https://snap.stanford.edu/class/cs224w-readings/tomita06cliques.pdf

The worst-case time complexity for generating all maximal cliques and computational experiments

We present an exact partial enumerative algorithm for the maximum clique problem. The pruning device used is derived from graph colorings. Pruning of the search tree is accomplished not only by the number of colors used to color a tree subproblem but also by using information gained in the process of coloring. This leads to increased pruning which translates into improved computational performance. Experimental results on test problems are presented.

An exact algorithm for the maximum clique problem with accelerated pruning

https://hal.univ-angers.fr/hal-02709508/document

A review on algorithms for maximum clique problems

We present an exact and efficient branch-and-bound algorithm for finding a maximum clique in an arbitrary graph. The algorithm is not specialized for any particular kind of graph. It employs approximate coloring and appropriate sorting of vertices to get an upper bound on the size of a maximum clique. We demonstrate by computational experiments on random graphs with up to 15,000 vertices and on DIMACS benchmark graphs that our algorithm remarkably outperforms other existing algorithms in general. It has been successfully applied to interesting problems in bioinformatics, image processing, the design of quantum circuits, and the design of DNA and RNA sequences for biomolecular computation.

An efficient branch-and-bound algorithm for finding a maximum clique

Motivation: Side-chain positioning is a central component of homology modeling and protein design. In a common formulation of the problem, the backbone is fixed, side-chain conformations come from a rotamer library, and a pairwise energy function is optimized. It is NP-complete to find even a reasonable approximate solution to this problem. We seek to put this hardness result into practical context.

Results: We present an integer linear programming (ILP) formulation of side-chain positioning that allows us to tackle large problem sizes. We relax the integrality constraint to give a polynomial-time linear programming (LP) heuristic. We apply LP to position side chains on native and homologous backbones and to choose side chains for protein design. Surprisingly, when positioning side chains on native and homologous backbones, optimal solutions using a simple, biologically relevant energy function can usually be found using LP. On the other hand, the design problem often cannot be solved using LP directly; however, optimal solutions for large instances can still be found using the computationally more expensive ILP procedure. While different energy functions also affect the difficulty of the problem, the LP/ILP approach is able to find optimal solutions. Our analysis is the first large-scale demonstration that LP-based approaches are highly effective in finding optimal (and successive near-optimal) solutions for the side-chain positioning problem.

Availability: The source code for generating the ILP given a file of pairwise energies between rotamers is available online at http://compbio.cs.princeton.edu/scplp

Contact: msingh@cs.princeton.edu

http://www.math.wsu.edu/math/faculty/bkrishna/FilesMath574/Papers/SideChainPosIP.pdf

Solving and analyzing side-chain positioning problems using linear and integer programming

Efficient branch-and-bound algorithms for finding a maximum clique

We developed maximum clique-based algorithms for spot matching for two-dimensional gel electrophoresis images, protein structure alignment and protein side-chain packing, where these problems are known to be NP-hard. Algorithms based on direct reductions to the maximum clique can find optimal solutions for instances of size (the number of points or residues) up to 50-150 using a standard PC. We also developed pre-processing techniques to reduce the sizes of graphs. Combined with some heuristics, many realistic instances can be solved approximately.

Point matching under non-uniform distortions and protein side chain packing based on an efficient maximum clique algorithm.

Point matching under non - uniform distortions and protein side chain packing based on efficient maximum clique algorithms

Protein side-chain packing has an important application in homology modeling, protein structure prediction, protein design, protein docking problems and many more.Protein side-chain packing problem is computationally known to be NP-hard (Akutsu, 1997) (Chazelle, Kingsford & Singh, 2003) (Pierce & Winfree, 2002). In the field of computer science, the notion of reduction of a problem to other problems is quite often used to design algorithms and to prove the complexity of a certain problem. In this work, we have used this notion of reduction to solve protein side-chain packing problem.We have developed a deterministic algorithm based approach to solve protein side-chain packing problem based on clique-based algorithms. For this, we reduced this problem to the maximum clique finding problem. Moreover, in order to incorporate the interaction preferences between the atoms, we have then extended this approach to maximum edge-weight clique finding problem by assigning weights based on probability discriminatory function. We have then solved this clique finding problem by using the clique finding algorithm developed by two of the authors (Tomita & Seki, 2003) and its variants (Suzuki, Tomita & Seki 2002).We have tested this approach to predict the side-chain conformations of a set of proteins and have compared the results with other existing methods. We have found considerable improvement in terms of the size of the proteins and in terms of the efficiency and accuracy of the prediction.

Protein side-chain packing problem: a maximum edge-weight clique algorithmic approach

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