Preparing North Philadelphia\u27s Teen Population for a Healthy Future

https://digitalcommons.pcom.edu/bridging_gaps2014/1025/thumbnail.jp

Precise Vacuum Stability Bound in the Standard Model

In the standard model, a lower bound to the Higgs mass (for a given top quark mass) exists if one requires that the standard model vacuum be stable. This bound is calculated as precisely as possible, including the most recent values of the gauge couplings, corrected two-loop beta functions and radiative corrections to the Higgs and top masses. In addition to being somewhat more precise, this work differs from previous calculations in that the bounds are given in terms of the poles of the Higgs and top quark propagators, rather than ''the MS-bar top quark mass''. This difference can be as large as 6-10 GeV for the top mass, which corresponds to as much as 15 GeV for the Higgs mass lower bound. Concentrating on the top quark mass region from 130 to 150 GeV, I find that for $\alpha_s=0.117$, $$m_H > 75 {\rm GeV} + 1.64 (m_t - 140 {\rm GeV}).$$ This result increases (decreases) by 3 GeV if the strong coupling decreases (increases) by 0.007, and is accurate to 2 GeV. If one allows for the standard model vacuum to be unstable, then weaker bounds can be obtained.Comment: 9 pages, WM-93-108, in Plain Tex, phyzzx macropackage added at the beginnin

Parameterised and Fine-Grained Subgraph Counting, Modulo 2

Given a class of graphs ?, the problem ?Sub(?) is defined as follows. The input is a graph H ? ? together with an arbitrary graph G. The problem is to compute, modulo 2, the number of subgraphs of G that are isomorphic to H. The goal of this research is to determine for which classes ? the problem ?Sub(?) is fixed-parameter tractable (FPT), i.e., solvable in time f(|H|)?|G|^O(1). Curticapean, Dell, and Husfeldt (ESA 2021) conjectured that ?Sub(?) is FPT if and only if the class of allowed patterns ? is matching splittable, which means that for some fixed B, every H ? ? can be turned into a matching (a graph in which every vertex has degree at most 1) by removing at most B vertices. Assuming the randomised Exponential Time Hypothesis, we prove their conjecture for (I) all hereditary pattern classes ?, and (II) all tree pattern classes, i.e., all classes ? such that every H ? ? is a tree. We also establish almost tight fine-grained upper and lower bounds for the case of hereditary patterns (I)

Parameterised and fine-grained subgraph counting, modulo 2

Given a class of graphs H, the problem ⊕Sub(H) is defined as follows. The input is a graph H ∈ H together with an arbitrary graph G. The problem is to compute, modulo 2, the number of subgraphs of G that are isomorphic to H. The goal of this research is to determine for which classes H the problem ⊕Sub(H) is fixed-parameter tractable (FPT), i.e., solvable in time f(|H|) · |G| O(1). Curticapean, Dell, and Husfeldt (ESA 2021) conjectured that ⊕Sub(H) is FPT if and only if the class of allowed patterns H is matching splittable, which means that for some fixed B, every H ∈ H can be turned into a matching (a graph in which every vertex has degree at most 1) by removing at most B vertices. Assuming the randomised Exponential Time Hypothesis, we prove their conjecture for (I) all hereditary pattern classes H, and (II) all tree pattern classes, i.e., all classes H such that every H ∈ H is a tree. We also establish almost tight fine-grained upper and lower bounds for the case of hereditary patterns (I)

The Weisfeiler-Leman dimension of conjunctive queries

A graph parameter is a function on graphs with the property that, for any pair of isomorphic graphs 1 and 2, (1) = (2). The Weisfeiler–Leman (WL) dimension of is the minimum such that, if 1 and 2 are indistinguishable by the -dimensional WL-algorithm then (1) = (2). The WL-dimension of is ∞ if no such exists. We study the WL-dimension of graph parameters characterised by the number of answers from a fixed conjunctive query to the graph. Given a conjunctive query , we quantify the WL-dimension of the function that maps every graph to the number of answers of in . The works of Dvorák (J. Graph Theory 2010), Dell, Grohe, and Rattan (ICALP 2018), and Neuen (ArXiv 2023) have answered this question for full conjunctive queries, which are conjunctive queries without existentially quantified variables. For such queries , the WL-dimension is equal to the treewidth of the Gaifman graph of . In this work, we give a characterisation that applies to all conjunctive queries. Given any conjunctive query , we prove that its WL-dimension is equal to the semantic extension width sew(), a novel width measure that can be thought of as a combination of the treewidth of and its quantified star size, an invariant introduced by Durand and Mengel (ICDT 2013) describing how the existentially quantified variables of are connected with the free variables. Using the recently established equivalence between the WL-algorithm and higher-order Graph Neural Networks (GNNs) due to Morris et al. (AAAI 2019), we obtain as a consequence that the function counting answers to a conjunctive query cannot be computed by GNNs of order smaller than sew(). The majority of the paper is concerned with establishing a lower bound of the WL-dimension of a query. Given any conjunctive query with semantic extension width , we consider a graph of treewidth obtained from the Gaifman graph of by repeatedly cloning the vertices corresponding to existentially quantified variables. Using a modification due to Fürer (ICALP 2001) of the Cai-Fürer-Immerman construction (Combinatorica 1992), we then obtain a pair of graphs ( ) and ˆ( ) that are indistinguishable by the ( − 1)- dimensional WL-algorithm since has treewidth . Finally, in the technical heart of the paper, we show that has a different number of answers in ( ) and ˆ( ). Thus, can distinguish two graphs that cannot be distinguished by the ( − 1)-dimensional WL-algorithm, so the WL-dimension of is at least

The Weisfeiler-Leman Dimension of Existential Conjunctive Queries

The Weisfeiler-Leman (WL) dimension of a graph parameter $f$ is the minimum $k$ such that, if $G_1$ and $G_2$ are indistinguishable by the $k$-dimensional WL-algorithm then $f(G_1)=f(G_2)$. The WL-dimension of $f$ is $\infty$ if no such $k$ exists. We study the WL-dimension of graph parameters characterised by the number of answers from a fixed conjunctive query to the graph. Given a conjunctive query $\varphi$, we quantify the WL-dimension of the function that maps every graph $G$ to the number of answers of $\varphi$ in $G$. The works of Dvor\'ak (J. Graph Theory 2010), Dell, Grohe, and Rattan (ICALP 2018), and Neuen (ArXiv 2023) have answered this question for full conjunctive queries, which are conjunctive queries without existentially quantified variables. For such queries $\varphi$, the WL-dimension is equal to the treewidth of the Gaifman graph of $\varphi$. In this work, we give a characterisation that applies to all conjunctive qureies. Given any conjunctive query $\varphi$, we prove that its WL-dimension is equal to the semantic extension width $\mathsf{sew}(\varphi)$, a novel width measure that can be thought of as a combination of the treewidth of $\varphi$ and its quantified star size, an invariant introduced by Durand and Mengel (ICDT 2013) describing how the existentially quantified variables of $\varphi$ are connected with the free variables. Using the recently established equivalence between the WL-algorithm and higher-order Graph Neural Networks (GNNs) due to Morris et al. (AAAI 2019), we obtain as a consequence that the function counting answers to a conjunctive query $\varphi$ cannot be computed by GNNs of order smaller than $\mathsf{sew}(\varphi)$.Comment: 36 pages, 4 figures, abstract shortened due to ArXiv requirement

Parameterised Approximation of the Fixation Probability of the Dominant Mutation in the Multi-Type Moran Process

The multi-type Moran process is an evolutionary process on a connected graph $G$ in which each vertex has one of $k$ types and, in each step, a vertex $v$ is chosen to reproduce its type to one of its neighbours. The probability of a vertex $v$ being chosen for reproduction is proportional to the fitness of the type of $v$. So far, the literature was almost solely concerned with the $2$-type Moran process in which each vertex is either healthy (type $0$) or a mutant (type $1$), and the main problem of interest has been the (approximate) computation of the so-called fixation probability, i.e., the probability that eventually all vertices are mutants. In this work we initiate the study of approximating fixation probabilities in the multi-type Moran process on general graphs. Our main result is an FPTRAS (fixed-parameter tractable randomised approximation scheme) for computing the fixation probability of the dominant mutation; the parameter is the number of types and their fitnesses. In the course of our studies we also provide novel upper bounds on the expected absorption time, i.e., the time that it takes the multi-type Moran process to reach a state in which each vertex has the same type.Comment: 14 page

Energy Choices Revisited : An Examination of the Costs and Benefits of Maine\u27s Energy Policy

https://digitalmaine.com/mainewatch_publications/1003/thumbnail.jp

Counting Subgraphs in Somewhere Dense Graphs

We study the problems of counting copies and induced copies of a small pattern graph $H$ in a large host graph $G$. Recent work fully classified the complexity of those problems according to structural restrictions on the patterns $H$. In this work, we address the more challenging task of analysing the complexity for restricted patterns and restricted hosts. Specifically we ask which families of allowed patterns and hosts imply fixed-parameter tractability, i.e., the existence of an algorithm running in time $f(H)\cdot |G|^{O(1)}$ for some computable function $f$. Our main results present exhaustive and explicit complexity classifications for families that satisfy natural closure properties. Among others, we identify the problems of counting small matchings and independent sets in subgraph-closed graph classes $\mathcal{G}$ as our central objects of study and establish the following crisp dichotomies as consequences of the Exponential Time Hypothesis: (1) Counting $k$-matchings in a graph $G\in\mathcal{G}$ is fixed-parameter tractable if and only if $\mathcal{G}$ is nowhere dense. (2) Counting $k$-independent sets in a graph $G\in\mathcal{G}$ is fixed-parameter tractable if and only if $\mathcal{G}$ is nowhere dense. Moreover, we obtain almost tight conditional lower bounds if $\mathcal{G}$ is somewhere dense, i.e., not nowhere dense. These base cases of our classifications subsume a wide variety of previous results on the matching and independent set problem, such as counting $k$-matchings in bipartite graphs (Curticapean, Marx; FOCS 14), in $F$-colourable graphs (Roth, Wellnitz; SODA 20), and in degenerate graphs (Bressan, Roth; FOCS 21), as well as counting $k$-independent sets in bipartite graphs (Curticapean et al.; Algorithmica 19).Comment: 35 pages, 3 figures, 4 tables, abstract shortened due to ArXiv requirement

Counting Homomorphisms to K4K_4-minor-free Graphs, modulo 2

We study the problem of computing the parity of the number of homomorphisms from an input graph $G$ to a fixed graph $H$. Faben and Jerrum [ToC'15] introduced an explicit criterion on the graph $H$ and conjectured that, if satisfied, the problem is solvable in polynomial time and, otherwise, the problem is complete for the complexity class $\oplus\mathrm{P}$ of parity problems. We verify their conjecture for all graphs $H$ that exclude the complete graph on $4$ vertices as a minor. Further, we rule out the existence of a subexponential-time algorithm for the $\oplus\mathrm{P}$-complete cases, assuming the randomised Exponential Time Hypothesis. Our proofs introduce a novel method of deriving hardness from globally defined substructures of the fixed graph $H$. Using this, we subsume all prior progress towards resolving the conjecture (Faben and Jerrum [ToC'15]; G\"obel, Goldberg and Richerby [ToCT'14,'16]). As special cases, our machinery also yields a proof of the conjecture for graphs with maximum degree at most $3$, as well as a full classification for the problem of counting list homomorphisms, modulo $2$