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2ima00 [2016/05/26 11:37]
bmpjansen [Student lectures]
2ima00 [2016/05/26 12:27] (current)
bmpjansen [Student lectures]
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 The literature linked to below can be accessed when logged into the TU/e network, either through VPN or by physically being present at TU/e. The literature linked to below can be accessed when logged into the TU/e network, either through VPN or by physically being present at TU/e.
  
-Kernels: +**Kernels: Chris & Leo.** 
-Chris & Leo. + 
-<href="http://​dx.doi.org/​10.1007/​s00224-009-9234-2">+The following paper gives kernel that is not as small, but is easier to compute, than the one given in the book. In particular, read Section 4 of this paper: 
 + 
 +[[http://​dx.doi.org/​10.1007/​s00224-009-9234-2|
 Hans L. Bodlaender, Thomas C. van Dijk: Hans L. Bodlaender, Thomas C. van Dijk:
-A Cubic Kernel for Feedback Vertex Set and Loop Cutset. Theory Comput. Syst. 46(3): 566-597 (2010)</a>+A Cubic Kernel for Feedback Vertex Set and Loop Cutset. Theory Comput. Syst. 46(3): 566-597 (2010)]] 
 + 
 +Instead of requiring an algorithm to find maximum matchings, it just needs an approximation algorithm for (weighted) feedback vertex set. It was developed independently by 2 sets of authors. Pick the one you find the easiest to read: 
 + 
 +Option 1: 
 + 
 +[[http://​dx.doi.org/​10.1137/​S0895480196305124|Vineet Bafna, Piotr Berman, Toshihiro Fujito: 
 +A 2-Approximation Algorithm for the Undirected Feedback Vertex Set Problem. SIAM J. Discrete Math. 12(3): 289-297 (1999)]] 
 + 
 +Option 2: 
 + 
 +[[https://​dx.doi.org/​10.1016%2F0004-3702%2895%2900004-6|Becker,​ Ann; Geiger, Dan (1996), "​Optimization of Pearl'​s method of conditioning and greedy-like approximation algorithms for the vertex feedback set problem.",​ Artificial Intelligence 83 (1): 167–188]] 
 + 
 +**Treewidth:​ Henk & Xi.** 
 + 
 +There are 2 approaches for solving the problem when you have a tree decomposition. The first one has a running time of 3^k * poly(n), harder to understand, but may be easier to implement:​ 
 + 
 +[[https://​arxiv.org/​pdf/​1103.0534v1.pdf|Cygan et al.: Solving connectivity problems parameterized by treewidth in single exponential time]] 
 + 
 +The next approach has a worse factor f(k), but a better polynomial term (linear), and is conceptually simpler: 
 + 
 +[[http://​link.springer.com/​chapter/​10.1007%2F3-540-36379-3_25|Ton Kloks, C.M. Lee, Jiping Liu. New Algorithms for k-Face Cover, k-Feedback Vertex Set, and k-Disjoint Cycles on Plane and Planar Graphs]] (page 8 and further) 
 + 
 +I don't know which of the 2 approaches will be faster on the inputs we have; one has a worse polynomial term, the other has a worse factor f(k). I don't think it is feasible to implement both, given the time. Pick one of the two and go for it. To **find** a tree decomposition,​ you can use one of the heuristics described in this paper: 
 + 
 +[[http://​dx.doi.org/​10.1016/​j.ic.2009.03.008|Hans L. Bodlaender, Arie M. C. A. Koster: 
 +Treewidth computations I. Upper bounds. Inf. Comput. 208(3): 259-275 (2010)]] 
 + 
 +Within the framework of iterative compression and you have *some* feedback vertex set X in your graph G, you can also make a tree decomposition by making a tree decomposition for the forest G - X (which is easy) and then adding all vertices of X to all bags. Afterwards, you can try to make it better by removing vertices from X from bags when the tree decomposition remains valid after removing a vertex from bag. 
 + 
 +**Parameterized by solution size (iterative compression & randomized):​ Huib & Stefan.**
  
-Treewidth:​ +For the iterative compression algorithm, I think the book gives sufficient details to build an implementation fromThere is theoretically faster algorithmwhich is linked belowYou could read it for inspirationbut it's not necessary ​and I do not think it is feasible to implement it fully because it needs subroutine with matroid computations.
-Henk & Xi. +
-<href="​https://​arxiv.org/​pdf/​1103.0534v1.pdf">​Cygan et al.: Solving connectivity problems parameterized by treewidth in single +
-exponential time</​a>​ +
-<a href="​http://​link.springer.com/​chapter/​10.1007%2F3-540-36379-3_25">​Ton KloksC.M. Lee, Jiping Liu. New Algorithms ​for k-Face Cover, k-Feedback Vertex Set, and k-Disjoint Cycles on Plane and Planar Graphs (page 8 and further)</​a>+
  
-Parameterized by solution size (iterative compression & randomized)+[[http://​dx.doi.org/​10.1016/​j.ipl.2014.05.001|Tomasz Kociumaka, Marcin Pilipczuk
-Huib & Stefan.+Faster deterministic Feedback Vertex SetInf. Process. Lett. 114(10): 556-560 (2014)]]
  
-<a href="​http://​dx.doi.org/​10.1016/​j.ipl.2014.05.001">​Tomasz KociumakaMarcin Pilipczuk:​ +For the randomized algorithmI don't know of any additional literatureYou could search by yourself.
-Faster deterministic Feedback Vertex Set. Inf. ProcessLett114(10): 556-560 (2014)</​a>​+
2ima00.1464255424.txt.gz · Last modified: 2016/05/26 11:37 by bmpjansen