graph Construction •  ...

### On Finding Subpaths With High Demand [chapter]

Stephan Schwartz, Leonardo Balestrieri, Ralf Borndörfer
2018 Operations Research Proceedings
We introduce a path-graph to prevent multiple generations of the same subpath and give a recursive approach to compute the demands of all subpaths.  ...  We study the problem of finding subpaths with high demand in a given network that is traversed by several users.  ...  If the subpath-graph is constructed, our recursive algorithm to solve the SLCP runs in O(s).  ...

### Optimality Program in Segment and String Graphs

Édouard Bonnet, Paweł Rzążewski
2019 Algorithmica
best on general graphs.  ...  Among our results, we show that, quite surprisingly, 3-Coloring can also be solved in time 2 O(n 2/3 log O(1) n) on string graphs while an algorithm running in time 2 o(n) for 4-Coloring even on axisparallel  ...  Theorem 10 For every fixed k 4, the k-Coloring problem of a 2-Dir graph cannot be solved in time 2 o(n) , unless the ETH fails. Proof We modify the construction from the proof of Theorem 9.  ...

### An efficient parallel algorithm for shortest paths in planar layered digraphs

P. K. Mishra
2004 Journal of Zhejiang University: Science A
The algorithms uses a divide and conquer approach and is based on the novel idea of a one-way separator, which has the property that any directed path can be crossed only once.  ...  This paper presents an efficient parallel algorithm for the shortest path problem in planar layered digraphs that runs in O(log 3 n) time with n processors.  ...  source and exactly one sink recursively constructed by series and parallel compositions).  ...

### A comparative study on three graph-based constructive algorithms for multi-stage scheduling with blocking

Pengyu Yan, Shi Qiang Liu, Cheng-Hu Yang, Mahmoud Masoud
2017 Journal of Industrial and Management Optimization
Based on an alternative graph that is an extension of classical disjunctive graph, a new and generic polynomial-time algorithm is proposed to construct a feasible schedule with a given job processing sequence  ...  The comparison shows that the proposed algorithm has the following advantages: i) it is more adaptive because it can be applied to three different types of scheduling problems (i.e., permutation flow-shop  ...  The authors are very grateful to the editors and the anonymous reviewers for their constructive comments and suggestions that have led to an improved version of this paper.  ...

### Spanning Trees of Recursive Scale-Free Graphs [article]

C. Tyler Diggans, Erik M. Bollt, Daniel ben-Avraham
2022 arXiv   pre-print
We present a link-by-link rule-based method for constructing all members of the ensemble of spanning trees for any recursively generated, finitely articulated graph, such as the DGM net.  ...  .), and thus provides solutions to several optimization problems. The analysis of spanning trees enhances the usefulness of recursive graphs as sophisticated models for everyday life complex networks.  ...  While finding this number is, in general, an intractable problem, we hope that the recursive properties of these graphs might provide a convenient foothold for such a problem.  ...

### Solving SDD linear systems in nearlymlog1/2ntime

Michael B. Cohen, Rasmus Kyng, Gary L. Miller, Jakub W. Pachocki, Richard Peng, Anup B. Rao, Shen Chen Xu
2014 Proceedings of the 46th Annual ACM Symposium on Theory of Computing - STOC '14
Our approach follows the recursive preconditioning framework, which aims to reduce graphs to trees using iterative methods.  ...  Both of these components are used in a variety of other algorithms, and our approach also extends to the dual problem of computing electrical flows.  ...  Recursive preconditioning that reduces a solve to several ones on smaller, but similar graphs. 2. Ultra-sparsifiers that lead to such smaller graphs. 3 .  ...
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