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Graph Theory

Finding k-edge-outerplanar graph embeddings ★★

Author(s): Bentz

Conjecture   It has been shown that a $ k $-outerplanar embedding for which $ k $ is minimal can be found in polynomial time. Does a similar result hold for $ k $-edge-outerplanar graphs?

Keywords: planar graph; polynomial algorithm

Posted by jcmeyer
updated April 18th, 2010
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Graph Theory

Approximation ratio for k-outerplanar graphs ★★

Author(s): Bentz

Conjecture   Is the approximation ratio for the Maximum Edge Disjoint Paths (MaxEDP) or the Maximum Integer Multiflow problem (MaxIMF) bounded by a constant in $ k $-outerplanar graphs or tree-width graphs?

Keywords: approximation algorithms; planar graph; polynomial algorithm

Posted by jcmeyer
updated April 18th, 2010
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Graph Theory

Approximation Ratio for Maximum Edge Disjoint Paths problem ★★

Author(s): Bentz

Conjecture   Can the approximation ratio $ O(\sqrt{n}) $ be improved for the Maximum Edge Disjoint Paths problem (MaxEDP) in planar graphs or can an inapproximability result stronger than $ \mathcal{APX} $-hardness?

Keywords: approximation algorithms; Disjoint paths; planar graph; polynomial algorithm

Posted by jcmeyer
updated April 18th, 2010
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Theoretical Comp. Sci. » Algorithms

P vs. NP ★★★★

Author(s): Cook; Levin

Problem   Is P = NP?

Keywords: Complexity Class; Computational Complexity; Millenium Problems; NP; P; polynomial algorithm

Posted by zitterbewegung
updated October 18th, 2007
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Theoretical Comp. Sci. » Complexity

Subset-sums equality (pigeonhole version) ★★★

Author(s):

Problem   Let $ a_1,a_2,\ldots,a_n $ be natural numbers with $ \sum_{i=1}^n a_i < 2^n - 1 $. It follows from the pigeon-hole principle that there exist distinct subsets $ I,J \subseteq \{1,\ldots,n\} $ with $ \sum_{i \in I} a_i = \sum_{j \in J} a_j $. Is it possible to find such a pair $ I,J $ in polynomial time?

Keywords: polynomial algorithm; search problem

Posted by mdevos
updated July 16th, 2007
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Graph Theory » Graph Algorithms

The stubborn list partition problem ★★

Author(s): Cameron; Eschen; Hoang; Sritharan

Problem   Does there exist a polynomial time algorithm which takes as input a graph $ G $ and for every vertex $ v \in V(G) $ a subset $ \ell(v) $ of $ \{1,2,3,4\} $, and decides if there exists a partition of $ V(G) $ into $ \{A_1,A_2,A_3,A_4\} $ so that $ v \in A_i $ only if $ i \in \ell(v) $ and so that $ A_1,A_2 $ are independent, $ A_4 $ is a clique, and there are no edges between $ A_1 $ and $ A_3 $?

Keywords: list partition; polynomial algorithm

Posted by mdevos
updated May 18th, 2010
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