“…See [8,44,45,57,60,66,67,151,158,160,162] for results that were directly linked with search parameters. For results on graph searching on special graph classes, see [60,76,77,80,81,116,128,129,157,170].…”
Section: Parameters Complexity and Algorithmsmentioning
Graph searching encompasses a wide variety of combinatorial problems related to the problem of capturing a fugitive residing in a graph using the minimum number of searchers. In this annotated bibliography, we give an elementary classification of problems and results related to graph searching and provide a source of bibliographical references on this field.
“…See [8,44,45,57,60,66,67,151,158,160,162] for results that were directly linked with search parameters. For results on graph searching on special graph classes, see [60,76,77,80,81,116,128,129,157,170].…”
Section: Parameters Complexity and Algorithmsmentioning
Graph searching encompasses a wide variety of combinatorial problems related to the problem of capturing a fugitive residing in a graph using the minimum number of searchers. In this annotated bibliography, we give an elementary classification of problems and results related to graph searching and provide a source of bibliographical references on this field.
“…• Deciding whether, for a ÿxed constant k, a graph G has treewidth 6k, i.e., whether a tree decomposition of width 6k for G exists [26].…”
Section: Results On Applicationsmentioning
confidence: 99%
“…The respective polynomial algorithms usually require the computation of a tree decomposition of bounded width. The following proposition summarizes important results by Wanke [26]. Proposition 5.2 (Wanke [26]).…”
Section: Computing Bounded-width Tree Decompositionsmentioning
The complexity class LOGCFL consists of all languages (or decision problems) which are logspace reducible to a context-free language. Since LOGCFL is included in AC 1 , the problems in LOGCFL are highly parallelizable.By results of Ruzzo (JCSS 21 (1980) 218), the complexity class LOGCFL can be characterized as the class of languages accepted by alternating Turing machines (ATMs) which use logarithmic space and have polynomially sized accepting computation trees. We show that for each such ATM M recognizing a language A in LOGCFL, it is possible to construct an L LOGCFL transducer TM such that TM on input w ∈ A outputs an accepting tree for M on w. It follows that computing single LOGCFL certiÿcates is feasible in functional AC 1 and is thus highly parallelizable. Wanke (J. Algorithms 16 (1994) 470) has recently shown that for any ÿxed k, deciding whether the treewidth of a graph is at most k is in the complexity-class LOGCFL. As an application of our general result, we show that the task of computing a tree-decomposition for a graph of constant treewidth is in functional LOGCFL, and thus in AC 1 . We also show that the following tasks are all highly parallelizable: Computing a solution to an acyclic constraint satisfaction problem; computing an m-coloring for a graph of bounded treewidth; computing the chromatic number and minimal colorings for graphs of bounded treewidth.
“…Although these algorithms are fast, they are extremely wasteful in terms of processors, in view of the linear-time sequential algorithm. A related result was obtained by Wanke [41], who showed that the problem of deciding whether the treewidth of an input graph is bounded by a constant k belongs to the complexity dass LOGCFLj this result also does not seem to lead to parallel algorithms that are efficient from the point of view of processor utilization. If we relax the requirements by allowing tree decompositions of width O( k), rather than exactly k, more algorithms come into play: Lagergren [31] finds a decomposition of width ::::; 6k + 5 in O((1og n)3) time using n processors, and we believe that Reed's sequential O( n log n )-time algorithm [38] for obtaining a decomposition of width ::::; 4k + 3 can be parallelized (using an algorithm of Khuller and Schieber [29] to solve a central path-finding problem) to yield an algorithm that works in O( (log n )2) time using O( no:( n) flog n) processors, where 0: is a very slowly growing "inverse Ackermann" function.…”
We describe the first parallel algorithm with optimal speedup for constructing minimum-width tree decompositions of graphs of bounded treewidth. On n-vertex input graphs, the algorithm works in O((logn)2) time using O(n) operations on the EREW PRAM. We also give faster parallel algorithms with optimal speedup for the problem of deciding whether the treewidth of an input graph is bounded by a given constant and for a vanety of problems on graphs of bounded treewidth, including all decision problems expressible in monadic second-order logic. On n-vertex input graphs, the algorithms use O(n) operations together with O(lognlog*n) time on the EREW PRAM, or 0 (log n) time on the CRCW PRAM.
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