By Avella P., Boccia M., Sforza A.

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Let the function G W NeC1 7! x0 ; : : : ; xe / we associate the vector . / D . x0 ; : : : ; xe / D i ˚ D max G . e i . //; G . 15) . / is given by the following lemma. 4. x0 ; x1 ; : : : ; xe / and . 15) above. Let j D minfi j ikD0 xk 3g. e i /: Proof. By definition we have e D G . /. Hence, for i D 0; 1; : : : ; e we also have e i . e i /g. e i / i D maxfG . e i G. //, or, equivalently e P G . e i . e i /, for all i j . Indeed, by hypothesis we have ikD0 xk 3, for all i j . 14) yields e D G. / G.

Suppose that yes D no D i C1;j ; hence, a fortiori, i Ä 2. Then we have the desired result by induction, since by hypothesis j 3I hence i 1 Ä 2j . 3j i 1/-state and a fortiori we have the desired result for i;j . Conversely, let yes D i 2;j 1 and no D i C1;j . 3j i 1/-states. 3j i 1/-state by induction. 3j i 1/-state. 3j i 1/-state. The proof is complete. i C1;j . 1 is a special case of the following more general construction. tj i /. In other words, any state in the table has a perfect winning strategy and is maximal in the previously defined terms.

0 / D ch. / 1. D satisfies the Volume Proof. x0 ; : : : ; xe /. a/. a/, by definition of character. For later purposes we record the following easy result. 2. A0 ; A1 ; : : : ; Ae /, with ej D0 Aj D fx; yg. Let i; j be such that x 2 Ai and y 2 Aj . i C j / C 1/-state. Proof. ”. i C j / C 1 times the question T . According to whether Carole gives e j C 1 positive answers or e i C 1 negative answers, Paul can safely conclude that the secret number is x or y respectively. For otherwise, Carole has lied more than she was allowed to.

### A Branch-and-Cut Algorithm for the Median-Path Problem by Avella P., Boccia M., Sforza A.

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