# Probabilistic hardness of approximation or solution of NP-hard optimization problems under a probabilistic generative model for input data

So in biology (DNA sequences), sequence alignment is a generalization of longest common subsequence where an alignment of two sequences is scored typically with a linear function of how many spaces are inserted into each sequence and how many times each possible pair of aligned characters appears in the alignment. Just like longest common subsequence, finding the optimal alignment of two strings under an arbitrary linear scoring scheme can be solved in quadratic time using dynamic programming. (Needleman-Wunsch algorithm). The longest subsequence problem and variants that use linear scoring schemes and ask for the optimal multiple sequence alignment are NP-hard when the number of input strings is not fixed.

However, in biology, there is a probabilistic generative model that generates related DNA sequences. Starting with an unknown root ancestor DNA sequence, bifurcations occur that create two daughter sequences (species) that are independently derived from the ancestral sequence by potentially adding some characters in random locations, deleting some characters, and changing some characters. Then the bifurcations continue with additional changes at each level until the modern day DNA sequences of extant species are obtained. Then we want to align the modern day species' sequences (e.g. find the longest common subsequence in the simplest case) without knowing the exact ancestral sequences. In this case, fossil records can help identify the bifurcation events and estimate the sequence mutation rates after each bifurcation. So a reasonable estimate of the generative model that generated the related modern day DNA sequences can sometimes be obtained.

Now, my question is, for such an NP-hard optimization problem with a well-defined probabilistic generative model that generates input data, has anyone studied the hardness of finding either an optimal or nearly optimal solution, where either the worst-case or expected running time depends on the parameters for the model that generates the input data? For example, if DNA mutation and insertion/deletion rates are very low for a particular group of species, then it should be fairly easy to get at least a nearly optimal alignment of all the DNA sequences using partial alignments and pruning and heuristics, without resorting to a full-blown exponential time solution.

• I assume you are after average-case performance, then? – Raphael Aug 21 '14 at 5:56

Here is a similar recent example due to Mossel et al. There are $n$ vertices which are partitioned randomly into two classes. Two vertices of the same type are connected with probability $p$, and two of opposite types with probability $q$. For what values of $p, q$ can we recover the partition with high probability (with respect to $n$)? It turns out that if $p$ and $q$ are too close then it is statistically impossible, and otherwise a simple algorithm succeeds with high probability.

The question you are asking is very specific, and the answer likely depends on the exact model and its parameters. It seems likely that for a low mutation rate, some iterative algorithm should work, perhaps even provably (with high probability).

One way to formulate this question more concretely is to come up with constraints on the relations between the sequences, constraints that hold with high probability. It could be that for an appropriate choice of constraints, the problem becomes feasible.

• This is a nice example but I don't see a closely related NP-hard optimization problem. If you could formulate one for this, that would be great. However it's nice to see that at least someone is doing complexity analysis with respect to non-trivial parametrized probabilistic generative models for input data. I've always wanted to obtain a nice result for multiple DNA sequence alignment, basically showing a polynomial time approximation or likely optimal solution if the parameters in the generative model are bounded by fixed constants (that potentially depend on the number of sequences). – user2566092 Aug 21 '14 at 16:25
• Related NP-hard problems are MAX-CUT and its variant MAX-BISECTION. – Yuval Filmus Aug 21 '14 at 21:51