1. An FPT Algorithm for Set Splitting Authors: Frank Dehne, Michael R.Fellows, and Frances A.Rosamond Presented By:Saurav Mohanty Graduate Student (Computer Science) Texas A & M University

2. Presentation Structure Problem Statement Definitions & Theorems(lemma) Approach to solution Algorithm Conclusion

3. Problem Statement F = {Collection of subsets of a finite set X} The Task is to find a partition of X into two disjoint subsets X 0 and X 1 which maximizes the number of subsets of F that are split by the partition i,e. not entirely contained in either X 0 or X 1 This Problem is called MAX SET Splitting and is NP-Complete

4. Parameterized Version: Can we find a partition of set X into two disjoint subsets X 0 and X 1 such that it splits at least k subsets of F ? Input : Set X, set of subsets F and Parameter k Is the problem Fixed ParameterTractable? If yes what is its running time? The Set Splitting problem is in FPT and its running time is O(n 4 +2 O(k) n 2.5 ) parameterized by the number of sets to be split.

5. Definitions and Theorems A set k-splitting SSP(X,F,K) for a collection of n subsets of a finite set X is a partition [X 0,X 1 ] such that atleast k sets in F are split by the partition. Predicate π ssp (X,F,k)=True if yes otherwise false. A sequence W = (b 1,...,b k, w 1,..., w k ) Є X 2k is a k-witness structure for F iff (b1,..., bk) ∩(w1,..., wk) = Ø and there exist k subsets S i Є F such that {b i, w i } is a subset of S i for all i = 1,..., k.

6. Lemma1 (a) Every k-splitting SSP(X, F, k) = [X0,X1] implies at least one k witness structure (b1,..., bk, w1,..., wk). (b) Every k-witness structure (b1,...,bk, w1,..., wk) implies at least one k-splitting SSP(X, F, k) = [X0,X1]. For any W = (b1,..., bk, w1,..., wk), we refer to the process of replacing all occurrences of an element bi or wj by another element bi or wj, respectively, as deleting bi or wj. A k-witness structure W = (b1,..., bk, w1,..., wk) is nonredundant if any W obtained from W by deleting any single element bi or wj is not a k-witness structure. Algorithm 1 Kernelization: Convert the given problem instance (X, F, k) into an equivalent reduced problem instance. (1) Apply the following rules as often as possible. Rule 1: IF there exists an element a Є X with deg F (a)> k THEN report SSP(X, F, k) = [{a},X - {a}] and STOP. Rule 2: IF there exists a set S Є F with |S| ≤ 1 THEN set F = F -{S}. Rule 3: IF there exists a set S Є F with |S| ≥ 2k THEN set F = F -{S} and k = k - 1. Rule 4: IF there exists a set S Є F, |S| ≥ 2, which contains an element a Є S with degF (a) = 1 THEN set F = F - {S} and k = k - 1.

7. Rule 5: IF there exist three different elements a1, a2, a3 Є X with Ø С F(a1) С F(a2) С (a3) THEN set S = S -{a1} for all S Є F. (Note: May need to re-apply Rule 2.) (2) Set X = U s Є F S — End of Algorithm After the end of this algorithm we will get a problem kernel. Let (X,F,k) be any reduced problem instance.If F ≥ 2k then π ssp (X,F,k)=True The above statement is true as it follows from the Boundary Lemma Boundary Lemma If (X, F, k) is reduced and πssp(X,F,k)=TRUE and πssp(X,F,k+1)= FALSE then |F| ≤ 2k.

8. Set k-Splitting Algorithm Two Phases Kernelization As given by the previous algorithm Running Time=O(n 4 ) If lFl ≥ 2k then π ssp (X,F,k)=True Else(lFl<2k) Search The search proceeds on the problem kernel obtained from the first phase.All possible k subsets are checked to find the solution. The process involves partitioning and sub partitioning each k subsets and constructing a bipartite graph to visualize the matching. Running time =O(n 4 +2 O(k) n 2.5 ) The n 2.5 factor comes from the Bipartite matching.The 2 O(k) factor comes from the selection of all possible k subsets from F.

9. References An Approximation algorithm for hypergraph max k-cut with given sizes of parts An n5/2 algorithm for maximum matching in Bipartite graph