The Epstein-Glaser approach to pQFT: graphs and Hopf algebras

The paper aims at investigating perturbative quantum field theory (pQFT) in the approach of Epstein and Glaser (EG) and, in particular, its formulation in the language of graphs and Hopf algebras (HAs). Various HAs are encountered, each one associated with a special combination of physical concepts such as normalization, localization, pseudo-unitarity, causality and an associated regularization, and renormalization. The algebraic structures, representing the perturbative expansion of the S-matrix, are imposed on the operator-valued distributions which are equipped with appropriate graph indices. Translation invariance ensures the algebras to be analytically well-defined and graded total symmetry allows to formulate bialgebras. The algebraic results are given embedded in the physical framework, which covers the two recent EG versions by Fredenhagen and Scharf that differ with respect to the concrete recursive implementation of causality. Besides, the ultraviolet divergences occuring in Feynman's representation are mathematically reasoned. As a final result, the change of the renormalization scheme in the EG framework is modeled via a HA which can be seen as the EG-analog of Kreimer's HA.Comment: 52 pages, 5 figure

Engineering Fused Lasso Solvers on Trees

The graph fused lasso optimization problem seeks, for a given input signal y=(y_i) on nodes i? V of a graph G=(V,E), a reconstructed signal x=(x_i) that is both element-wise close to y in quadratic error and also has bounded total variation (sum of absolute differences across edges), thereby favoring regionally constant solutions. An important application is denoising of spatially correlated data, especially for medical images. Currently, fused lasso solvers for general graph input reduce the problem to an iteration over a series of "one-dimensional" problems (on paths or line graphs), which can be solved in linear time. Recently, a direct fused lasso algorithm for tree graphs has been presented, but no implementation of it appears to be available. We here present a simplified exact algorithm and additionally a fast approximation scheme for trees, together with engineered implementations for both. We empirically evaluate their performance on different kinds of trees with distinct degree distributions (simulated trees; spanning trees of road networks, grid graphs of images, social networks). The exact algorithm is very efficient on trees with low node degrees, which covers many naturally arising graphs, while the approximation scheme can perform better on trees with several higher-degree nodes when limiting the desired accuracy to values that are useful in practice

On the algorithmic complexity of twelve covering and independence parameters of graphs

The definitions of four previously studied parameters related to total coverings and total matchings of graphs can be restricted, thereby obtaining eight parameters related to covering and independence, each of which has been studied previously in some form. Here we survey briefly results concerning total coverings and total matchings of graphs, and consider the aforementioned 12 covering and independence parameters with regard to algorithmic complexity. We survey briefly known results for several graph classes, and obtain new NP-completeness results for the minimum total cover and maximum minimal total cover problems in planar graphs, the minimum maximal total matching problem in bipartite and chordal graphs, and the minimum independent dominating set problem in planar cubic graphs

무지개 집합 문제에서의 위상수학적 조합론

학위논문(박사)--서울대학교 대학원 :자연과학대학 수리과학부,2019. 8. 국웅.$\mathcal{F}=\{S_1,\ldots,S_m\}$를 $V$의 공집합이 아닌 부분 집합들의 모임이라 할 때, $\mathcal{F}$의 무지개 집합이란 공집합이 아니며 $S=\{s_{i_1},\ldots,s_{i_k}\} \subset V$와 같은 형태로 주어지는 것으로 다음 조건을 만족하는 것을 말한다. $1\leq i_1<\cdots<i_k \leq m$이고 $j \ne j$이면 $s_{i_j} \ne s_{i_j'}$를 만족하며 각 $j \in [m]$에 대해 $s_{i_j} \in S_{i_j}$이다. 특히 $k=m$인 경우, 즉 모든 $S_i$들이 표현되면, 무지개 집합 $S$를 $\mathcal{F}$의 완전 무지개 집합이라고 한다. 주어진 집합계가 특정 조건을 만족하는 무지개 집합을 가지기 위한 충분 조건을 찾는 문제는 홀의 결혼 정리에서 시작되어 최근까지도 조합수학에서 가장 대표적 문제 중 하나로 여겨져왔다. 이러한 방향으로의 문제를 무지개 집합 문제라고 부른다. 본 학위논문에서는 무지개 집합 문제와 관련하여 위상수학적 홀의 정리와 위상수학적 다색 헬리 정리를 소개하고, (하이퍼)그래프에서의 무지개 덮개와 무지개 독립 집합에 관한 결과들을 다루고자 한다.Let $\mathcal{F}=\{S_1,\ldots,S_m\}$ be a finite family of non-empty subsets on the ground set $V$. A rainbow set of $\mathcal{F}$ is a non-empty set of the form $S=\{s_{i_1},\ldots,s_{i_k}\} \subset V$ with $1 \leq i_1 < \cdots < i_k \leq m$ such that $s_{i_j} \neq s_{i_{j'}}$ for every $j \neq j'$ and $s_{i_j} \in S_{i_j}$ for each $j \in [k]$. If $k = m$, namely if all $S_i$ is represented, then the rainbow set $S$ is called a full rainbow set of $\mathcal{F}$. Originated from the celebrated Hall's marriage theorem, it has been one of the most fundamental questions in combinatorics and discrete mathematics to find sufficient conditions on set-systems to guarantee the existence of certain rainbow sets. We call problems in this direction the rainbow set problems. In this dissertation, we give an overview on two topological tools on rainbow set problems, Aharoni and Haxell's topological Hall theorem and Kalai and Meshulam's topological colorful Helly theorem, and present some results on and rainbow independent sets and rainbow covers in (hyper)graphs.Abstract i 1 Introduction 1 1.1 Topological Hall theorem . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Topological colorful Helly theorem . . . . . . . . . . . . . . . . . 3 1.2.1 Collapsibility and Lerayness of simplicial complexes . . . 4 1.2.2 Nerve theorem and topological Helly theorem . . . . . . . 5 1.2.3 Topological colorful Helly theorem . . . . . . . . . . . . 6 1.3 Domination numbers and non-cover complexes of hypergraphs . . 7 1.3.1 Domination numbers of hypergraphs . . . . . . . . . . . . 10 1.3.2 Non-cover complexes of hypergraphs . . . . . . . . . . . . 10 1.4 Rainbow independent sets in graphs . . . . . . . . . . . . . . . . 12 1.5 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2 Collapsibility of non-cover complexes of graphs 16 2.1 The minimal exclusion sequences . . . . . . . . . . . . . . . . . . 16 2.2 Independent domination numbers and collapsibility numbers of non-cover complexes of graphs . . . . . . . . . . . . . . . . . . . 21 3 Domination numbers and non-cover complexes of hypergraphs 24 3.1 Proof of Theorem 1.3.4 . . . . . . . . . . . . . . . . . . . . . . . 25 3.1.1 Edge-annihilation . . . . . . . . . . . . . . . . . . . . . . 25 3.1.2 Non-cover complexes for hypergraphs . . . . . . . . . . . 27 3.2 Lerayness of non-cover complexes . . . . . . . . . . . . . . . . . 30 3.2.1 Total domination numbers . . . . . . . . . . . . . . . . . 30 3.2.2 Independent domination numbers . . . . . . . . . . . . . 33 3.2.3 Edgewise-domination numbers . . . . . . . . . . . . . . . 34 3.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3.1 Independent domination numbers of hypergraphs . . . . . 35 3.3.2 Independence complexes of hypergraphs . . . . . . . . . . 36 3.3.3 General position complexes . . . . . . . . . . . . . . . . . 37 3.3.4 Rainbow covers of hypergraphs . . . . . . . . . . . . . . 39 3.3.5 Collapsibility of non-cover complexes of hypergraphs . . . 40 4 Rainbow independent sets 42 4.1 Graphs avoiding certain induced subgraphs . . . . . . . . . . . . 42 4.1.1 Claw-free graphs . . . . . . . . . . . . . . . . . . . . . . 42 4.1.2 $\{C_4,C_5, . . . ,C_s\}$-free graphs . . . . . . . . . . . . . . . . . 44 4.1.3 Chordal graphs . . . . . . . . . . . . . . . . . . . . . . . 49 4.1.4 $K_r$-free graphs and $K^{−}_r$-free graphs . . . . . . . . . . . . . 50 4.2 $k$-colourable graphs . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.3 Graphs with bounded degrees . . . . . . . . . . . . . . . . . . . . 55 4.3.1 The case $m < n$ . . . . . . . . . . . . . . . . . . . . . . . 56 4.4 A topological approach . . . . . . . . . . . . . . . . . . . . . . . 64 4.5 Concluding remark . . . . . . . . . . . . . . . . . . . . . . . . . 67 Abstract (in Korean) 69 Acknowledgement (in Korean) 70Docto

On maximal chain subgraphs and covers of bipartite graphs

In this paper, we address three related problems. One is the enumeration of all the maximal edge induced chain subgraphs of a bipartite graph, for which we provide a polynomial delay algorithm. We give bounds on the number of maximal chain subgraphs for a bipartite graph and use them to establish the input-sensitive complexity of the enumeration problem. The second problem we treat is the one of finding the minimum number of chain subgraphs needed to cover all the edges a bipartite graph. For this we provide an exact exponential algorithm with a non trivial complexity. Finally, we approach the problem of enumerating all minimal chain subgraph covers of a bipartite graph and show that it can be solved in quasi-polynomial time

Identifying codes of corona product graphs

For a vertex $x$ of a graph $G$, let $N_G[x]$ be the set of $x$ with all of its neighbors in $G$. A set $C$ of vertices is an {\em identifying code} of $G$ if the sets $N_G[x]\cap C$ are nonempty and distinct for all vertices $x$. If $G$ admits an identifying code, we say that $G$ is identifiable and denote by $\gamma^{ID}(G)$ the minimum cardinality of an identifying code of $G$. In this paper, we study the identifying code of the corona product $H\odot G$ of graphs $H$ and $G$. We first give a necessary and sufficient condition for the identifiable corona product $H\odot G$, and then express $\gamma^{ID}(H\odot G)$ in terms of $\gamma^{ID}(G)$ and the (total) domination number of $H$. Finally, we compute $\gamma^{ID}(H\odot G)$ for some special graphs $G$

Subgraph covers -- An information theoretic approach to motif analysis in networks

Many real world networks contain a statistically surprising number of certain subgraphs, called network motifs. In the prevalent approach to motif analysis, network motifs are detected by comparing subgraph frequencies in the original network with a statistical null model. In this paper we propose an alternative approach to motif analysis where network motifs are defined to be connectivity patterns that occur in a subgraph cover that represents the network using minimal total information. A subgraph cover is defined to be a set of subgraphs such that every edge of the graph is contained in at least one of the subgraphs in the cover. Some recently introduced random graph models that can incorporate significant densities of motifs have natural formulations in terms of subgraph covers and the presented approach can be used to match networks with such models. To prove the practical value of our approach we also present a heuristic for the resulting NP-hard optimization problem and give results for several real world networks.Comment: 10 pages, 7 tables, 1 Figur