By Cécile Murat, Vangelis Th. Paschos

This accomplished survey calls for just some mathematical figuring out and information approximately complexity and approximation thought and covers essentially the most paradigmatic combinatorial difficulties on graphs, akin to the maximum-independent set, minimum-vertex overlaying, longest course, and minimal coloring.

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G) = 3, then α(G) ∗ E (G, S , M4) ˆ M4 E G, S, n/4 and: 1 1 − (1 − p)3 + (1 − p)3 4 ˆ M4) If in addition p = 1/2, then E(G, S ∗ , M4)/E(G, S, 1 4 11/32. 13 can be easily extended to the case where vertexprobabilities are distinct and S¯ = argmax{ vi ∈S pi : S independent set of G} is used as an a priori solution. Moreover, without loss of generality, one can suppose that S¯ is maximal. 21] ¯ v i ∈S 1. Recall once more that such an independent set cannot be computed in polynomial time. The Probabilistic Maximum Independent Set 57 Combining the above expressions, we obtain: ¯ M4 E G, S, ¯ v i ∈S ˆ M4 E G, S, pi pi 1 ∆(G) + 1 vi ∈V where the last inequality is the weighted version of Turàn’s theorem.

However, it is not taken into account that shots realized under strong cloud-covering are not operational. Consequently, in order to compute an exploitable an operational solution, it is essential to also model weather forecasting. This can be done by associating probability p i with vertex vi of the conflict graph; the higher the vertex-probability, the more operational the shot taken. In this way, we naturally obtain a model leading to a probabilistic combinatorial optimization problem. Such a model for the satellite shots planning problem allows, given an a priori MAX INDEPENDENT SET-solution, computation of the expected number of operational shots.

An }; c) D′ = {b1 , . . , bn }. Based upon the above, it is proved in [BEL 93] that, in the case of identical vertexprobabilities, if C is a small matrix with distinct values, then, setting q = 1 − p: – E(Kn , T ∗ , MS) = p(1 − q n−1 )d, if and only if conditions 2 and 3 are verified; – if conditions 1, 2 and 3 are not verified then: - E(Kn , T ∗ , MS) = p(1 − q n−1 )d′ , if and only if D ′ verifies conditions a) and b); - E(Kn , T ∗ , MS) = p(1 − q n−1 ) min{d + dn+2 − dn , d − dn−1 + dn+1 }, if and only if D ′ does not verify conditions a), b) and c) and either one of the following conditions (c1), (c2) is satisfied: (c1) D ∪ {dn+2 } \ {dn } satisfies condition b) or c) and d+dn+2 −dn d−dn−1 +dn+1 ; (c2) D∪{dn+1 }\{dn−1 } satisfies condition b) or c) and d + dn+2 − dn d − dn−1 + dn+1 .

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