Potential Maximal Clique Algorithms for Perfect Phylogeny Problems

Kloks, Kratsch, and Spinrad showed how treewidth and minimum-fill, NP-hard combinatorial optimization problems related to minimal triangulations, are broken into subproblems by block subgraphs defined by minimal separators. These ideas were expanded on by Bouchitt\'e and Todinca, who used potential maximal cliques to solve these problems using a dynamic programming approach in time polynomial in the number of minimal separators of a graph. It is known that solutions to the perfect phylogeny problem, maximum compatibility problem, and unique perfect phylogeny problem are characterized by minimal triangulations of the partition intersection graph. In this paper, we show that techniques similar to those proposed by Bouchitt\'e and Todinca can be used to solve the perfect phylogeny problem with missing data, the two- state maximum compatibility problem with missing data, and the unique perfect phylogeny problem with missing data in time polynomial in the number of minimal separators of the partition intersection graph

Unique Perfect Phylogeny Characterizations via Uniquely Representable Chordal Graphs

The perfect phylogeny problem is a classic problem in computational biology, where we seek an unrooted phylogeny that is compatible with a set of qualitative characters. Such a tree exists precisely when an intersection graph associated with the character set, called the partition intersection graph, can be triangulated using a restricted set of fill edges. Semple and Steel used the partition intersection graph to characterize when a character set has a unique perfect phylogeny. Bordewich, Huber, and Semple showed how to use the partition intersection graph to find a maximum compatible set of characters. In this paper, we build on these results, characterizing when a unique perfect phylogeny exists for a subset of partial characters. Our characterization is stated in terms of minimal triangulations of the partition intersection graph that are uniquely representable, also known as ur-chordal graphs. Our characterization is motivated by the structure of ur-chordal graphs, and the fact that the block structure of minimal triangulations is mirrored in the graph that has been triangulated

Approximately counting locally-optimal structures

A locally-optimal structure is a combinatorial structure such as a maximal independent set that cannot be improved by certain (greedy) local moves, even though it may not be globally optimal. It is trivial to construct an independent set in a graph. It is easy to (greedily) construct a maximal independent set. However, it is NP-hard to construct a globally-optimal (maximum) independent set. In general, constructing a locally-optimal structure is somewhat more difficult than constructing an arbitrary structure, and constructing a globally-optimal structure is more difficult than constructing a locally-optimal structure. The same situation arises with listing. The differences between the problems become obscured when we move from listing to counting because nearly everything is #P-complete. However, we highlight an interesting phenomenon that arises in approximate counting, where the situation is apparently reversed. Specifically, we show that counting maximal independent sets is complete for #P with respect to approximation-preserving reductions, whereas counting all independent sets, or counting maximum independent sets is complete for an apparently smaller class, $\mathrm{\#RH}\Pi_1$ which has a prominent role in the complexity of approximate counting. Motivated by the difficulty of approximately counting maximal independent sets in bipartite graphs, we also study the problem of approximately counting other locally-optimal structures that arise in algorithmic applications, particularly problems involving minimal separators and minimal edge separators. Minimal separators have applications via fixed-parameter-tractable algorithms for constructing triangulations and phylogenetic trees. Although exact (exponential-time) algorithms exist for listing these structures, we show that the counting problems are #P-complete with respect to both exact and approximation-preserving reductions.Comment: Accepted to JCSS, preliminary version accepted to ICALP 2015 (Track A

Information content and aerosol property retrieval potential for different types of in situ polar nephelometer data

Polar nephelometers are in situ instruments used to measure the angular distribution of light scattered by aerosol particles. These types of measurements contain substantial information about the properties of the aerosol being probed (e.g. concentrations, sizes, refractive indices, shape parameters), which can be retrieved through inversion algorithms. The aerosol property retrieval potential (i.e. information content) of a given set of measurements depends on the spectral, polarimetric, and angular characteristics of the polar nephelometer that was used to acquire the measurements. To explore this issue quantitatively, we applied Bayesian information content analysis and calculated the metric degrees of freedom for signal (DOFS) for a range of simulated polar nephelometer instrument configurations, aerosol models and test cases, and assumed levels of prior knowledge about the variances of specific aerosol properties. Assuming a low level of prior knowledge consistent with an unconstrained ambient/field measurement setting, we demonstrate that even very basic polar nephelometers (single wavelength, no polarization capability) will provide informative measurements with a very high retrieval potential for the size distribution and refractive index state parameters describing simple unimodal, spherical test aerosols. As expected, assuming a higher level of prior knowledge consistent with well-constrained laboratory applications leads to a reduction in potential for information gain via performing the polarimetric measurement. Nevertheless, we show that in this situation polar nephelometers can still provide informative measurements: e.g. it can be possible to retrieve the imaginary part of the refractive index with high accuracy if the laboratory setting makes it possible to keep the probed aerosol sample simple. The analysis based on a high level of prior knowledge also allows us to better assess the impact of different polar nephelometer instrument design features in a consistent manner for retrieved aerosol parameters. The results indicate that the addition of multi-wavelength and/or polarimetric measurement capabilities always leads to an increase in information content, although in some cases the increase is negligible, e.g. when adding a fourth, near-IR measurement wavelength for the retrieval of unimodal size distribution parameters or if the added polarization component has high measurement uncertainty. By considering a more complex bimodal, non-spherical-aerosol model, we demonstrate that performing more comprehensive spectral and/or polarimetric measurements leads to very large benefits in terms of the achieved information content. We also investigated the impact of angular truncation (i.e. the loss of measurement information at certain scattering angles) on information content. Truncation at extreme angles (i.e. in the near-forward or near-backward directions) results in substantial decreases in information content for coarse-aerosol test cases. However for fine-aerosol test cases, the sensitivity of DOFS to extreme-angle truncation is noticeably smaller and can be further reduced by performing more comprehensive measurements. Side angle truncation has very little effect on information content for both the fine and coarse test cases. Furthermore, we demonstrate that increasing the number of angular measurements generally increases the information content. However, above a certain number of angular measurements (∼20–40) the observed increases in DOFS plateau out. Finally, we demonstrate that the specific placement of angular measurements within a nephelometer can have a large impact on information content. As a proof of concept, we show that a reductive greedy algorithm based on the DOFS metric can be used to find optimal angular configurations for given target aerosols and applications.</p

Information content and aerosol property retrieval potential for different types of in situ polar nephelometer data

Polar nephelometers are in situ instruments used to measure the angular distribution of light scattered by aerosol particles. These type of measurements contain substantial information about the properties of the aerosol being probed (e.g. concentrations, sizes, refractive indices, shape parameters), which can be retrieved through inversion algorithms. The aerosol property retrieval potential (i.e., information content) of a given set of measurements depends on the spectral, polarimetric and angular characteristics of the polar nephelometer that was used to acquire it. To explore this issue quantitatively, we applied Bayesian information content analysis and calculated the metric Degrees of Freedom for Signal (DOFS) for a range of simulated polar nephelometer instrument configurations, aerosol models and test cases, and assumed levels of prior knowledge about the variances of specific aerosol properties. Assuming a low level of prior knowledge consistent with an unconstrained ambient/field measurement setting, we demonstrate that even very basic polar nephelometers (single wavelength, no polarization capability) will provide informative measurements with very high retrieval potential for the size distribution and refractive index state parameters describing simple unimodal, spherical test aerosols. As expected, assuming a higher level of prior knowledge consistent with well constrained laboratory applications leads to a reduction in potential for information gain via performing the polarimetric measurement. This analysis allows us to better assess the impact of different polar nephelometer instrument design features in a consistent manner for retrieved aerosol parameters. The results indicate that the addition of multi-wavelength and/or polarimetric measurement capabilities always leads to an increase in information content, although in some cases the increase is negligible: e.g. when adding a fourth, near-IR measurement wavelength for the retrieval of unimodal size distribution parameters, or if the added polarization component has high measurement uncertainty. By considering a more complex bimodal, non-spherical aerosol model, we demonstrate that performing the more comprehensive spectral and/or polarimetric measurements leads to very large benefits in terms of the achieved information content. We also investigated the impact of angular truncation (i.e., the loss of measurement information at certain scattering angles) on information content. Truncation at extreme angles (i.e., in the near-forward or &ndash;backward directions) results in substantial decreases in information content for coarse aerosol test cases. However for fine aerosol test cases, the sensitivity of DOFS to extreme angle truncation is noticeably smaller and can be further reduced by performing more comprehensive measurements. Side-angle truncation has very little effect on information content for both the fine and coarse test cases. Furthermore, we demonstrate that increasing the number of angular measurements generally increases the information content. However, above a certain number of angular measurements (~20&ndash;40) the observed increases in DOFS plateau out. Finally, we demonstrate that the specific placement of angular measurements within a nephelometer can have a large impact on information content. As a proof-of-concept, we show that a reductive greedy algorithm based on the DOFS metric can be used to find optimal angular configurations for given target aerosols and applications.</p

Unique Perfect Phylogeny Characterizations via Uniquely Representable Chordal Graphs

The perfect phylogeny problem is a classic problem in computational biology, where we seek an unrooted phylogeny that is compatible with a set of qualitative characters. Such a tree exists precisely when an intersection graph associated with the character set, called the partition intersection graph, can be triangulated using a restricted set of fill edges. Semple and Steel used the partition intersection graph to characterize when a character set has a unique perfect phylogeny. Bordewich, Huber, and Semple showed how to use the partition intersection graph to find a maximum compatible set of characters. In this paper, we build on these results, characterizing when a unique perfect phylogeny exists for a subset of partial characters. Our characterization is stated in terms of minimal triangulations of the partition intersection graph that are uniquely representable, also known as ur-chordal graphs. Our characterization is motivated by the structure of ur-chordal graphs, and the fact that the block structure of minimal triangulations is mirrored in the graph that has been triangulated

Potential Maximal Clique Algorithms for Perfect Phylogeny Problems

Kloks, Kratsch, and Spinrad showed how treewidth and minimum-fill, NP-hard combinatorial optimization problems related to minimal triangulations, are broken into subproblems by block subgraphs defined by minimal separators. These ideas were expanded on by Bouchitt\'e and Todinca, who used potential maximal cliques to solve these problems using a dynamic programming approach in time polynomial in the number of minimal separators of a graph. It is known that solutions to the perfect phylogeny problem, maximum compatibility problem, and unique perfect phylogeny problem are characterized by minimal triangulations of the partition intersection graph. In this paper, we show that techniques similar to those proposed by Bouchitt\'e and Todinca can be used to solve the perfect phylogeny problem with missing data, the two- state maximum compatibility problem with missing data, and the unique perfect phylogeny problem with missing data in time polynomial in the number of minimal separators of the partition intersection graph

Constructing perfect phylogenies and proper triangulations for three-state characters

Abstract In this paper, we study the problem of constructing perfect phylogenies for three-state characters. Our work builds on two recent results. The first result states that for three-state characters, the local condition of examining all subsets of three characters is sufficient to determine the global property of admitting a perfect phylogeny. The second result applies tools from minimal triangulation theory to the partition intersection graph to determine if a perfect phylogeny exists. Despite the wealth of combinatorial tools and algorithms stemming from the chordal graph and minimal triangulation literature, it is unclear how to use such approaches to efficiently construct a perfect phylogeny for three-state characters when the data admits one. We utilize structural properties of both the partition intersection graph and the original data in order to achieve a competitive time bound.</p

Constructing perfect phylogenies and proper triangulations for three-state characters

Abstract In this paper, we study the problem of constructing perfect phylogenies for three-state characters. Our work builds on two recent results. The first result states that for three-state characters, the local condition of examining all subsets of three characters is sufficient to determine the global property of admitting a perfect phylogeny. The second result applies tools from minimal triangulation theory to the partition intersection graph to determine if a perfect phylogeny exists. Despite the wealth of combinatorial tools and algorithms stemming from the chordal graph and minimal triangulation literature, it is unclear how to use such approaches to efficiently construct a perfect phylogeny for three-state characters when the data admits one. We utilize structural properties of both the partition intersection graph and the original data in order to achieve a competitive time bound