Bidimensionality and Geometric Graphs

In this paper we use several of the key ideas from Bidimensionality to give a new generic approach to design EPTASs and subexponential time parameterized algorithms for problems on classes of graphs which are not minor closed, but instead exhibit a geometric structure. In particular we present EPTASs and subexponential time parameterized algorithms for Feedback Vertex Set, Vertex Cover, Connected Vertex Cover, Diamond Hitting Set, on map graphs and unit disk graphs, and for Cycle Packing and Minimum-Vertex Feedback Edge Set on unit disk graphs. Our results are based on the recent decomposition theorems proved by Fomin et al [SODA 2011], and our algorithms work directly on the input graph. Thus it is not necessary to compute the geometric representations of the input graph. To the best of our knowledge, these results are previously unknown, with the exception of the EPTAS and a subexponential time parameterized algorithm on unit disk graphs for Vertex Cover, which were obtained by Marx [ESA 2005] and Alber and Fiala [J. Algorithms 2004], respectively. We proceed to show that our approach can not be extended in its full generality to more general classes of geometric graphs, such as intersection graphs of unit balls in R^d, d >= 3. Specifically we prove that Feedback Vertex Set on unit-ball graphs in R^3 neither admits PTASs unless P=NP, nor subexponential time algorithms unless the Exponential Time Hypothesis fails. Additionally, we show that the decomposition theorems which our approach is based on fail for disk graphs and that therefore any extension of our results to disk graphs would require new algorithmic ideas. On the other hand, we prove that our EPTASs and subexponential time algorithms for Vertex Cover and Connected Vertex Cover carry over both to disk graphs and to unit-ball graphs in R^d for every fixed d

Robust and Efficient Algorithms for Protein 3-D Structure Alignment and Genome Sequence Comparison

Sequence analysis and structure analysis are two of the fundamental areas of bioinformatics research. This dissertation discusses, specifically, protein structure related problems including protein structure alignment and query, and genome sequence related problems including haplotype reconstruction and genome rearrangement. It first presents an algorithm for pairwise protein structure alignment that is tested with structures from the Protein Data Bank (PDB). In many cases it outperforms two other well-known algorithms, DaliLite and CE. The preliminary algorithm is a graph-theory based approach, which uses the concept of \stars to reduce the complexity of clique-finding algorithms. The algorithm is then improved by introducing \double-center stars in the graph and applying a self-learning strategy. The updated algorithm is tested with a much larger set of protein structures and shown to be an improvement in accuracy, especially in cases of weak similarity. A protein structure query algorithm is designed to search for similar structures in the PDB, using the improved alignment algorithm. It is compared with SSM and shows better performance with lower maximum and average Q-score for missing proteins. An interesting problem dealing with the calculation of the diameter of a 3-D sequence of points arose and its connection to the sublinear time computation is discussed. The diameter calculation of a 3-D sequence is approximated by a series of sublinear time deterministic, zero-error and bounded-error randomized algorithms and we have obtained a series of separations about the power of sublinear time computations. This dissertation also discusses two genome sequence related problems. A probabilistic model is proposed for reconstructing haplotypes from SNP matrices with incomplete and inconsistent errors. The experiments with simulated data show both high accuracy and speed, conforming to the theoretically provable e ciency and accuracy of the algorithm. Finally, a genome rearrangement problem is studied. The concept of non-breaking similarity is introduced. Approximating the exemplar non-breaking similarity to factor n1..f is proven to be NP-hard. Interestingly, for several practical cases, several polynomial time algorithms are presented

A hybrid breakout local search and reinforcement learning approach to the vertex separator problem

The Vertex Separator Problem (VSP) is an NP-hard problem which arises from several important domains and applications. In this paper, we present an improved Breakout Local Search for VSP (named BLS-RLE). The distinguishing feature of BLS-RLE is a new parameter control mechanism that draws upon ideas from reinforcement learning theory for an interdependent decision on the number and on the type of perturbation moves. The mechanism complies with the principle “intensification first, minimal diversification only if needed”, and uses a dedicated sampling strategy for a rapid convergence towards a limited set of parameter values that appear to be the most convenient for the given state of search. Extensive experimental evaluations and statistical comparisons on a wide range of benchmark instances show significant improvement in performance of the proposed algorithm over the existing BLS algorithm for VSP. Indeed, out of the 422 tested instances, BLS-RLE was able to attain the best-known solution in 93.8% of the cases, which is around 20% higher compared to the existing BLS. In addition, we provide detailed analyses to evaluate the importance of the key elements of the proposed method and to justify the degree of diversification introduced during perturbation

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum