Inverting the Viterbi algorithm: an abstract framework for structure design

ABSTRACT

Probabilistic grammatical formalisms such as hidden Markov models (HMMs) and stochastic context-free grammars (SCFGs) have been extensively studied and widely applied in a number of fields. Here, we introduce a new algorithmic problem on HMMs and SCFGs that arises naturally from protein and RNA design, and which has not been previously studied. The problem can be viewed as an inverse to the one solved by the Viterbi algorithm on HMMs or by the CKY algorithm on SCFGs. We study this problem theoretically and obtain the first algorithmic results. We prove that the problem is NP-complete, even for a 3-letter emission alphabet, via a reduction from 3-SAT, a result that has implications for the hardness of RNA secondary structure design. We then develop a number of approaches for making the problem tractable. In particular, for HMMs we develop a branch-and-bound algorithm, which can be shown to have fixed-parameter tractable worst-case running time, exponential in the number of states of the HMM but linear in the length of the structure. We also show how to cast the problem as a Mixed Integer Linear Program.

REFERENCES

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INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.3 PROBABILITY AND STATISTICS
      Subjects: Markov processes

Additional Classification:
  F. Theory of Computation
  F.1 COMPUTATION BY ABSTRACT DEVICES
      F.1.3 Complexity Measures and Classes
          Subjects: Reducibility and completeness

  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.6 Optimization
          Subjects: Integer programming; Linear programming

  I. Computing Methodologies
  I.2 ARTIFICIAL INTELLIGENCE
      I.2.6 Learning
  I.6 SIMULATION AND MODELING
      I.6.5 Model Development
          Subjects: Modeling methodologies

General Terms:
Algorithms, Design, Theory