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By Sbihi A.

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Show that the problem admits a kernel with at most 2k sets and O(k2 ) universe size. 7. In the Minimum Maximal Matching problem, we are given an undirected graph G and a positive integer k, and the objective is to decide whether there exists a maximal matching in G on at most k edges. Obtain a polynomial kernel for the problem (parameterized by k). 8. In the Min-Ones-2-SAT problem, we are given a 2-CNF formula φ and an integer k, and the objective is to decide whether there exists a satisfying assignment for φ with at most k variables set to true.

Let F be an inclusion-wise minimal feedback arc set of G. Assume to the contrary that G F has a directed cycle C. Then C cannot contain only edges of E(G)\F , as that would contradict the fact that F is a feedback arc set. Let f1 , f2 , · · · , f be the edges of C ∩ rev(F ) in the order of their appearance on the cycle C, and let ei ∈ F be the edge fi reversed. Since F is inclusion-wise minimal, for every ei , there exists a directed cycle Ci in G such that F ∩ Ci = {ei }. Now consider the following closed walk W in G: we follow the cycle C, but whenever we are to traverse an edge fi ∈ rev(F ) (which is not present in G), we instead traverse the path Ci − ei .

For finding a crown decomposition in polynomial time, we use the following well known structural and algorithmic results. The first is a mini-max theorem due to Kőnig. 3 Crown decomposition 27 C H R Fig. 1: Example of a crown decomposition. 11 (Kőnig’s theorem, [303]). In every undirected bipartite graph the size of a maximum matching is equal to the size of a minimum vertex cover. Let us recall that a matching saturates a set of vertices S when every vertex in S is incident to an edge in the matching.

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A best first search exact algorithm for the Multiple-choice Multidimensional Knapsack Problem by Sbihi A.


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