Counting patterns in strings and graphs

We study problems related to finding and counting patterns in strings and graphs. In the string-regime, we are interested in counting how many substring of a text are at Hamming (or edit) distance at most to a pattern . Among others, we are interested in the fully-compressed setting, where both and are given in a compressed representation. For both distance measures, we give the first algorithm that runs in (almost) linear time in the size of the compressed representations. We obtain the algorithms by new and tight structural insights into the solution structure of the problems. In the graph-regime, we study problems related to counting homomorphisms between graphs. In particular, we study the parameterized complexity of the problem #IndSub(), where we are to count all -vertex induced subgraphs of a graph that satisfy the property . Based on a theory of Lovász, Curticapean et al., we express #IndSub() as a linear combination of graph homomorphism numbers to obtain #W[1]-hardness and almost tight conditional lower bounds for properties that are monotone or that depend only on the number of edges of a graph. Thereby, we prove a conjecture by Jerrum and Meeks. In addition, we investigate the parameterized complexity of the problem #Hom(ℋ → ) for graph classes ℋ and . In particular, we show that for any problem in the class #W[1], there are classes ℋ_ and _ such that is equivalent to #Hom(ℋ_ → _ ).Wir untersuchen Probleme im Zusammenhang mit dem Finden und Zählen von Mustern in Strings und Graphen. Im Stringbereich ist die Aufgabe, alle Teilstrings eines Strings zu bestimmen, die eine Hamming- (oder Editier-)Distanz von höchstens zu einem Pattern haben. Unter anderem sind wir am voll-komprimierten Setting interessiert, in dem sowohl , als auch in komprimierter Form gegeben sind. Für beide Abstandsbegriffe entwickeln wir die ersten Algorithmen mit einer (fast) linearen Laufzeit in der Größe der komprimierten Darstellungen. Die Algorithmen nutzen neue strukturelle Einsichten in die Lösungsstruktur der Probleme. Im Graphenbereich betrachten wir Probleme im Zusammenhang mit dem Zählen von Homomorphismen zwischen Graphen. Im Besonderen betrachten wir das Problem #IndSub(), bei dem alle induzierten Subgraphen mit Knoten zu zählen sind, die die Eigenschaft haben. Basierend auf einer Theorie von Lovász, Curticapean, Dell, and Marx drücken wir #IndSub() als Linearkombination von Homomorphismen-Zahlen aus um #W[1]-Härte und fast scharfe konditionale untere Laufzeitschranken zu erhalten für , die monoton sind oder nur auf der Kantenanzahl der Graphen basieren. Somit beweisen wir eine Vermutung von Jerrum and Meeks. Weiterhin beschäftigen wir uns mit der Komplexität des Problems #Hom(ℋ → ) für Graphklassen ℋ und . Im Besonderen zeigen wir, dass es für jedes Problem in #W[1] Graphklassen ℋ_ und _ gibt, sodass äquivalent zu #Hom(ℋ_ → _ ) ist

Clique-Based Lower Bounds for Parsing Tree-Adjoining Grammars

up to lower order factors

Faster Approximate Pattern Matching: A Unified Approach

Approximate pattern matching is a natural and well-studied problem on strings: Given a text $T$, a pattern $P$, and a threshold $k$, find (the starting positions of) all substrings of $T$ that are at distance at most $k$ from $P$. We consider the two most fundamental string metrics: the Hamming distance and the edit distance. Under the Hamming distance, we search for substrings of $T$ that have at most $k$ mismatches with $P$, while under the edit distance, we search for substrings of $T$ that can be transformed to $P$ with at most $k$ edits. Exact occurrences of $P$ in $T$ have a very simple structure: If we assume for simplicity that $|T| \le 3|P|/2$ and trim $T$ so that $P$ occurs both as a prefix and as a suffix of $T$, then both $P$ and $T$ are periodic with a common period. However, an analogous characterization for the structure of occurrences with up to $k$ mismatches was proved only recently by Bringmann et al. [SODA'19]: Either there are $O(k^2)$ $k$-mismatch occurrences of $P$ in $T$, or both $P$ and $T$ are at Hamming distance $O(k)$ from strings with a common period $O(m/k)$. We tighten this characterization by showing that there are $O(k)$ $k$-mismatch occurrences in the case when the pattern is not (approximately) periodic, and we lift it to the edit distance setting, where we tightly bound the number of $k$-edit occurrences by $O(k^2)$ in the non-periodic case. Our proofs are constructive and let us obtain a unified framework for approximate pattern matching for both considered distances. We showcase the generality of our framework with results for the fully-compressed setting (where $T$ and $P$ are given as a straight-line program) and for the dynamic setting (where we extend a data structure of Gawrychowski et al. [SODA'18]).Comment: 74 pages, 7 figures, FOCS'2

Counting Small Induced Subgraphs Satisfying Monotone Properties

Given a graph property $\Phi$, the problem $\#\mathsf{IndSub}(\Phi)$ asks, on input a graph $G$ and a positive integer $k$, to compute the number of induced subgraphs of size $k$ in $G$ that satisfy $\Phi$. The search for explicit criteria on $\Phi$ ensuring that $\#\mathsf{IndSub}(\Phi)$ is hard was initiated by Jerrum and Meeks [J. Comput. Syst. Sci. 15] and is part of the major line of research on counting small patterns in graphs. However, apart from an implicit result due to Curticapean, Dell and Marx [STOC 17] proving that a full classification into "easy" and "hard" properties is possible and some partial results on edge-monotone properties due to Meeks [Discret. Appl. Math. 16] and D\"orfler et al. [MFCS 19], not much is known. In this work, we fully answer and explicitly classify the case of monotone, that is subgraph-closed, properties: We show that for any non-trivial monotone property $\Phi$, the problem $\#\mathsf{IndSub}(\Phi)$ cannot be solved in time $f(k)\cdot |V(G)|^{o(k/ {\log^{1/2}(k)})}$ for any function $f$, unless the Exponential Time Hypothesis fails. By this, we establish that any significant improvement over the brute-force approach is unlikely; in the language of parameterized complexity, we also obtain a $\#\mathsf{W}[1]$-completeness result.Comment: 33 pages, 2 figure

Counting Small Induced Subgraphs with Edge-monotone Properties

We study the parameterized complexity of #IndSub($\Phi$), where given a graph $G$ and an integer $k$, the task is to count the number of induced subgraphs on $k$ vertices that satisfy the graph property $\Phi$. Focke and Roth [STOC 2022] completely characterized the complexity for each $\Phi$ that is a hereditary property (that is, closed under vertex deletions): #IndSub($\Phi$) is #W[1]-hard except in the degenerate cases when every graph satisfies $\Phi$ or only finitely many graphs satisfy $\Phi$. We complement this result with a classification for each $\Phi$ that is edge monotone (that is, closed under edge deletions): #IndSub($\Phi$) is #W[1]-hard except in the degenerate case when there are only finitely many integers $k$ such that $\Phi$ is nontrivial on $k$-vertex graphs. Our result generalizes earlier results for specific properties $\Phi$ that are related to the connectivity or density of the graph. Further, we extend the #W[1]-hardness result by a lower bound which shows that #IndSub($\Phi$) cannot be solved in time $f(k) \cdot |V(G)|^{o(\sqrt{\log k/\log\log k})}$ for any function $f$, unless the Exponential-Time Hypothesis (ETH) fails. For many natural properties, we obtain even a tight bound $f(k) \cdot |V(G)|^{o(k)}$; for example, this is the case for every property $\Phi$ that is nontrivial on $k$-vertex graphs for each $k$ greater than some $k_0$

Faster Minimization of Tardy Processing Time on a Single Machine

This paper is concerned with the $1||\sum p_jU_j$ problem, the problem of minimizing the total processing time of tardy jobs on a single machine. This is not only a fundamental scheduling problem, but also a very important problem from a theoretical point of view as it generalizes the Subset Sum problem and is closely related to the 0/1-Knapsack problem. The problem is well-known to be NP-hard, but only in a weak sense, meaning it admits pseudo-polynomial time algorithms. The fastest known pseudo-polynomial time algorithm for the problem is the famous Lawler and Moore algorithm which runs in $O(P \cdot n)$ time, where $P$ is the total processing time of all $n$ jobs in the input. This algorithm has been developed in the late 60s, and has yet to be improved to date. In this paper we develop two new algorithms for $1||\sum p_jU_j$, each improving on Lawler and Moore's algorithm in a different scenario. Both algorithms rely on basic primitive operations between sets of integers and vectors of integers for the speedup in their running times. The second algorithm relies on fast polynomial multiplication as its main engine, while for the first algorithm we define a new "skewed" version of $(\max,\min)$-convolution which is interesting in its own right

Tight Complexity Bounds for Counting Generalized Dominating Sets in Bounded-Treewidth Graphs

We investigate how efficiently a well-studied family of domination-type problems can be solved on bounded-treewidth graphs. For sets $\sigma,\rho$ of non-negative integers, a $(\sigma,\rho)$-set of a graph $G$ is a set $S$ of vertices such that $|N(u)\cap S|\in \sigma$ for every $u\in S$, and $|N(v)\cap S|\in \rho$ for every $v\not\in S$. The problem of finding a $(\sigma,\rho)$-set (of a certain size) unifies standard problems such as Independent Set, Dominating Set, Independent Dominating Set, and many others. For all pairs of finite or cofinite sets $(\sigma,\rho)$, we determine (under standard complexity assumptions) the best possible value $c_{\sigma,\rho}$ such that there is an algorithm that counts $(\sigma,\rho)$-sets in time $c_{\sigma,\rho}^{\sf tw}\cdot n^{O(1)}$ (if a tree decomposition of width ${\sf tw}$ is given in the input). For example, for the Exact Independent Dominating Set problem (also known as Perfect Code) corresponding to $\sigma=\{0\}$ and $\rho=\{1\}$, we improve the $3^{\sf tw}\cdot n^{O(1)}$ algorithm of [van Rooij, 2020] to $2^{\sf tw}\cdot n^{O(1)}$. Despite the unusually delicate definition of $c_{\sigma,\rho}$, we show that our algorithms are most likely optimal, i.e., for any pair $(\sigma, \rho)$ of finite or cofinite sets where the problem is non-trivial, and any $\varepsilon>0$, a $(c_{\sigma,\rho}-\varepsilon)^{\sf tw}\cdot n^{O(1)}$-algorithm counting the number of $(\sigma,\rho)$-sets would violate the Counting Strong Exponential-Time Hypothesis (#SETH). For finite sets $\sigma$ and $\rho$, our lower bounds also extend to the decision version, showing that our algorithms are optimal in this setting as well. In contrast, for many cofinite sets, we show that further significant improvements for the decision and optimization versions are possible using the technique of representative sets

Tight Complexity Bounds for Counting Generalized Dominating Sets in Bounded-Treewidth Graphs

We investigate how efficiently a well-studied family of domination-type problems can be solved on bounded-treewidth graphs. For sets $\sigma,\rho$ of non-negative integers, a $(\sigma,\rho)$-set of a graph $G$ is a set $S$ of vertices such that $|N(u)\cap S|\in \sigma$ for every $u\in S$, and $|N(v)\cap S|\in \rho$ for every $v\not\in S$. The problem of finding a $(\sigma,\rho)$-set (of a certain size) unifies standard problems such as \textsc{Independent Set}, \textsc{Dominating Set}, \textsc{Independent Dominating Set}, and many others. For almost all pairs of finite or cofinite sets $(\sigma,\rho)$, we determine (under standard complexity assumptions) the best possible value $c_{\sigma,\rho}$ such that there is an algorithm that counts $(\sigma,\rho)$-sets in time c_{\sigma,\rho}^\tw\cdot n^{\O(1)} (if a tree decomposition of width \tw is given in the input). Let \sigMax denote the largest element of $\sigma$ if $\sigma$ is finite, or the largest missing integer $+1$ if $\sigma$ is cofinite; \rhoMax is defined analogously for $\rho$. Surprisingly, $c_{\sigma,\rho}$ is often significantly smaller than the natural bound \sigMax+\rhoMax+2 achieved by existing algorithms [van Rooij, 2020]. Toward defining $c_{\sigma,\rho}$, we say that $(\sigma, \rho)$ is \mname-structured if there is a pair $(\alpha,\beta)$ such that every integer in $\sigma$ equals $\alpha$ mod \mname, and every integer in $\rho$ equals $\beta$ mod \mname. Then, setting \begin{itemize} \item c_{\sigma,\rho}=\max\{\sigMax,\rhoMax\}+1 if $(\sigma,\rho)$ is \mname-structured for some \mname \ge 3, or 2-structured with \sigMax\neq \rhoMax, or 2-structured with \sigMax=\rhoMax being odd, \item c_{\sigma,\rho}=\max\{\sigMax,\rhoMax\}+2 if $(\sigma,\rho)$ is 2-structured, but not \mname-structured for any \mname \ge 3, and \sigMax=\rhoMax is even, and \item c_{\sigma,\rho}=\sigMax+\rhoMax+2 if $(\sigma,\rho)$ is not \mname-structured for any \mname\ge 2, \end{itemize} we provide algorithms counting $(\sigma,\rho)$-sets in time c_{\sigma,\rho}^\tw\cdot n^{\O(1)}. For example, for the \textsc{Exact Independent Dominating Set} problem (also known as \textsc{Perfect Code}) corresponding to $\sigma=\{0\}$ and $\rho=\{1\}$, this improves the 3^\tw\cdot n^{\O(1)} algorithm of van Rooij to 2^\tw\cdot n^{\O(1)}. Despite the unusually delicate definition of $c_{\sigma,\rho}$, we show that our algorithms are most likely optimal, i.e., for any pair $(\sigma, \rho)$ of finite or cofinite sets where the problem is non-trivial (except those having cofinite $\sigma$ with $\rho=\mathbb Z_{\ge0}$), and any $\varepsilon>0$, a (c_{\sigma,\rho}-\varepsilon)^\tw\cdot n^{\O(1)}-algorithm counting the number of $(\sigma,\rho)$-sets would violate the Counting Strong Exponential-Time Hypothesis (\#SETH). For finite sets $\sigma$ and $\rho$, our lower bounds also extend to the decision version, showing that our algorithms are optimal in this setting as well. In contrast, for many cofinite sets, we show that further significant improvements for the decision and optimization versions are possible using the technique of representative sets

Neurologic Involvement in Children and Adolescents Hospitalized in the United States for COVID-19 or Multisystem Inflammatory Syndrome

This article is made available for unrestricted research re-use and secondary analysis in any form or by any means with acknowledgement of the original source. These permissions are granted for the duration of the World Health Organization (WHO) declaration of COVID-19 as a global pandemic.Importance Coronavirus disease 2019 (COVID-19) affects the nervous system in adult patients. The spectrum of neurologic involvement in children and adolescents is unclear. Objective To understand the range and severity of neurologic involvement among children and adolescents associated with COVID-19. Setting, Design, and Participants Case series of patients (age <21 years) hospitalized between March 15, 2020, and December 15, 2020, with positive severe acute respiratory syndrome coronavirus 2 test result (reverse transcriptase-polymerase chain reaction and/or antibody) at 61 US hospitals in the Overcoming COVID-19 public health registry, including 616 (36%) meeting criteria for multisystem inflammatory syndrome in children. Patients with neurologic involvement had acute neurologic signs, symptoms, or diseases on presentation or during hospitalization. Life-threatening involvement was adjudicated by experts based on clinical and/or neuroradiologic features. Exposures Severe acute respiratory syndrome coronavirus 2. Main Outcomes and Measures Type and severity of neurologic involvement, laboratory and imaging data, and outcomes (death or survival with new neurologic deficits) at hospital discharge. Results Of 1695 patients (909 [54%] male; median [interquartile range] age, 9.1 [2.4-15.3] years), 365 (22%) from 52 sites had documented neurologic involvement. Patients with neurologic involvement were more likely to have underlying neurologic disorders (81 of 365 [22%]) compared with those without (113 of 1330 [8%]), but a similar number were previously healthy (195 [53%] vs 723 [54%]) and met criteria for multisystem inflammatory syndrome in children (126 [35%] vs 490 [37%]). Among those with neurologic involvement, 322 (88%) had transient symptoms and survived, and 43 (12%) developed life-threatening conditions clinically adjudicated to be associated with COVID-19, including severe encephalopathy (n = 15; 5 with splenial lesions), stroke (n = 12), central nervous system infection/demyelination (n = 8), Guillain-Barré syndrome/variants (n = 4), and acute fulminant cerebral edema (n = 4). Compared with those without life-threatening conditions (n = 322), those with life-threatening neurologic conditions had higher neutrophil-to-lymphocyte ratios (median, 12.2 vs 4.4) and higher reported frequency of D-dimer greater than 3 μg/mL fibrinogen equivalent units (21 [49%] vs 72 [22%]). Of 43 patients who developed COVID-19–related life-threatening neurologic involvement, 17 survivors (40%) had new neurologic deficits at hospital discharge, and 11 patients (26%) died. Conclusions and Relevance In this study, many children and adolescents hospitalized for COVID-19 or multisystem inflammatory syndrome in children had neurologic involvement, mostly transient symptoms. A range of life-threatening and fatal neurologic conditions associated with COVID-19 infrequently occurred. Effects on long-term neurodevelopmental outcomes are unknown