Lagrangian Relaxation - Solving NP-hard Problems in Computational Biology via Combinatorial Optimization

This thesis is devoted to two NP-complete combinatorial optimization problems arising in computational biology, the well-studied multiple sequence alignment problem and the new formulated interval constraint coloring problem. It shows that advanced mathematical programming techniques are capable of solving large scale real-world instances from biology to optimality. Furthermore, it reveals alternative methods that provide approximate solutions. In the first part of the thesis, we present a Lagrangian relaxation approach for the multiple sequence alignment (MSA) problem. The multiple alignment is one common mathematical abstraction of the comparison of multiple biological sequences, like DNA, RNA, or protein sequences. If the weight of a multiple alignment is measured by the sum of the projected pairwise weights of all pairs of sequences in the alignment, then finding a multiple alignment of maximum weight is NP-complete if the number of sequences is not fixed. The majority of the available tools for aligning multiple sequences implement heuristic algorithms ; no current exact method is able to solve moderately large instances or instances involving sequences exhibiting a lower degree of similarity. We present a branch-and-bound (B&B) algorithm for the MSA problem. We approximate the optimal integer solution in the nodes of the B&B tree by a Lagrangian relaxation of an ILP formulation for MSA relative to an exponential large class of inequalities, that ensure that all pairwise alignments can be incorporated to a multiple alignment. By lifting these constraints prior to dualization the Lagrangian subproblem becomes an extended pairwise alignment (EPA) problem : Compute the longest path in an acyclic graph, that is penalized a charge for entering “obstacles”. We describe an efficient algorithm that solves the EPA problem repetitively to determine near-optimal Lagrangian multipliers via subgradient optimization. The reformulation of the dualized constraints with respect to additionally introduced variables improves the convergence rate dramatically. We account for the exponential number of dualized constraints by starting with an empty constraint pool in the first iteration to which we add cuts in each iteration, that are most violated by the convex combination of a small number of preceding Lagrangian solutions (including the current solution). In this relax-and-cut scheme, only inequalities from the constraint pool are dualized. i te l-0 03 88 52 1, v er si on 1 26 M ay 2 00 9