The Conversion Set Problem on Graphs

The dominating set problem asks for a small subset D of nodes in a graph such that every node is either in D or adjacent to a node in D. This problem arises in a number of distributed network applications, where it is important to locate a small number of centers in the network such that every node is nearby at least one center. Finding a dominating set of minimum size is NP-complete, and the best known approximation is logarithmic in the maximum degree of the graph and is provided by the same simple greedy approach that gives the well-known logarithmic approximation result for the closely related set cover problem.We describe and analyze new randomized distributed algorithms for the dominating set problem that run in polylogarithmic time, independent of the diameter of the network, and that return a dominating set of size within a logarithmic factor from optimal, with high probability. In particular, our best algorithm runs in O(log n log Δ) rounds with high probability, where n is the number of nodes, Δ is one plus the maximum degree of any node, and each round involves a constant number of message exchanges among any two neighbors; the size of the dominating set obtained is within O (log Δ) of the optimal in expectation and within O(log n) of the optimal with high probability. We also describe generalizations to the weighted case and the case of multiple covering requirements.

An improved algorithm for the red–blue hitting set problem with the consecutive ones property

PurposeQuality control plays an increasingly important role in quantitative PET imaging and is typically performed using phantoms. The purpose of this work was to develop and validate a fully-automated analysis method for two common PET/CT quality assurance phantoms: the NEMA NU-2 IQ and SNMMI/CTN oncology phantom. The algorithm was designed to only utilize the PET scan to enable the analysis of phantoms with thin-walled inserts.MethodsWe introduce a model-based method for automated analysis of phantoms with spherical inserts. Models are first constructed for each type of phantom to be analyzed. A robust insert detection algorithm uses the model to locate all inserts inside the phantom. First, candidates for inserts are detected using a scale-space detection approach. Second, candidates are given an initial label using a score-based optimization algorithm. Third, a robust model fitting step aligns the phantom model to the initial labeling and fixes incorrect labels. Finally, the detected insert locations are refined and measurements are taken for each insert and several background regions. In addition, an approach for automated selection of NEMA and CTN phantom models is presented.The method was evaluated on a diverse set of 15 NEMA and 20 CTN phantom PET/CT scans. NEMA phantoms were filled with radioactive tracer solution at 9.7:1 activity ratio over background, and CTN phantoms were filled with 4:1 and 2:1 activity ratio over background. For quantitative evaluation, an independent reference standard was generated by two experts using PET/CT scans of the phantoms. In addition, the automated approach was compared against manual analysis, which represents the current clinical standard approach, of the PET phantom scans by four experts.ResultsThe automated analysis method successfully detected and measured all inserts in all test phantom scans. It is a deterministic algorithm (zero variability), and the insert detection RMS error (i.e., bias) was 0.97, 1.12, and 1.48 mm for phantom activity ratios 9.7:1, 4:1, and 2:1, respectively. For all phantoms and at all contrast ratios, the average RMS error was found to be significantly lower for the proposed automated method compared to the manual analysis of the phantom scans. The uptake measurements produced by the automated method showed high correlation with the independent reference standard (R2 ≥ 0.9987). In addition, the average computing time for the automated method was 30.6 seconds and was found to be significantly lower (p ≪ 0.001) compared to manual analysis (mean: 247.8 seconds).ConclusionsThe proposed automated approach was found to have less error when measured against the independent reference than the manual approach. It can be easily adapted to other phantoms with spherical inserts. In addition, it eliminates inter- and intra-operator variability in PET phantom analysis and is significantly more time efficient, and therefore, represents a promising approach to facilitate and simplify PET standardization and harmonization efforts.

We consider the problem of inferring the local transition functions of discrete dynamical systems from observed behavior. Our focus is on synchronous systems whose local transition functions are threshold functions. We assume that the topology of the system is known and that the goal is to infer a threshold value for each node so that the system produces the observed behavior. We show that some of these inference problems are efficiently solvable while others are NP-complete, even when the underlying graph of the dynamical system is a simple path. We also identify a fixed parameter tractable problem in this context.

A vertex subset I of a graph G is called a k-path vertex cover if every path on k vertices in G contains at least one vertex from I. The k -Path Vertex Cover Reconfiguration (k -PVCR) problem asks if one can transform one k-path vertex cover into another via a sequence of k-path vertex covers where each intermediate member is obtained from its predecessor by applying a given reconfiguration rule exactly once. We investigate the computational complexity of k -PVCR from the viewpoint of graph classes under the well-known reconfiguration rules: \(\mathsf {TS}\), \(\mathsf {TJ}\), and \(\mathsf {TAR}\). The problem for \(k=2\), known as the Vertex Cover Reconfiguration (VCR) problem, has been well-studied in the literature. We show that certain known hardness results for VCR on different graph classes including planar graphs, bounded bandwidth graphs, chordal graphs, and bipartite graphs, can be extended for k -PVCR. In particular, we prove a complexity dichotomy for k -PVCR on general graphs: on those whose maximum degree is 3 (and even planar), the problem is \(\mathtt {PSPACE}\)-complete, while on those whose maximum degree is 2 (i.e., paths and cycles), the problem can be solved in polynomial time. Additionally, we also design polynomial-time algorithms for k -PVCR on trees under each of \(\mathsf {TJ}\) and \(\mathsf {TAR}\). Moreover, on paths, cycles, and trees, we describe how one can construct a reconfiguration sequence between two given k-path vertex covers in a yes-instance. In particular, on paths, our constructed reconfiguration sequence is shortest.

Continuous interpolates are described for classical dynamical systems defined by discrete time-steps. Functional conjugation methods play a central role in obtaining the interpolations. The interpolates correspond to particle motion in an underlying potential, $V$. Typically, $V$ has no lower bound and can exhibit switchbacks wherein $V$ changes form when turning points are encountered by the particle. The Beverton-Holt and Skellam models of population dynamics, and particular cases of the logistic map are used to illustrate these features.

Finding a maximum independent set is a fundamental NP-hard problem that is used in many real-world applications. Given an unweighted graph, this problem asks for a maximum cardinality set of pairwise non-adjacent vertices. In recent years, some of the most successful algorithms for solving this problem are based on the branch-and-bound or branch-and-reduce paradigms. In particular, branch-and-reduce algorithms, which combine branch-and-bound with reduction rules, have been able to achieve substantial results, solving many previously infeasible real-world instances. These results were to a large part achieved by developing new, more practical reduction rules. However, other components that have been shown to have a significant impact on the performance of these algorithms have not received as much attention. One of these is the branching strategy, which determines what vertex is included or excluded in a potential solution. Even now, the most commonly used strategy selects vertices solely based on their degree and does not take into account other factors that contribute to the performance of the algorithm. In this work, we develop and evaluate several novel branching strategies for both branch-and-bound and branch-and-reduce algorithms. Our strategies are based on one of two approaches which are motivated by existing research. They either (1) aim to decompose the graph into two or more connected components which can then be solved independently, or (2) try to remove vertices that hinder the application of a reduction rule which can lead to smaller graphs. Our experimental evaluation on a large set of real-world instances indicates that our strategies are able to improve the performance of the state-of-the-art branch-and-reduce algorithm by Akiba and Iwata. To be more specific, our reduction-based packing branching rule is able to outperform the default branching strategy of selecting a vertex of highest degree on 65% of all instances tested. Furthermore, our decomposition-based strategy based on edge cuts is able to achieve a speedup of 2.29 on sparse networks (1.22 on all instances).

Given a planar graph G=(V,E) and a vertex set W ⊆ V, the subgraph induced planar connectivity augmentation problem asks for a minimum cardinality set F of additional edges with end vertices in W such that G’ = (V, E ∪ F) is planar and the subgraph of G’ induced by W is connected. The problem arises in automatic graph drawing in the context of c-planarity testing of clustered graphs. We describe a linear time algorithm based on SPQR-tress that tests if a subgraph induced planar connectivity augmentation exists and, if so, constructs a minimum cardinality augmenting edge set.

The largest common embeddable subtree problem asks for the largest possible tree embeddable into two input trees and generalizes the classical maximum common subtree problem. Several variants of the problem in labeled and unlabeled rooted trees have been studied, e.g., for the comparison of evolutionary trees. We consider a generalization, where the sought embedding is maximal with regard to a weight function on pairs of labels. We support rooted and unrooted trees with vertex and edge labels as well as distance penalties for skipping vertices. This variant is important for many applications such as the comparison of chemical structures and evolutionary trees. Our algorithm computes the solution from a series of bipartite matching instances, which are solved efficiently by exploiting their structural relation and imbalance. Our analysis shows that our approach improves or matches the running time of the formally best algorithms for several problem variants. Specifically, we obtain a running time of O(|T| |T'|Delta) for two rooted or unrooted trees T and T', where Delta=min{Delta(T),Delta(T')} with Delta(X) the maximum degree of X. If the weights are integral and at most C, we obtain a running time of O(|T| |T'|sqrt Delta log (C min{|T|,|T'|})) for rooted trees.

Abstract. It is well known that a fractal set is not a submanifold of the ambient space. However, fractals arise as invariant subsets even in infinitely smooth conditions and the Minkowski dimension serves in this case as a characteristic of complexity of this scale. For example, when the equilibrium state during the Andronov-Hopf bifurcation losses its stability, the closure of the non-singular trajectory is a parametrically defined curve of the fractal type. In this work the fractal dimension of such curves is calculated. In addition, special two-parameter family of functions is studied such that Minkowski dimension of their graphs varies from 1 to 2. The obtained result allows us to implement a regular dynamic system with an isolated hyperbolic point such that the closure of two-dimensional stable manifold of this point may have Minkowski dimension greater than 2. To calculate the graph dimension, the segment of the argument defining the graph is split into two parts. The dimension of the first part of the graph can be estimated from above by direct calculation of the corresponding curve’s length. The upper estimation of the other part’s dimension is provided by means of the area of rectangle containing this curve. The lower estimation of the Minkowski dimension is based on calculating the cardinality of ε -distinguishable set of graph points.

The complexity of finding harmless individuals in social networks

Characterization of general position sets and its applications to cographs and bipartite graphs

A vertex subset S of a graph G is a 2-dominating set of G if every vertex not in S is adjacent to two vertices of S. The 2-domination number γ2(G) is the minimum cardinality of a 2-dominating set of G. The 2-reinforcement number r2(G) is the smallest number of extra edges whose addition to G results in a graph Gwith γ2(G � )<γ 2(G) .L etT be a tree. It is showed by Lu, Hu, and Xu that r2(T ) ≤ 3. In this paper, we will show that r2(T ) = 3 if and only if there is a 2-dominating set S of T such that T contains neither S-vulnerable vertices nor S-vulnerable paths. Keywords 2-Domination · 2-Reinforcement number · S-vulnerable · Trees

Fixed-parameter algorithms for vertex cover [formula omitted

On Acyclic Orientations and Sequential Dynamical Systems