On the Maximum Connectivity Improvement Problem

In this paper, a hierarchical ensemble classification approach that utilizes a Directed Acyclic Graph (DAG) structure is proposed as a solution to the multi-class classification problem. Two main DAG structures are considered: (i) rooted DAG, and (ii) non-rooted DAG. The main challenges that are considered in this paper are: (i) the successive misclassification issue associated with hierarchical classification, and (i) identification of the starting node within the non-rooted DAG approach. To address these issues the idea is to utilize Bayesian probability values to: select the best starting DAG node, and to dictate whether single or multiple paths should be followed within the DAG structure. The reported experimental results indicated that the proposed DAG structure is more effective than when using a simple binary tree structure for generating a hierarchical classification model.

BackgroundDirected acyclic graphs (DAGs) are an effective means of presenting expert-knowledge assumptions when selecting adjustment variables in epidemiology, whereas the change-in-estimate procedure is a common statistics-based approach. As DAGs imply specific empirical relationships which can be explored by the change-in-estimate procedure, it should be possible to combine the two approaches. This paper proposes such an approach which aims to produce well-adjusted estimates for a given research question, based on plausible DAGs consistent with the data at hand, combining prior knowledge and standard regression methods.MethodsBased on the relationships laid out in a DAG, researchers can predict how a collapsible estimator (e.g. risk ratio or risk difference) for an effect of interest should change when adjusted on different variable sets. Implied and observed patterns can then be compared to detect inconsistencies and so guide adjustment-variable selection.ResultsThe proposed approach involves i. drawing up a set of plausible background-knowledge DAGs; ii. starting with one of these DAGs as a working DAG, identifying a minimal variable set, S, sufficient to control for bias on the effect of interest; iii. estimating a collapsible estimator adjusted on S, then adjusted on S plus each variable not in S in turn (“add-one pattern”) and then adjusted on the variables in S minus each of these variables in turn (“minus-one pattern”); iv. checking the observed add-one and minus-one patterns against the pattern implied by the working DAG and the other prior DAGs; v. reviewing the DAGs, if needed; and vi. presenting the initial and all final DAGs with estimates.ConclusionThis approach to adjustment-variable selection combines background-knowledge and statistics-based approaches using methods already common in epidemiology and communicates assumptions and uncertainties in a standardized graphical format. It is probably best suited to areas where there is considerable background knowledge about plausible variable relationships. Researchers may use this approach as an additional tool for selecting adjustment variables when analyzing epidemiological data.

A tournament T = (V , A) is a directed graph in which there is exactly one arc between every pair of distinct vertices. Given a digraph on n vertices and an integer parameter k, the Feedback Arc Set problem asks whether the given digraph has a set of k arcs whose removal results in an acyclic digraph. The Feedback Arc Set problem restricted to tournaments is known as the k-Feedback Arc Set in Tournaments (k-FAST) problem. In this paper we obtain a linear vertex kernel for k-FAST. That is, we give a polynomial time algorithm which given an input instance T to k-FAST obtains an equivalent instance T ′ on O(k) vertices. In fact, given any fixed ϵ > 0, the kernelized instance has at most (2 + ϵ)k vertices. Our result improves the previous known bound of O(k2 ) on the kernel size for k-FAST. Our kernelization algorithm solves the problem on a subclass of tournaments in polynomial time and uses a known polynomial time approximation scheme for k- FAST.

The Directed Acyclic Graph (DAG) is a graph representing causal pathways for informing the conduct of an observational study. The use of DAGs allows transparent communication of a causal model between researchers and can prevent over-adjustment biases when conducting causal inference, permitting greater confidence and transparency in reported causal estimates. In the era of ‘big data’ and increasing number of observational studies, the role of the DAG is becoming more important. Recent best-practice guidance for constructing a DAG with reference to the literature has been published in the ‘Evidence synthesis for constructing DAGs’ (ESC-DAG) protocol. We aimed to assess adherence to these principles for DAGs constructed within perioperative literature. Following registration on the International Prospective Register of Systematic Reviews (PROSPERO) and with adherence to the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) reporting framework for systematic reviews, we searched the Excerpta Medica dataBASE (Embase), the Medical Literature Analysis and Retrieval System Online (MEDLINE) and Cochrane databases for perioperative observational research incorporating a DAG. Nineteen studies were included in the final synthesis. No studies demonstrated any evidence of following the mapping stage of the protocol. Fifteen (79%) fulfilled over half of the translation and integration one stages of the protocol. Adherence with one stage did not guarantee fulfilment of the other. Two studies (11%) undertook the integration two stage. Unmeasured variables were handled inconsistently between studies. Only three (16%) studies included unmeasured variables within their DAG and acknowledged their implication within the main text. Overall, DAGs that were constructed for use in perioperative observational literature did not consistently adhere to best practice, potentially limiting the benefits of subsequent causal inference. Further work should focus on exploring reasons for this deviation and increasing methodological transparency around DAG construction.

Rooted phylogenetic networks, or more generally, directed acyclic graphs (DAGs), are widely used to model species or gene relationships that traditional rooted trees cannot fully capture, especially in the presence of reticulate processes or horizontal gene transfers. Such networks or DAGs are typically inferred from observable data (e.g., genomic sequences of extant species), providing only an estimate of the true evolutionary history. However, these inferred DAGs are often complex and difficult to interpret. In particular, many contain vertices that do not serve as least common ancestors (LCAs) for any subset of the underlying genes or species, thus may lack direct support from the observable data. In contrast, LCA vertices are witnessed by historical traces justifying their existence and thus represent ancestral states substantiated by the data. To reduce unnecessary complexity and eliminate unsupported vertices, we aim to simplify a DAG to retain only LCA vertices while preserving essential evolutionary information. In this paper, we characterize LCA-relevant and lca-relevant DAGs, defined as those in which every vertex serves as an LCA (or unique LCA) for some subset of taxa. We introduce methods to identify LCAs in DAGs and efficiently transform any DAG into an LCA-relevant or lca-relevant one while preserving key structural properties of the original DAG or network. This transformation is achieved using a simple operator “⊖” that mimics vertex suppression.

Chapter 11 Bayesian Networks

Consider the longest path problem for directed acyclic graphs (DAGs), where a mutually independent random variable is associated with each of the edges as its edge length. Given a DAG G and any distributions that the random variables obey, let F MAX (x ) be the distribution function of the longest path length. We first represent F MAX (x ) by a repeated integral that involves n *** 1 integrals, where n is the order of G . We next present an algorithm to symbolically execute the repeated integral, provided that the random variables obey the standard exponential distribution. Although there can be *** (2 n ) paths in G , its running time is bounded by a polynomial in n , provided that k , the cardinality of the maximum anti-chain of the incidence graph of G , is bounded by a constant. We finally propose an algorithm that takes x and *** > 0 as inputs and approximates the value of repeated integral of x , assuming that the edge length distributions satisfy the following three natural conditions: (1) The length of each edge (v i ,v j ) *** E is non-negative, (2) the Taylor series of its distribution function F ij (x ) converges to F ij (x ), and (3) there is a constant *** that satisfies $\sigma^p \le \left|\left(\frac{d}{dx}\right)^p F_{ij}(x)\right|$ for any non-negative integer p . It runs in polynomial time in n , and its error is bounded by *** , when x , *** , *** and k can be regarded as constants.

Directed acyclic graphs (DAGs) play a large role in the modern approach to causal inference. DAGs describe the relationship between measurements taken at various discrete times including the effect of interventions. The causal mechanisms, on the other hand, would naturally be assumed to be a continuous process operating over time in a cause–effect fashion. How does such immediate causation, that is causation occurring over very short time intervals, relate to DAGs constructed from discrete observations? We introduce a time-continuous model and simulate discrete observations in order to judge the relationship between the DAG and the immediate causal model. We find that there is no clear relationship; indeed the Bayesian network described by the DAG may not relate to the causal model. Typically, discrete observations of a process will obscure the conditional dependencies that are represented in the underlying mechanistic model of the process. It is therefore doubtful whether DAGs are always suited to describe causal relationships unless time is explicitly considered in the model. We relate the issues to mechanistic modeling by using the concept of local (in)dependence. An example using data from the Swiss HIV Cohort Study is presented.

Cancer progression models such as Oncogenetic Tress [1], Conjunctive Bayesian Networks [2], or CAPRI [3], try to infer possible restrictions in the order of accumulation of mutations using cancer cross-sectional data. These restrictions would arise from epistatic interactions between mutations. The Directed Acyclic Graphs (DAGs) returned by these methods predict the genotypes that can and cannot be observed. Since fitness landscapes are shaped by those same epistatics interactions and also specify what genotypes are accessible and inaccessible [4] we would expect a close correspondence between a fitness landscape and the DAGs inferred from evolutionary processes in that landscape. I will show here, however, that there is a many-to- many mapping between fitness landscapes and DAGs inferred from cancer progression models. I show that many fitness landscapes cannot be represented by a cancer progression DAG in the sense that no DAG can codify the genotypes that are accessible under that landscape and I propose two measures of “DAG-non-representability”. Next, using 500 different random fitness lanscapes (with 100 begin perfectly DAG-representable) I simulate cancer progression using different mutation rates and sampling regimes; I analyze 20 replicates (of 1000 genotypes) under each condition with CAPRI[3] and CBN[2]. I show that, even under constant mutation and sampling regime conditions, data obtained from the same landscape lead to inferring widely different DAGs (e.g., median Gini-Simpson index of 0.5); this one-landscape-to-many-DAGs phenomenon increases with DAG-non-representability. I also show the one-DAG-to-many-landscapes phenomenon where the same DAG is frequently inferred from very different fitness landscapes (e.g., > 50% difference in accessible genotypes). Finally, I “reverse landscape engineer” the pancreatic cancer data used in [6]: using 190 random fitness landscapes, I show that genotype frequencies similar to the empirically observed one can be obtained under fitness landscapes that are very different from each other and that produce data that lead to the inference of DAGs that are also widely different among themselves (e.g., pairwise difference in edges of DAGs > 50%). I discuss the implications of these results for the interpretation and applied use of cancer progression models. Novelty and significance This paper explicitly establishes a connection between the inference of cancer progression models and the fitness landscapes that condition that cancer progression. It shows there is a many-to- many mapping between them, even under large sample sizes, that can limit their interpretability.

Adjusting for Selection Effects in Epidemiologic Studies

Directed Gaussian graphical models with toric vanishing ideals

We propose a self-stabilizing algorithm to construct a minimal weakly $\mathcal{ST}$-reachable directed acyclic graph (DAG), which is suited for routing messages on wireless networks. Given an arbitrary, simple, connected, and undirected graph $G=(V, E)$ and two sets of nodes, senders $\mathcal{S} (\subset V)$ and targets $\mathcal{T} (\subset V)$, a directed subgraph $\vec{G}$ of $G$ is a weakly $\mathcal{ST}$-reachable DAG on $G$, if $\vec{G}$ is a DAG and every sender can reach at least one target, and every target is reachable from at least one sender in $\vec{G}$. We say that a weakly $\mathcal{ST}$-reachable DAG $\vec{G}$ on $G$ is minimal if any proper subgraph of $\vec{G}$ is no longer a weakly $\mathcal{ST}$-reachable DAG. This DAG is a relaxed version of the original (or strongly) $\mathcal{ST}$-reachable DAG, where every target is reachable from every sender. This is because a strongly $\mathcal{ST}$-reachable DAG $G$ does not always exist; some graph has no strongly $\mathcal{ST}$-reachable DAG even in the case $|\mathcal{S}|=|\mathcal{T}|=2$. On the other hand, the proposed algorithm always constructs a weakly $\mathcal{ST}$-reachable DAG for any $|\mathcal{S}|$ and $|\mathcal{T}|$. Furthermore, the proposed algorithm is self-stabilizing; even if the constructed DAG deviates from the reachability requirement by a breakdown or exhausting the battery of a node having an arc in the DAG, this algorithm automatically reconstructs the DAG to satisfy the requirement again. The convergence time of the algorithm is $O(D)$ asynchronous rounds, where $D$ is the diameter of a given graph. We conduct small simulations to evaluate the performance of the proposed algorithm. The simulation result indicates that its execution time decreases when the number of sender nodes or target nodes is large.

We reduce (in polynomial time) the enumeration of minimal dominating sets in interval and permutation graphs to the enumeration of paths in directed acyclic graphs(DAGs). As a consequence, we can enumerate with linear delay, after a polynomial time pre-processing, minimal dominating sets in interval and permutation graphs. We can also count them in polynomial time. This improves considerably upon previously known results on interval graphs, and up to our knowledge no output polynomial time algorithm for the enumeration of minimal dominating sets and their counting were known for permutation graphs.

We provide a comprehensive study on network flow problems with arc reversal capabilities. The problem is to identify the arcs to be reversed in order to achieve a maximum flow from source(s) to sink(s). The problem finds its applications in emergency transportation management, where the lanes of a road network could be reversed to enable flow in the opposite direction. We study several network flow problems with the arc reversal capability and discuss their complexity. More specifically, we discuss the polynomial time algorithms for the maximum dynamic flow problem with arc reversal capability having a single source and a single sink, and for the maximum (static) flow problem. The presented algorithms are based on graph transformations and reductions to polynomially solvable flow problems. In addition, we show that the quickest transshipment problem with arc reversal capability and the problem of minimizing the total cost resulting from arc switching costs are \(\mathcal{NP}\) -hard.

ABSTRACTThe Minimum Path Cover (MPC) problem consists of finding a minimum‐cardinality set of node‐disjoint paths that cover all nodes in a given graph. We explore a variant of the MPC problem on directed acyclic graphs (DAGs) where, given a subset of arcs, each path within the MPC should contain at least one arc from this subset. We prove that the feasibility problem is strongly ‐hard on arbitrary DAGs, but the problem can be solved in polynomial time when the DAG is the transitive closure of a path. Given that the problem may not always be feasible, our solution focuses on covering a maximum number of nodes with a minimum number of node‐disjoint paths, such that each path includes at least one arc from the predefined subset of arcs. This paper introduces and investigates two integer programming formulations for this problem. We propose several valid inequalities to enhance the linear programming relaxations, employing them as cutting planes in a branch‐and‐cut approach. The procedure is implemented and tested on a wide range of instances, including real‐world instances derived from an airline crew scheduling problem, demonstrating the effectiveness of the proposed approach.