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Phylogenetic CSPs are Approximation Resistant

Vaggos ChatziafratisKonstantin Makarychev
Dec 2022
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) informationabout $n$ items of interest, and we present optimal hardness of approximationresults under the Unique Games Conjecture. The class of problems studied herecan be described as Constraint Satisfaction Problems (CSPs) over infinitedomains, where instead of values $\{0,1\}$ or a fixed-size domain, thevariables can be mapped to any of the $n$ leaves of a phylogenetic tree. Thetopology of the tree then determines whether a given constraint on thevariables is satisfied or not, and the resulting CSPs are called PhylogeneticCSPs. Prominent examples of Phylogenetic CSPs with a long history andapplications in various disciplines include: Triplet Reconstruction, QuartetReconstruction, Subtree Aggregation (Forbidden or Desired). For example, inTriplet 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, thatrespects the constraints as much as possible. Despite more than four decades ofresearch, the basic question of maximizing the number of satisfied constraintsis not well-understood. The current best approximation is achieved byoutputting a random tree (for triplets, this achieves a 1/3 approximation). Ourmain 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 inGuruswami, Hastad, Manokaran, Raghavendra, and Charikar (2011), who showed thatordering CSPs are approximation resistant (e.g., Max Acyclic Subgraph,Betweenness).