We present a polynomial time approximation scheme (PTAS) for the minimum feedback arc set problem on tournaments. A simple weighted generalization gives a PTAS for Kemeny-Young rank aggregation.

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How to rank with few errors

We present a randomized subexponential time, polynomial space parameterized algorithm for the k -Weighted Feedback Arc Set in Tournaments (k -FAST ) problem. We also show that our algorithm can be derandomized by slightly increasing the running time. To derandomize our algorithm we construct a new kind of universal hash functions, that we coin universal coloring families . For integers m ,k and r , a family ${\mathcal F}$ of functions from [m ] to [r ] is called a universal (m ,k ,r )-coloring family if for any graph G on the set of vertices [m ] with at most k edges, there exists an $f \in {\mathcal F}$ which is a proper vertex coloring of G . Our algorithm is the first non-trivial subexponential time parameterized algorithm outside the framework of bidimensionality.

Fast FAST

i Acknowledgements First of all, I would like to thank my supervisor Pinar Heggernes for her help and guidance, both with my reasearch and with all other aspects of being a graduate student. This thesis would not have existed without her. Thomassen and Yngve Villanger. A special thanks goes to Saket Saurabh, with whom I have shared many sleepless nights and cups of coffee preparing this work. Thanks to my family, especially to my mother and father. In more than one way, i am here largely because of you. I would also like to thank my friends for filling my life with things other than discrete mathematics. You mean a lot to me, hugs to all of you and kisses to one in particular. Thanks also to the Algorithms group at the University of Bergen, you are a great lot. An extra thanks goes to my table-tennis opponents, even though this thesis exists despite of your efforts to keep me away from it. Finally, thanks to Pasha for being furry and having whiskers.

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New Methods in Parameterized Algorithms and Complexity

Hadamard matrices exist for infinitely many orders 4m, m ≧ 1, m integer, including all 4m < 100, cf. [3], [2]. They are conjectured to exist for all such orders. Skew Hadamard matrices have been constructed for all orders 4m < 100 except for 36, 52, 76, 92, cf. the table in [4]. Recently Szekeres [6] found skew Hadamard matrices of the order 2(pt +1)≡ 12 (mod 16), p prime, thus covering the case 76. In addition, Blatt and Szekeres [1] constructed one of order 52. The present note contains a skew Hadamard matrix of order 36 (and one of order 52), thus leaving 92 as the smallest open case.

E.T. Parker

Papers

Disproof of a conjecture of Erdös and moser on tournaments