# Fast FAST By Noga Alon, Daniel Lokshtanov And Saket Saurabh Presentation by Gil Einziger

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- Slide 1
- Fast FAST By Noga Alon, Daniel Lokshtanov And Saket Saurabh Presentation by Gil Einziger
- Slide 2
- Fast FAST? Fast A relative fast algorithm for an NP- Complete problem. FAST (minimal) Feedback Arc Set in Tournaments.
- Slide 3
- Feedback Arc Set Given a Directed Graph We want to find a set of arcs such as: is a DAG. We want F to be minimal, how?
- Slide 4
- Feedback Arc Set The Problem: 1.NP Complete but we can expect it. 2.In a general Directed graph, un-weighted Feedback arc set is APX- Hard meaning that there exist a constant k such as there is no polynomial time k approximation algorithm for this problem.
- Slide 5
- What is A Tournament? A tournament is a directed graph (digraph) obtained by assigning a direction for each edge in an undirected complete graph.directed graph undirectedcomplete graph A tournament is, a directed graph in which every pair of vertices is connected by a single directed edge.
- Slide 6
- Tournament Important Observations Let T(V,A) be a Tournament 1. for is a Tournament 2. if a Tournament is DAG, it have a unique topological order. 1-2 3 Not a tournament, 2 possible topological orders 1 3 Tournament, only one topological order 2 1-2
- Slide 7
- Tournament and FAS Assume we have n tennis players. each tennis player is playing 1 game against all other tennis player. How can we decide who the best tennis player is? How can we rank the players?
- Slide 8
- Tournament and FAS 1 If the results of the tournament are acyclic, we can use topological ordering to determine both the winner and the full rank of the players. No player have any reason to complain since all the players I won, are always ranked lower then me.
- Slide 9
- Tournament and FAS 2 If the results arent acyclic, we cant satisfy ALL the players. So we want a solution satisfying as many players as possible. Given a minimal feedback arc set, we have such solution. Why?
- Slide 10
- K-Weighted Feedback Arc Set On Tournaments Given a tournament T=(V,A). A weight function And an Integer k. Question: Is there an arc set such that and T=(V,A\S) is a DAG.
- Slide 11
- K-FAST NP-Complete FPT (the parameter will be k.) Article improves a previous result in this problem from: to: Interested?
- Slide 12
- Preliminaries: w* For an arc weighted tournament we define the weight function w*
- Slide 13
- Preliminaries: D{F} Let D=(V,A) a directed graph. And a set F of arcs in A. We define D{F} to be a directed graph obtained from D by reversing all arcs of F.
- Slide 14
- D{F} And FAS 1 Claim: let T=(V,A) be a tournament. F is a minimal FAS of T=(V,A) if and only if F is a minimal set of arcs such that T{F} is a DAG. In other words, you dont have to remove FAS arcs in a minimal solution, you can simply REVERSE them.
- Slide 15
- D{F} And FAS 2 Explanation/Prove: Given a minimal feedback arc set F of a tournament T, the ordering corresponding to F is the unique topological ordering of T{F}
- Slide 16
- D{F} And FAS 3 Conversely, given an ordering of the vertices of T, the feedback arc set F corresponding to is the set of arcs whose endpoint appears before their start point in
- Slide 17
- D{F} And FAS conclusion We showed that every vertex ordering define a FAS and that every FAS define a vertex ordering. The cost of an arc set F is: And the cost of a vertex ordering is the cost of the corresponding FAS.
- Slide 18
- The Algorithm 1 Perform a data reduction to obtain a tournament T of size 2. Let. Color the vertices of T uniformly at random with colors from {1,,t} 3.Let be the set of arcs whose endpoints have different color, find a minimum FAS contained in, or conclude that no such FAS exist

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