Logspace Algorithms for Computing Shortest and Longest Paths in Series-Parallel Graphs

Jakoby, Andreas; Tantau, Till

doi:10.1007/978-3-540-77050-3_18

Cited by 12 publications

(21 citation statements)

References 23 publications

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“…For (2) note that the problem for a given graph G, a node s of G and an integer n to decide whether the longest path in G starting in s has the length n is NL-complete [13]. guess nondeterministically an integer n < i 4: if lp M (w) = n then accept else reject 5: else reject Algorithm 1 decides FPL 0 -KMc with the resources of LOGCFL.…”

Section: Optimality Of the Bounds Of The Numbers Of Variablesmentioning

confidence: 99%

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Intuitionistic implication makes model checking hard

Mundhenk¹,

Weiss²

2012

Logical Methods in Computer Science

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We investigate the complexity of the model checking problem for intuitionistic and modal propositional logics over transitive Kripke models. More specific, we consider intuitionistic logic IPC, basic propositional logic BPL, formal propositional logic FPL, and Jankov's logic KC. We show that the model checking problem is P-complete for the implicational fragments of all these intuitionistic logics. For BPL and FPL we reach P-hardness even on the implicational fragment with only one variable. The same hardness results are obtained for the strictly implicational fragments of their modal companions. Moreover, we investigate whether formulas with less variables and additional connectives make model checking easier. Whereas for variable free formulas outside of the implicational fragment, FPL model checking is shown to be in LOGCFL, the problem remains P-complete for BPL.Comment: 29 pages, 10 figure

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Section: Optimality Of the Bounds Of The Numbers Of Variablesmentioning

confidence: 99%

“…For a PrL 0 instance ϕ, M, w one can compute lp M (w) with the resources of NL (see [13] It is not known whether AC 1 also is the lower bound of PrL 0 -KMc. But from Lemmas 2.2 and 3.8, the lower bound NL follows, even for the strictly implicational fragment.…”

Section: Lower Bounds For Modal Logicsmentioning

confidence: 99%

Intuitionistic implication makes model checking hard

Mundhenk¹,

Weiss²

2012

Logical Methods in Computer Science

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“…On the other hand, very little is known about classes of planar graphs that admit log-space algorithms. Jacoby et al show that various reachability and optimization questions for series-parallel graphs admit deterministic log-space algorithms [JLR06,JT07]. Series-parallel graphs are a very restricted subclass of planar directed acyclic graphs (DAGs).…”

Section: Introductionmentioning

confidence: 99%

A Log-Space Algorithm for Reachability in Planar Acyclic Digraphs with Few Sources

Stolee

Bourke

Vinodchandran

2010

2010 IEEE 25th Annual Conference on Computational Complexity

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Designing algorithms that use logarithmic space for graph reachability problems is fundamental to complexity theory. It is well known that for general directed graphs this problem is equivalent to the NL vs L problem. This paper focuses on the reachability problem over planar graphs where the complexity is unknown. Showing that the planar reachability problem is NLcomplete would show that nondeterministic log-space computations can be made unambiguous. On the other hand, very little is known about classes of planar graphs that admit log-space algorithms. We present a new 'source-based' structural decomposition method for planar DAGs. Based on this decomposition, we show that reachability for planar DAGs with m sources can be decided deterministically in O(m + log n) space. This leads to a log-space algorithm for reachability in planar DAGs with O(log n) sources. Our result drastically improves the class of planar graphs for which we know how to decide reachability in deterministic log-space. Specifically, the class extends from planar DAGs with at most two sources to at most O(log n) sources.

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“…A more direct proof is given by Kulkarni [Kul09]. Jacoby and Tantau [JT07] showed for series-parallel graphs that reachability is complete for L. They also showed that the problem to compute distances between vertices or longest paths are complete for L. Thierauf and Wagner [TW08] proved that the distance problem for planar graphs is in UL ∩ coUL. For general graphs and even undirected planar graphs, the longest path problem is complete for NP.…”

Section: Introductionmentioning

confidence: 99%

Reachability in K 3,3-Free Graphs and K 5-Free Graphs Is in Unambiguous Log-Space

Thierauf

Wagner

2009

Fundamentals of Computation Theory

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We show that the reachability problem for directed graphs that are either K 3,3 -free or K 5 -free is in unambiguous log-space, UL ∩ coUL. This significantly extends the result of Bourke, Tewari, and Vinodchandran that the reachability problem for directed planar graphs is in UL ∩ coUL.Our algorithm decomposes the graphs into biconnected and triconnected components. This gives a tree structure on these components. The non-planar components are replaced by planar components that maintain the reachabilty properties. For K 5 -free graphs we also need a decomposition into fourconnected components. A careful analysis finally gives a polynomial size planar graph which can be computed in log-space.We show the same upper bound for computing distances in K 3,3 -free and K 5 -free directed graphs and for computing longest paths in K 3,3 -free and K 5 -free directed acyclic graphs. IntroductionIn this paper we consider the reachability problem which is defined as follows. Given is a graph G and vertices s, t, is there a path connecting s with t. It is a widely studied problem in the complexity theory, especially in the space setting. It characterizes complexity classes, for example L or NL when considering edges as undirected or directed, respectively. It also has its importance in the graph theory.For undirected graphs, the reachability problem is L-complete [Rei05]. For general graphs, reachability is NL-complete. Bourke, Tewari and Vinodchandran [BTV07] proved that reachability on planar graphs is in UL ∩ coUL and is hard for L. They built on work of Reinhard and Allender [RA00] and Allender, Datta, and Roy [ADR05]. A more direct proof is given by Kulkarni [Kul09]. Jacoby and Tantau [JT07] showed for series-parallel graphs that reachability is complete for L. They also showed that the problem to compute distances between vertices or longest paths are complete for L. Thierauf and Wagner [TW08] proved that the distance problem for planar graphs is in UL ∩ coUL. For general graphs and even undirected * Supported by DFG grant TO 200/2-2.

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Logspace Algorithms for Computing Shortest and Longest Paths in Series-Parallel Graphs

Cited by 12 publications

References 23 publications

Intuitionistic implication makes model checking hard

Intuitionistic implication makes model checking hard

A Log-Space Algorithm for Reachability in Planar Acyclic Digraphs with Few Sources

Reachability in K 3,3-Free Graphs and K 5-Free Graphs Is in Unambiguous Log-Space

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