Cliquewidth and knowledge compilation

In this paper we study the role of cliquewidth in succinct representation of Boolean functions. Our main statement is the following: Let Z be a Boolean circuit having cliquewidth k. Then there is another circuit Z * computing the same function as Z having treewidth at most 18k + 2 and which has at most 4|Z| gates where |Z| is the number of gates of Z. In this sense, cliquewidth is not more ‘powerful’ than treewidth for the purpose of representation of Boolean functions. We believe this is quite a surprising fact because it contrasts the situation with graphs where an upper bound on the treewidth implies an upper bound on the cliquewidth but not vice versa. We demonstrate the usefulness of the new theorem for knowledge compilation. In particular, we show that a circuit Z of cliquewidth k can be compiled into a Decomposable Negation Normal Form (dnnf) of size O(918k k 2|Z|) and the same runtime. To the best of our knowledge, this is the first result on efficient knowledge compilation parameterized by cliquewidth of a Boolean circuit

Computing small pivot-minors.

A graph G contains a graph H as a pivot-minor if H can be obtained from G by applying a sequence of vertex deletions and edge pivots. Pivot-minors play an important role in the study of rank-width. However, so far, pivot-minors have only been studied from a structural perspective. We initiate a systematic study into their complexity aspects. We first prove that the PIVOT-MINOR problem, which asks if a given graph G contains a given graph H as a pivot-minor, is NP-complete. If H is not part of the input, we denote the problem by H-PIVOT-MINOR. We give a certifying polynomial-time algorithm for H -PIVOT-MINOR for every graph H with |V(H)|≤4|V(H)|≤4 except when H∈{K4,C3+P1,4P1}H∈{K4,C3+P1,4P1}, via a structural characterization of H-pivot-minor-free graphs in terms of a set FHFH of minimal forbidden induced subgraphs

Convexity in partial cubes: the hull number

We prove that the combinatorial optimization problem of determining the hull number of a partial cube is NP-complete. This makes partial cubes the minimal graph class for which NP-completeness of this problem is known and improves some earlier results in the literature. On the other hand we provide a polynomial-time algorithm to determine the hull number of planar partial cube quadrangulations. Instances of the hull number problem for partial cubes described include poset dimension and hitting sets for interiors of curves in the plane. To obtain the above results, we investigate convexity in partial cubes and characterize these graphs in terms of their lattice of convex subgraphs, improving a theorem of Handa. Furthermore we provide a topological representation theorem for planar partial cubes, generalizing a result of Fukuda and Handa about rank three oriented matroids.Comment: 19 pages, 4 figure

Tree pivot-minors and linear rank-width.

Treewidth and its linear variant path-width play a central role for the graph minor relation. Rank-width and linear rank-width do the same for the graph pivot-minor relation. Robertson and Seymour (1983) proved that for every tree T there exists a constant cT such that every graph of path-width at least cT contains T as a minor. Motivated by this result, we examine whether for every tree T there exists a constant dT such that every graph of linear rank-width at least dT contains T as a pivot-minor. We show that this is false if T is not a caterpillar, but true if T is the claw

Computing small pivot-minors

A graph G contains a graph H as a pivot-minor if H can be obtained from G by applying a sequence of vertex deletions and edge pivots. Pivot-minors play an important role in the study of rank-width. However, so far, pivot-minors have only been studied from a structural perspective. We initiate a systematic study into their complexity aspects. We first prove that the PIVOT-MINOR problem, which asks if a given graph G contains a given graph H as a pivot-minor, is NP-complete. If H is not part of the input, we denote the problem by H-PIVOT-MINOR. We give a certifying polynomial-time algorithm for H -PIVOT-MINOR for every graph H with |V(H)|≤4|V(H)|≤4 except when H∈{K4,C3+P1,4P1}H∈{K4,C3+P1,4P1}, via a structural characterization of H-pivot-minor-free graphs in terms of a set FHFH of minimal forbidden induced subgraphs

The ASOS Surgical Risk Calculator: development and validation of a tool for identifying African surgical patients at risk of severe postoperative complications

Background: The African Surgical Outcomes Study (ASOS) showed that surgical patients in Africa have a mortality twice the global average. Existing risk assessment tools are not valid for use in this population because the pattern of risk for poor outcomes differs from high-income countries. The objective of this study was to derive and validate a simple, preoperative risk stratification tool to identify African surgical patients at risk for in-hospital postoperative mortality and severe complications. Methods: ASOS was a 7-day prospective cohort study of adult patients undergoing surgery in Africa. The ASOS Surgical Risk Calculator was constructed with a multivariable logistic regression model for the outcome of in-hospital mortality and severe postoperative complications. The following preoperative risk factors were entered into the model; age, sex, smoking status, ASA physical status, preoperative chronic comorbid conditions, indication for surgery, urgency, severity, and type of surgery. Results: The model was derived from 8799 patients from 168 African hospitals. The composite outcome of severe postoperative complications and death occurred in 423/8799 (4.8%) patients. The ASOS Surgical Risk Calculator includes the following risk factors: age, ASA physical status, indication for surgery, urgency, severity, and type of surgery. The model showed good discrimination with an area under the receiver operating characteristic curve of 0.805 and good calibration with c-statistic corrected for optimism of 0.784. Conclusions: This simple preoperative risk calculator could be used to identify high-risk surgical patients in African hospitals and facilitate increased postoperative surveillance. © 2018 British Journal of Anaesthesia. Published by Elsevier Ltd. All rights reserved.Medical Research Council of South Africa gran