a b s t r a c tWe are interested in a problem introduced by Vassilvitskii and Yannakakis (2005), the computation of a minimum set of solutions that approximates within an accuracy ε the Pareto set of a multi-objective optimization problem. We mainly establish a new 3-approximation algorithm for the bi-objective case. We also propose a study of the greedy algorithm performance for the tri-objective case when the points are given explicitly, answering an open question raised by Koltun and Papadimitriou in (2007).

In this paper, we study three connection games among the most widely played: havannah, twixt, and slither. We show that determining the outcome of an arbitrary input position is PSPACE-complete in all three cases. Our reductions are based on the popular graph problem generalized geography and on hex itself. We also consider the complexity of generalizations of hex parameterized by the length of the solution and establish that while short generalized hex is W[1]-hard, short hex is FPT. Finally, we prove that the ultra-weak solution to the empty starting position in hex cannot be fully adapted to any of these three games.

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