Algorithmic Decision Theory: Second International by Dorothea Baumeister, Gábor Erdélyi, Jörg Rothe (auth.),

By Dorothea Baumeister, Gábor Erdélyi, Jörg Rothe (auth.), Ronen I. Brafman, Fred S. Roberts, Alexis Tsoukià s (eds.)

This ebook constitutes the refereed lawsuits of the second one foreign convention on Algorithmic determination idea, ADT 2011, held in Piscataway, NJ, united states, in October 2011. The 24 revised complete papers awarded have been conscientiously reviewed and chosen from 50 submissions.

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Additional resources for Algorithmic Decision Theory: Second International Conference, ADT 2011, Piscataway, NJ, USA, October 26-28, 2011. Proceedings

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N. Therefore (|S1 |, . . , |SC |) (|S1 |, . . , |SC |) by definition of . Conversely, we now show that (|S1 |, . . , |SC |) (|S1 |, . . , |SC |) ⇒ S S. Assume that |Sc | ≥ |Sc | for all c. Since |S1 | ≥ |S1 |, there exists an injection π1 : S1 → S1 . Obviously, ∀i ∈ S1 , γ π1 (i) γ i . For any c > 1, one can then define by mutual recursion: – an injection πc : Sc \Sc−1 → Sc \πc−1 (Sc−1 ) – an injection πc : Sc → Sc by πc (i) = πc−1 (i) if i ∈ Sc−1 and πc (i) = πc (i) otherwise. Injection πc exists for any c > 1 because |Sc \πc−1 (Sc−1 )| ≥ |Sc \Sc−1 |.

We have indeed |Sc \πc−1 (Sc−1 )| = |Sc | − |πc−1 (Sc−1 )| since πc−1 (Sc−1 ) ⊆ Sc , |Sc | − |πc−1 (Sc−1 )| = |Sc | − |Sc−1 | since πc−1 is an injection, |Sc | − |Sc−1 | ≥ |Sc \Sc−1 | since |Sc | ≥ |Sc |. Note that by construction, for any c, ∀i ∈ Sc , γ πc (i) γ i . For c = C this is precisely the definition, therefore S S . Coming back again to the example of the introduction, cumulative vector (1, 2, 2) is associated to {1, 3}, and (0, 2, 2) to {2, 3}. Note then, that (1, 2, 2) (0, 2, 2), consistently with {1, 3} {2, 3}.

We then propose two polynomial algorithms to solve it, one based on a multiobjective dynamic programming scheme and the other on a multiobjective branch and bound procedure. We conclude by providing numerical tests to compare both approaches. Keywords: Committee selection, Ordinal combinatorial optimization, Multiobjective combinatorial optimization, Knapsack with cardinality constraint, Polynomial algorithms. 1 Introduction Ranking sets of objects based on a ranking relation on objects has been extensively studied in social choice theory within an axiomatic approach [1].

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