Companion based matrix functions: description and minimal factorization

Companion based matrix functions are rational matrix functions admitting a minimal realization involving state space matrices that are first companions. Necessary and sufficient conditions are given for a rational matrix function to be companion based. Minimal factorization of such functions is discussed in detail. It is shown that the property of being companion based is hereditary with respect to minimal factorization. Also, the issue of minimal factorization is reduced to a division problem for pairs of monic polynomials of the same degree. In this context, a connection with the Euclidean algorithm is made. The results apply to canonical Wiener-Hopf factorization as well as to complete factorization. The analysis of the latter leads to a combinatorial problem involving the eigenvalues of the state space matrices. The algorithmic aspects of this problem are intimately related to the two machine flow shop problem and Johnson's rule from job scheduling theory

Variants of the Two Machine Flow Shop Problem connected with factorization of matrix functions

In this paper we consider a number of variants of the Two Machine Flow Shop Problem. In these variants the makespan is given and the problem is to find a schedule that meets this makespan, thereby minimizing the infeasibilities of the jobs in a prescribed sense: In the max-variant the maximum infeasibility of the jobs is to be minimized, whereas in the sum-variant the objective is to minimize the sum of the infeasibilities of the jobs. For both variants observations about the structure of the optimal schedules are presented. In particular, it is proved that every instance of these problems has an optimal permutation schedule. It is also shown that the max-variant can be solved by Johnson's Rule. For the sum-variant this is not the case: For solving this problem to optimality something quite different is necessary. Both variants are connected with factorization problems for certain rational matrix functions. The factorizations involved are optimal in some sense and generalize the notion of complete factorization. In this way a connection is established between job scheduling theory on one hand, and mathematical systems theory on the other

Factorization and job scheduling: a connection via companion based matrix functions

A connection is made between two sets of problems. The first set involves factorization problems of specific rational matrix functions, the companion based matrix functions. The second set is concerned with variants of the two machine flow shop problem (2MFSP) from job scheduling theory. In particular, it is shown that with each companion based matrix function one can associate an instance of 2MFSP and vice versa. The latter can be done in such a way that the factorization properties of the companion based matrix function correspond to the combinatorial properties of the instance of 2MFSP

Finding k-secluded trees faster

We revisit the k-SECLUDED TREE problem. Given a vertex-weighted undirected graph G, its objective is to find a maximum-weight induced subtree T whose open neighborhood has size at most k. We present a fixed-parameter tractable algorithm that solves the problem in time 2 O(klog⁡k)⋅n O(1), improving on a double-exponential running time from earlier work by Golovach, Heggernes, Lima, and Montealegre. Starting from a single vertex, our algorithm grows a k-secluded tree by branching on vertices in the open neighborhood of the current tree T. To bound the branching depth, we prove a structural result that can be used to identify a vertex that belongs to the neighborhood of any k-secluded supertree T ′⊇T once the open neighborhood of T becomes sufficiently large. We extend the algorithm to enumerate compact descriptions of all maximum-weight k-secluded trees, which allows us to count them as well.</p

Finding k-secluded trees faster

We revisit the k-SECLUDED TREE problem. Given a vertex-weighted undirected graph G, its objective is to find a maximum-weight induced subtree T whose open neighborhood has size at most k. We present a fixed-parameter tractable algorithm that solves the problem in time 2 O(klog⁡k)⋅n O(1), improving on a double-exponential running time from earlier work by Golovach, Heggernes, Lima, and Montealegre. Starting from a single vertex, our algorithm grows a k-secluded tree by branching on vertices in the open neighborhood of the current tree T. To bound the branching depth, we prove a structural result that can be used to identify a vertex that belongs to the neighborhood of any k-secluded supertree T ′⊇T once the open neighborhood of T becomes sufficiently large. We extend the algorithm to enumerate compact descriptions of all maximum-weight k-secluded trees, which allows us to count them as well.</p

5-Approximation for H\mathcal{H}-Treewidth Essentially as Fast as H\mathcal{H}-Deletion Parameterized by Solution Size

The notion of $\mathcal{H}$-treewidth, where $\mathcal{H}$ is a hereditary graph class, was recently introduced as a generalization of the treewidth of an undirected graph. Roughly speaking, a graph of $\mathcal{H}$-treewidth at most $k$ can be decomposed into (arbitrarily large) $\mathcal{H}$-subgraphs which interact only through vertex sets of size $O(k)$ which can be organized in a tree-like fashion. $\mathcal{H}$-treewidth can be used as a hybrid parameterization to develop fixed-parameter tractable algorithms for $\mathcal{H}$-deletion problems, which ask to find a minimum vertex set whose removal from a given graph $G$ turns it into a member of $\mathcal{H}$. The bottleneck in the current parameterized algorithms lies in the computation of suitable tree $\mathcal{H}$-decompositions. We present FPT approximation algorithms to compute tree $\mathcal{H}$-decompositions for hereditary and union-closed graph classes $\mathcal{H}$. Given a graph of $\mathcal{H}$-treewidth $k$, we can compute a 5-approximate tree $\mathcal{H}$-decomposition in time $f(O(k)) \cdot n^{O(1)}$ whenever $\mathcal{H}$-deletion parameterized by solution size can be solved in time $f(k) \cdot n^{O(1)}$ for some function $f(k) \geq 2^k$. The current-best algorithms either achieve an approximation factor of $k^{O(1)}$ or construct optimal decompositions while suffering from non-uniformity with unknown parameter dependence. Using these decompositions, we obtain algorithms solving Odd Cycle Transversal in time $2^{O(k)} \cdot n^{O(1)}$ parameterized by $\mathsf{bipartite}$-treewidth and Vertex Planarization in time $2^{O(k \log k)} \cdot n^{O(1)}$ parameterized by $\mathsf{planar}$-treewidth, showing that these can be as fast as the solution-size parameterizations and giving the first ETH-tight algorithms for parameterizations by hybrid width measures.Comment: Conference version to appear at the European Symposium on Algorithms (ESA 2023

Single-Exponential FPT Algorithms for Enumerating Secluded F\mathcal{F}-Free Subgraphs and Deleting to Scattered Graph Classes

The celebrated notion of important separators bounds the number of small $(S,T)$-separators in a graph which are 'farthest from $S$' in a technical sense. In this paper, we introduce a generalization of this powerful algorithmic primitive that is phrased in terms of $k$-secluded vertex sets: sets with an open neighborhood of size at most $k$. In this terminology, the bound on important separators says that there are at most $4^k$ maximal $k$-secluded connected vertex sets $C$ containing $S$ but disjoint from $T$. We generalize this statement significantly: even when we demand that $G[C]$ avoids a finite set $\mathcal{F}$ of forbidden induced subgraphs, the number of such maximal subgraphs is $2^{O(k)}$ and they can be enumerated efficiently. This allows us to make significant improvements for two problems from the literature. Our first application concerns the 'Connected $k$-Secluded $\mathcal{F}$-free subgraph' problem, where $\mathcal{F}$ is a finite set of forbidden induced subgraphs. Given a graph in which each vertex has a positive integer weight, the problem asks to find a maximum-weight connected $k$-secluded vertex set $C \subseteq V(G)$ such that $G[C]$ does not contain an induced subgraph isomorphic to any $F \in \mathcal{F}$. The parameterization by $k$ is known to be solvable in triple-exponential time via the technique of recursive understanding, which we improve to single-exponential. Our second application concerns the deletion problem to scattered graph classes. Here, the task is to find a vertex set of size at most $k$ whose removal yields a graph whose each connected component belongs to one of the prescribed graph classes $\Pi_1, \ldots, \Pi_d$. We obtain a single-exponential algorithm whenever each class $\Pi_i$ is characterized by a finite number of forbidden induced subgraphs. This generalizes and improves upon earlier results in the literature.Comment: To appear at ISAAC'2

5-Approximation for ?-Treewidth Essentially as Fast as ?-Deletion Parameterized by Solution Size

The notion of ?-treewidth, where ? is a hereditary graph class, was recently introduced as a generalization of the treewidth of an undirected graph. Roughly speaking, a graph of ?-treewidth at most k can be decomposed into (arbitrarily large) ?-subgraphs which interact only through vertex sets of size ?(k) which can be organized in a tree-like fashion. ?-treewidth can be used as a hybrid parameterization to develop fixed-parameter tractable algorithms for ?-deletion problems, which ask to find a minimum vertex set whose removal from a given graph G turns it into a member of ?. The bottleneck in the current parameterized algorithms lies in the computation of suitable tree ?-decompositions. We present FPT-approximation algorithms to compute tree ?-decompositions for hereditary and union-closed graph classes ?. Given a graph of ?-treewidth k, we can compute a 5-approximate tree ?-decomposition in time f(?(k)) ? n^?(1) whenever ?-deletion parameterized by solution size can be solved in time f(k) ? n^?(1) for some function f(k) ? 2^k. The current-best algorithms either achieve an approximation factor of k^?(1) or construct optimal decompositions while suffering from non-uniformity with unknown parameter dependence. Using these decompositions, we obtain algorithms solving Odd Cycle Transversal in time 2^?(k) ? n^?(1) parameterized by bipartite-treewidth and Vertex Planarization in time 2^?(k log k) ? n^?(1) parameterized by planar-treewidth, showing that these can be as fast as the solution-size parameterizations and giving the first ETH-tight algorithms for parameterizations by hybrid width measures

Search-Space Reduction via Essential Vertices

We investigate preprocessing for vertex-subset problems on graphs. While the notion of kernelization, originating in parameterized complexity theory, is a formalization of provably effective preprocessing aimed at reducing the total instance size, our focus is on finding a non-empty vertex set that belongs to an optimal solution. This decreases the size of the remaining part of the solution which still has to be found, and therefore shrinks the search space of fixed-parameter tractable algorithms for parameterizations based on the solution size. We introduce the notion of a c-essential vertex as one that is contained in all c-approximate solutions. For several classic combinatorial problems such as Odd Cycle Transversal and Directed Feedback Vertex Set, we show that under mild conditions a polynomial-time preprocessing algorithm can find a subset of an optimal solution that contains all 2-essential vertices, by exploiting packing/covering duality. This leads to FPT algorithms to solve these problems where the exponential term in the running time depends only on the number of non-essential vertices in the solution

Incidence of gynaecological cancer during the COVID-19 pandemic:A population-based study in the Netherlands

Objective: To study the impact of the COVID-19 pandemic and consequent lockdown on the number of diagnoses of gynaecological malignancies in the Netherlands. Methods: We performed a retrospective cohort study using data from the Netherlands Cancer Registry (NCR) on women of 18 years and older diagnosed with invasive endometrial, ovarian, cervical or vulvar cancer in the period 2017–2021. Analyses were stratified for age, socioeconomical status (SES) and region. Results: The incidence rate of gynaecological cancer was 67/100.000 (n = 4832) before (2017–2019) and 68/100.000 (n = 4833) during (2020) the COVID-19 pandemic. Comparing the number of diagnoses of the two periods for the four types of cancer separately showed no significant difference. During the first wave of COVID-19 (March-June 2020), a clear decrease in number of gynaecological cancer diagnoses was visible (20–34 %). Subsequently, large increases in number of diagnoses were visible (11–29 %). No significant differences in incidence were found between different age groups, SES and regions. In 2021 an increase of 5.9 % in number of diagnoses was seen. Conclusion: In the Netherlands, a clear drop in number of diagnoses was visible for all four types of gynaecological cancers during the first wave, with a subsequent increase in number of diagnoses in the second part of 2020 and in 2021. No differences between SES groups were found. This illustrates good organisation of and access to health care in the Netherlands.</p