The longest path problem is polynomial on interval graphs.

The longest path problem is the problem of finding a path of maximum length in a graph. Polynomial solutions for this problem are known only for small classes of graphs, while it is NP-hard on general graphs, as it is a generalization of the Hamiltonian path problem. Motivated by the work of Uehara and Uno in [20], where they left the longest path problem open for the class of interval graphs, in this paper we show that the problem can be solved in polynomial time on interval graphs. The proposed algorithm runs in O(n 4) time, where n is the number of vertices of the input graph, and bases on a dynamic programming approach

Computational methods for finding long simple cycles in complex networks

© 2017 Elsevier B.V. Detection of long simple cycles in real-world complex networks finds many applications in layout algorithms, information flow modelling, as well as in bioinformatics. In this paper, we propose two computational methods for finding long cycles in real-world networks. The first method is an exact approach based on our own integer linear programming formulation of the problem and a data mining pipeline. This pipeline ensures that the problem is solved as a sequence of integer linear programs. The second method is a multi-start local search heuristic, which combines an initial construction of a long cycle using depth-first search with four different perturbation operators. Our experimental results are presented for social network samples, graphs studied in the network science field, graphs from DIMACS series, and protein-protein interaction networks. These results show that our formulation leads to a significantly more efficient exact approach to solve the problem than a previous formulation. For 14 out of 22 networks, we have found the optimal solutions. The potential of heuristics in this problem is also demonstrated, especially in the context of large-scale problem instances

Contact replacement for NMR resonance assignment

Motivation: Complementing its traditional role in structural studies of proteins, nuclear magnetic resonance (NMR) spectroscopy is playing an increasingly important role in functional studies. NMR dynamics experiments characterize motions involved in target recognition, ligand binding, etc., while NMR chemical shift perturbation experiments identify and localize protein–protein and protein–ligand interactions. The key bottleneck in these studies is to determine the backbone resonance assignment, which allows spectral peaks to be mapped to specific atoms. This article develops a novel approach to address that bottleneck, exploiting an available X-ray structure or homology model to assign the entire backbone from a set of relatively fast and cheap NMR experiments

A Streaming Algorithm for the Undirected Longest Path Problem

We present the first streaming algorithm for the longest path problem in undirected graphs. The input graph is given as a stream of edges and RAM is limited to only a linear number of edges at a time (linear in the number of vertices n). We prove a per-edge processing time of O(n), where a naive solution would have required Omega(n^2). Moreover, we give a concrete linear upper bound on the number of bits of RAM that are required. On a set of graphs with various structure, we experimentally compare our algorithm with three leading RAM algorithms: Warnsdorf (1823), Pohl-Warnsdorf (1967), and Pongrasz (2012). Although conducting only a small constant number of passes over the input, our algorithm delivers competitive results: with the exception of preferential attachment graphs, we deliver at least 71% of the solution of the best RAM algorithm. The same minimum relative performance of 71% is observed over all graph classes after removing the 10% worst cases. This comparison has strong meaning, since for each instance class there is one algorithm that on average delivers at least 84% of a Hamilton path. In some cases we deliver even better results than any of the RAM algorithms

On the Performance of a Simple Approximation Algorithm for the Longest Path Problem

The longest path problem is known to be NP-hard. Moreover, they cannot be approximated within a constant ratio, unless ${\rm P=NP}$. The best known polynomial time approximation algorithms for this problem essentially find a path of length that is the logarithm of the optimum.In this paper we investigate the performance of an approximation algorithm for this problem in almost every case. We show that a simple algorithm, based on depth-first search, finds on almost every undirected graph $G=(V,E)$ a path of length more than $|V|-3\sqrt{|V| \log |V|}$ and so has performance ratio less than $1+4\sqrt{\log |V|/|V|}$.

Circumference of 3-connected claw-free graphs and large Eulerian subgraphs of 3-edge-connected graphs

AbstractThe circumference of a graph is the length of its longest cycles. Results of Jackson, and Jackson and Wormald, imply that the circumference of a 3-connected cubic n-vertex graph is Ω(n0.694), and the circumference of a 3-connected claw-free graph is Ω(n0.121). We generalize and improve the first result by showing that every 3-edge-connected graph with m edges has an Eulerian subgraph with Ω(m0.753) edges. We use this result together with the Ryjáček closure operation to improve the lower bound on the circumference of a 3-connected claw-free graph to Ω(n0.753). Our proofs imply polynomial time algorithms for finding large Eulerian subgraphs of 3-edge-connected graphs and long cycles in 3-connected claw-free graphs

Triangles, Long Paths, and Covered Sets

In chapter 2, we consider a generalization of the well-known Maker-Breaker triangle game for uniform hypergraphs in which Maker tries to build a triangle by choosing one edge in each round and Breaker tries to prevent her from doing so by choosing q edges in each round. The main result is the analysis of a new Breaker strategy using potential functions, introduced by Glazik and Srivastav (2019). Both bounds are of the order Θ(n3/2) so they are asymptotically optimal. The constant for the lower bound is 2-o(1) and for the upper bound it is 3√2. In chapter 3, we describe another Maker-Breaker game, namely the P3-game in which Maker tries to build a path of length 3. First, we show that the methods of chapter 2 are not applicable in this scenario and give an intuition why that might be the case. Then, we give a more simple counting argument to bound the threshold bias. In chapter 4, we consider the longest path problem which is a classic NP-hard problem that arises in many contexts. Our motivation to investigate this problem in a big-data context was the problem of genome-assembly, where a long path in a graph that is constructed of the reads of a genome potentially represents a long contiguous sequence of the genome. We give a semi-streaming algorithm. Our algorithm delivers results competitive to algorithms that do not have a restriction on the amount of memory. In chapter 5, we investigate the b-SetMultiCover problem, a classic combinatorial problem which generalizes the set cover problem. Using an LP-relaxation and analysis with the bounded differences inequality of C. McDiarmid (1989), we show that there is a strong concentration around the expectation

Graph algorithms for NMR resonance assignment and cross-link experiment planning

The study of three-dimensional protein structures produces insights into protein function at the molecular level. Graphs provide a natural representation of protein structures and associated experimental data, and enable the development of graph algorithms to analyze the structures and data. This thesis develops such graph representations and algorithms for two novel applications: structure-based NMR resonance assignment and disulfide cross-link experiment planning for protein fold determination. The first application seeks to identify correspondences between spectral peaks in NMR data and backbone atoms in a structure (from x-ray crystallography or homology modeling), by computing correspondences between a contact graph representing the structure and an analogous but very noisy and ambiguous graph representing the data. The assignment then supports further NMR studies of protein dynamics and protein-ligand interactions. A hierarchical grow-and-match algorithm was developed for smaller assignment problems, ensuring completeness of assignment, while a random graph approach was developed for larger problems, provably determining unique matches in polynomial time with high probability. Test results show that our algorithms are robust to typical levels of structural variation, noise, and missings, and achieve very good overall assignment accuracy. The second application aims to rapidly determine the overall organization of secondary structure elements of a target protein by probing it with a set of planned disulfide cross-links. A set of informative pairs of secondary structure elements is selected from graphs representing topologies of predicted structure models. For each pair in this ``fingerprint\u27\u27, a set of informative disulfide probes is selected from graphs representing residue proximity in the models. Information-theoretic planning algorithms were developed to maximize information gain while minimizing experimental complexity, and Bayes error plan assessment frameworks were developed to characterize the probability of making correct decisions given experimental data. Evaluation of the approach on a number of structure prediction case studies shows that the optimized plans have low risk of error while testing only a very small portion of the quadratic number of possible cross-link candidates