Feedback Vertex Set

Feedback Vertex Set

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  Feedback Vertex Set AAGAProject University of Pierre et Marie Curie NADARAJAH Mag-Stellon
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  Contents 1 Introduction 1 2Strategies 2 2.1 Naive Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Fixed Parameter Tractable . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2.1 Definitions and notations . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Disjoint-FVS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3.1 Definitions et notations . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 2-Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4.1 Definitions and notations . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.5 Randomized Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.5.1 Definitions et notations . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.5.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3 Conclusion 13 4 References 14
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  1. Introduction This paperdeals with one of the first NP complete problems : FeedBack Vertex Set. NP complete category problems are hard to solve. In general, algorithms providing a solution to a problem of these are rather slow. That’s why, these solutions are unusable in practice (although on a reduced instance of a problem). However, it is possible to verify a solution to a NP complete problems in polynomial time. At present, we know that it is sufficient to find a polynomial solution to a NP-complete problem in order to solve them all. This is the subject of the biggest computer science prob- lem : P versus NP. In the rest of this paper, we will analyze different algorithms to solve this problem. FVS is the subject of much research (still ongoing), solutions provide an improvement in time complexity (they tend to Θ(polynomial)). What is FVS ? This is a problem related to cyclic graph. The goal is to find a mini- mum number of nodes "playing a role" in cycles of a graph. By removing these nodes of the graph is obtained an acyclic graph. More formally, FVS is the set of N nodes of a cyclic graph G whose deletion makes the graph G acyclic. This problem is a major problem in combinatorial optimization. It is frequently encoun- tered in operating systems, databases (management of deadlocks), creation of printed circuits (VLSI), compilers optimizing ...
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  2. Strategies 2.1 NaiveAlgorithm A naive algorithm is often an intuitive solution. Here, it would be to test all combinations C of vertices of graph G. If G − C makes G acyclic then C is a FV S of G. This approach is set aside in view of its time complexity elevated. 2.2 Fixed Parameter Tractable 2.2.1 Definitions and notations u a b c d e f Figure 2.1: u is an u-flower of order 2 : u has two disjoints cycles but u in common. This cycles are petals and u is a "flower with two petals".
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  2.2 Fixed ParameterTractable 3 ts a b c d e f Figure 2.2: u-flower computation : We split u into two nodes t and s. The subgraph s contains all neighbors nodes v of s such that (vs) is a directed "incoming" arc (there is a oriented path from v to s). The subgraph t contains all neighbors nodes with "outgoing" arc. The maximal number of flower is the number of disjoint path (st), called petal(u). 2.2.2 Algorithm A Fixed Parameter Tractable algorithm (FPT) solves a problem in Θ(f(k)polynomial(n)). Therefore, It does not only depend on the input size of the problem. Thus, for relatively small values of k, we hope to obtain a solution in Θ(polynomial(n)). FPT Algorithm ----- Input : Graph G = (V,E) Output : k-FVS of G Choose a (k+1) nodes subgraph H of G Choose a (k) nodes subgraph I of H while H != G Add a node of G into H and I Compute FVS -Reduction on I if FVS -Reduction fails return G does not have k-FVS return I To improve the quality of this algorithm, we can define heuristics in order to get better subgraph H and I. Let’s give some heuristic ideas : • Select by greedy method, maximum degree nodes which make up G acyclic. • We compute an -approximation (see 2.4) of G. Then, we use previous heuristic on -approximation nodes.
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  2.2 Fixed ParameterTractable 4 FV S − Reduction is a set of rules to apply on a graph. This algorithm performs FV S − Reduction on I (I is a (k + 1)FV S of H) to reduce it to k. Let I = ∅ and we going to give rules to carry out during FV S − Reduction : u Figure 2.3: Add u to I, decrease k and remove u and adjacent edges of u from the graph. u v u v Figure 2.4: Remove all edge of (uv) but one. u Graphe Graphe Figure 2.5: If u does not have any incoming edge or outgoing edge then remove u and its incident edges from the graph. u v merge Figure 2.6: If there is only single path between u and v then merge u and v. uGraphe a b c
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  2.2 Fixed ParameterTractable 5 Graphe a b c Figure 2.7: If petal(u) = 1, remove u and its incident edges from Graphe. Create an edge between u neighbors in Graphe and nodes of the cycle. u Graphe a b c d e f Figure 2.8: If petal(u) > |k|, add u to I, decrease k, remove u and its incident edges from Graphe. 2.2.3 Example A B C D E F Figure 2.9: We set FTP to k = 2 for this graph G.
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  2.2 Fixed ParameterTractable 6 A B C Figure 2.10: We choose a k + 1 = 3 nodes subgraph H of G following the first heuristic (see 2.2.2). A C Figure 2.11: We choose a k = 2 nodes subgraph I of H and we execute rules 4 of FV S − Reduction, we merge C and A (see 2.6). B A D E F Figure 2.12: The graph G is equals to H and FV S − Reduction has succeeded on I. Then, I is a 2 − FV S of G.
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  2.3 Disjoint-FVS 7 2.3Disjoint-FVS 2.3.1 Definitions et notations Let a graph G(V, E) and FV S F of G. If we can compute a k nodes FV S L of G such as L V − F then F and L are Disjoint-FV S. Node set of graph G FVS F k-FVS L 2.3.2 Algorithm This algorithm is based on nodes swap. Let’s explain how it works. We find two sets of nodes which are disjoint-FV S on G and we call them V 1 and V 2. First of all, we remove all degree-0 and degree-1 nodes. It’s a kind of clean up in order to compute FV S faster. Then, for all degree-2 v in V 1 nodes, we have to follow two rules. B A C Figure 2.13: Let B ∈ V 1 and (C, A) ∈ V 22 . B is a degree-2 and its neighbors A, C are into V 2. So, we include B into V 1 − FV S, we remove B from G and we decrease k.
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  2.3 Disjoint-FVS 8 B A C Figure2.14: Let (B, A) ∈ V 12 and C ∈ V 2. B is a degree-2 and only C is into V 2. So, we move B from V 1 to V 2. The complete algorithm is written below :
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  2.4 2-Approximation 9 2.42-Approximation 2.4.1 Definitions and notations Let the function w : V (G) → N denotes the weight of a node in the graph G. A graph G is clean if it contains no vertex of degree less than two. A cycle C is semidisjoint if ∀u ∈ C, d(u) = 2 with at most one exception. 2.4.2 Algorithm An approximation algorithm gives an approximate solution to a complex problem. The qual- ity of an approximation algorithm is given by the ratio r of the solution approached on the optimal solution. We will use a 2-approximation algorithm that applies only to weighted and undirected graphs. This algorithm will be based on the principle of local searches. This principle focuses on a subgraph H of G and attempts to resolve the problem on H. Thus, if we can make H acyclic that means that we can remove cycles on G. This process is repeated until G has no more cycles. The 2-approximation algorithm is bounded by Θ(min|E|log|V |, |V |2 ).
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  2.4 2-Approximation 10 2.4.3Example A B C D E F Figure 2.15: A graph G whose nodes are weighted by 1 A B C Figure 2.16: The CleanUp remove all degree-1 nodes. We move A, B, C from G to the Stack and F. At step 1 et 2 (pop of C and B), F − u is a FV S of G. A the third step (pop of A), F − u is not a FV S of G then F = A is a FV S of G. B C D E F Figure 2.17: The graph G after 2-approximation
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  2.5 Randomized Algorithm11 2.5 Randomized Algorithm 2.5.1 Definitions et notations A leaf is degree-1 node. A link point is a degree-2 node. A branch point is a node of degree more than three. A graph is rich if it contains only branch point and no node with edges on itself (self node) 2.5.2 Algorithm This algorithm is based on the fact that in a rich graph, if we randomly take a node s, there is a probability of at least 1/2 that s is a neighbor of a FV S node. Indeed, let a graph G(V, E) and a graph F (FV S of G), then G − F is a forest. Let, X = V − F. We have, |E(X)| < |X|. Every node of X is a branch point and 3|X| < w∈X deg(v) = |E(F, X)| + 2|E(X)|. We use this property with the algorithm below : We can notice that this algorithm achieves the FV S with the minimum of probability (1/4j ). But we can, however, increase artificially the probability of this algorithm.
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  2.5 Randomized Algorithm12 We have a FV S with better probability than Single Guess due to the product of proba- bilities. Nevertheless, the worries of the Repeat Guess algorithm is that each call to Single Guess is made independently. So we may have already calculating the Single Guess that one is trying to achieve. It was a small but real probability that the calculation of FV S ends in a very long time and that even with a very small FV S. We will change this algorithm to improve this weakness. A theorem on branchy graphs states that w∈X deg(v) 6x w∈X deg(v). So we will limit our iterations by Max given to arbitrarily set the number of iteration and c6w(F) where w(F) is the weight of FV S lowest found by the algorithm.
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  3. Conclusion In thispaper, I have presented some great algorithms for the FV S problem which are re- ally tricky. All of those algorithms give a solution with a great time complexity for the Feedback V ertex Set problem. As we can see on this paper On Feedback Vertex Set New Measure and New Structures, this problem is far from being closed and is still relevant for a good time. Figure 3.1: History of parameterized algorithms for FV S problem. Acknowledgment. Thank Binh Minh Bui Xuan (see 4) for the help provided through- out this project.
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  4. References • Binh-MinhBui-Xuan | Algorithms, Programs and Resolution Team, LIP6 | http://www- apr.lip6.fr/ buixuan/ • Irit Dinur, Samuel Safra, On the hardness of approximating vertex cover, 2005 Annals of Mathematics, Pages 439-485 from Volume 162 • Fedor V. Fomin, Serge Gaspers, Artem V. Pyatkin, Igor Razgon, On the Minimum Feedback Vertex Set Problem: Exact and Enumeration Algorithms, 2007 , Algorith- mica, Volume 52, Issue 2 , pp 293-307 • Igor Razgon , Computing Minimum Directed Feedback Vertex Set in O(1.9977n) , 2007 , WSPC • Jianer Chen, Yang Liu, Songjian Lu, Barry O’sullivan, Igor Razgon , A fixed-parameter algorithm for the directed feedback vertex set problem , Volume 55 Issue 5, October 2008 Article No. 21 , Journal of the ACM (JACM) • Jianer Chena, Fedor V. Fominb, Yang Liua, Songjian Lua, Yngve Villangerb , Improved algorithms for feedback vertex set problems , 2008 , Journal of Computer and System Sciences, Volume 74, Issue 7 • Ann Becker, Dan Geiger , Optimization of Pearl’s method of conditioning and greedy- like approximation algorithms for the vertex feedback set problem , 1996 , Artificial Intelligence Volume 83, Issue 1 • Bafna, V., Berman, P., and Fujito, T. (1995). Constant ratio approximations of the weighted feedback vertex set problem for undirected graphs. In Proceedings of the Sixth Annual Symposium on Algorithms and Computation (ISAAC95) • Bar-Yehuda, R., Geiger, D., Naor, J., Roth, R. (1994). Approximation algorithms for the feedback vertex set problems with applications to constraint satisfaction and Bayesian inference. In Proceedings of the 5th Annual ACM-Siam Symposium On Discrete Algorithms, pp. 344,354. • Fedor V. Fomin Serge Gaspers Artem V. Pyatkin Igor Razgon, On the minimum feed- back vertex set problem : Exact and enumeration algorithms. (2007).
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  15 • Rudolf Fleischer,Xi Wu, Liwei Yuan, Experimental Study of FPT Algorithms for the Directed Feedback Vertex Set Problem (2009). 17th Annual European Symposium, Copenhagen, Denmark, September 7-9, 2009. Proceedings • A. Becker, R. Bar-Yehuda and D. Geiger (2000) "Randomized Algorithms for the Loop Cutset Problem", Volume 12, pages 219-234 • Yixin Cao, Jianer Chen, Yang Liu, On Feedback Vertex Set: New Measure and New Structures, Algorithm Theory - SWAT 2010, 12th Scandinavian Symposium and Work- shops on Algorithm Theory, Bergen, Norway, June 21-23, 2010. Proceedings • Vineet Bafna, Piotr Berman, Toshihiro Fujito, A 2-Approximation Algorithm for the Undirected Feedback Vertex Set Problem, SIAM Journal on Discrete Mathematics, Volume 12 Issue 3, Sept. 1999, Pages 289-297