Reconciling gene trees with a species tree is a fundamental problem to understand the evolution of gene families. Many existing approaches reconcile each gene tree independently. However, it is well-known that the evolution of gene families is interconnected. In this paper, we extend a previous approach to reconcile a set of gene trees with a species tree based on segmental macro-evolutionary events, where segmental duplication events and losses are associated with cost $delta$ and $lambda$, respectively. We show that the problem is polynomial-time solvable when $delta leq lambda$ (via LCA-mapping), while if $delta > lambda$ the problem is NP-hard, even when $lambda = 0$ and a single gene tree is given, solving a long standing open problem on the complexity of the reconciliation problem. On the positive side, we give a fixed-parameter algorithm for the problem, where the parameters are $delta/lambda$ and the number $d$ of segmental duplications, of time complexity $O(lceil frac{delta}{lambda} rceil^{d} cdot n cdot frac{delta}{lambda})$. Finally, we demonstrate the usefulness of this algorithm on two previously studied real datasets: we first show that our method can be used to confirm or refute hypothetical segmental duplications on a set of 16 eukaryotes, then show how we can detect whole genome duplications in yeast genomes.
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