Intractability of the Minimum-Flip Supertree problem and its variants

The supertree method that is by far the most widely used today was independently proposed by Baum [3] and Ragan [26] in 1992; it is called Matrix Representation with Parsimony analysis (MRP) [4]. From a theoretical point of view, implementing MRP is designing an algorithm for an NP-hard optimization problem [14,18], so the running times of MRP algorithms are sometimes prohibitive for large data sets.

In 2002, Chen et al. proposed a variant of MRP [11], which was later called Matrix Representation using Flipping (MRF) [9]. MRF is arguably more relevant than MRP [4] (see also [12,16]), and various efficient implementations of MRF have been presented [10,13,16]. However, as in the case of MRP, implementing MRF is designing an algorithm for an NP-hard optimization problem [12], namely Minimum-Flip Supertree. The aim of the present paper is to study the approximability and the fixed-parameter tractability [17] of Minimum-Flip Supertree. We prove strongly negative results.

Let S be a finite set. A (rooted) phylogeny for S is a subset T of the power set of S that satisfies the following properties: ∅ ∈ T , S ∈ T , {s} ∈ T for all s ∈ S, and X ∩ Y ∈ {∅, X, Y } for all X, Y ∈ T . The elements of S are the leaves of T . The elements of T are the clusters of T . The most natural representation of T is, of course, a rooted graph-theoretic tree with |T | -1 nodes (the empty cluster does not correspond to any vertex).

Given two phylogenies T 1 and T 2 for S, T 1 is a subset of T 2 if, and only if, the graph representation of T 1 can be obtained from the graph representation of T 2 by contracting (internal) edges. If T 1 is a subset of T 2 and if we assume that hard polytomies never occur then T 2 is at least as informative as T 1 . Let M(G) denote the set of all quintuples (s, c, s

∈ E, and (c ′ , s) / ∈ E. The latter conditions state that the bipartite graph depicted in [25,Figure 4] is an induced subgraph of G. A perfect phylogeny for G is a phylogeny T for S such that N G (c) is a cluster of T for every c ∈ C. We say that G is M-free [5,[11][12][13]22] (or Σ-free) [4,9,25]) if the following three equivalent conditions are met:

For each e ∈ C × S, the magnitude of F (e) is the edit cost of e in H. An edition of H is a bipartite graph G of the form G = (C, S, E) for some subset E ⊆ C × S. A conflict between G and H is an element e ∈ C × S that satisfies one of the following two conditions:

1. e is an edge of G and e is a non-edge of H or 2. e is a non-edge of G and e is an edge of H.

The sum of the edit costs in H over all conflicts between G and H is denoted ∆(G, H):

The following minimization problem and its (parameterized) decision version generalize several previously studied problems:

Name: M-free Edition or Edit. Input: A bipartite draft-graph H and an integer k ≥ 0. Question: Is there an M-free edition G of H such that ∆(G, H) ≤ k? Parameter : k.

For each subset X ⊆ Z, define Min Edit-X as the restriction of Min Edit to those bipartite draft-graphs whose weight ranges are subsets of X, and similarly, define Edit-X as the restriction of Edit to those instances (H, k) such that the weight range of H is a subset of X. Notably, Min Edit-{-1, +1} is the Minimum-Flip Supertree problem and its restiction Min Edit-{-1, 0, +1} is the Minimum-Flip Consensus Tree problem [4, 5, 9-13, 16, 22].

Modelization. Incomplete and/or possibly erroneous character data sets are naturally modeled by bipartite draft-graphs: joker-edges represent incompletenesses and edit costs allow parsimonious error-corrections.

Supertrees. The most interesting feature of Min Edit is that it can be thought as a supertree construction problem, and more precisely, the optimization problem underlying MRF [4,9,11,12].

Min Edit-X has been studied for several subsets X ⊆ Z [4,5,9,10,12,13,16,[20][21][22]25], sometimes implicitely. Let H = (C, S, F ) be a bipartite draft-graph and let k be a nonnegative integer.

Put Z + = n ∈ Z : n ≥ 0 and Z -= n ∈ Z : n ≤ 0 . If H has no non-edge, or equivalently, if the weight range of H is a subset of Z + then the complete bipartite graph Edit-I, Edit-D, and Edit-U are NP-complete [12].

Put Z * = Z \ {0}. Min Edit-Z * is the restriction of Min Edit to those bipartite draftgraphs that have no joker-edge. The most positive r

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