We study the approximability of a broad class of computational problems --
originally motivated in evolutionary biology and phylogenetic reconstruction --
concerning the aggregation of potentially inconsistent (local) information
about n items of interest, and we present optimal hardness of approximation
results under the Unique Games Conjecture. The class of problems studied here
can be described as Constraint Satisfaction Problems (CSPs) over infinite
domains, where instead of values {0,1} or a fixed-size domain, the
variables can be mapped to any of the n leaves of a phylogenetic tree. The
topology of the tree then determines whether a given constraint on the
variables is satisfied or not, and the resulting CSPs are called Phylogenetic
CSPs. Prominent examples of Phylogenetic CSPs with a long history and
applications in various disciplines include: Triplet Reconstruction, Quartet
Reconstruction, Subtree Aggregation (Forbidden or Desired). For example, in
Triplet Reconstruction, we are given m triplets of the form ij∣k
(indicating that ``items i,j are more similar to each other than to k'')
and we want to construct a hierarchical clustering on the n items, that
respects the constraints as much as possible. Despite more than four decades of
research, the basic question of maximizing the number of satisfied constraints
is not well-understood. The current best approximation is achieved by
outputting a random tree (for triplets, this achieves a 1/3 approximation). Our
main result is that every Phylogenetic CSP is approximation resistant, i.e.,
there is no polynomial-time algorithm that does asymptotically better than a
(biased) random assignment. This is a generalization of the results in
Guruswami, Hastad, Manokaran, Raghavendra, and Charikar (2011), who showed that
ordering CSPs are approximation resistant (e.g., Max Acyclic Subgraph,
Betweenness).Comment: 45 pages, 11 figures, Abstract shortened for arxi