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

Operations research: from computational biology to sensor network

In this dissertation we discuss the deployment of combinatorial optimization methods for modeling and solve real life problemS, with a particular emphasis to two biological problems arising from a common scenario: the reconstruction of the three-dimensional shape of a biological molecule from Nuclear Magnetic Resonance (NMR) data. The fi rst topic is the 3D assignment pathway problem (APP) for a RNA molecule. We prove that APP is NP-hard, and show a formulation of it based on edge-colored graphs. Taking into account that interactions between consecutive nuclei in the NMR spectrum are diff erent according to the type of residue along the RNA chain, each color in the graph represents a type of interaction. Thus, we can represent the sequence of interactions as the problem of fi nding a longest (hamiltonian) path whose edges follow a given order of colors (i.e., the orderly colored longest path). We introduce three alternative IP formulations of APP obtained with a max flow problem on a directed graph with packing constraints over the partitions, which have been compared among themselves. Since the last two models work on cyclic graphs, for them we proposed an algorithm based on the solution of their relaxation combined with the separation of cycle inequalities in a Branch & Cut scheme. The second topic is the discretizable distance geometry problem (DDGP), which is a formulation on discrete search space of the well-known distance geometry problem (DGP). The DGP consists in seeking the embedding in the space of a undirected graph, given a set of Euclidean distances between certain pairs of vertices. DGP has two important applications: (i) fi nding the three dimensional conformation of a molecule from a subset of interatomic distances, called Molecular Distance Geometry Problem, and (ii) the Sensor Network Localization Problem. We describe a Branch & Prune (BP) algorithm tailored for this problem, and two versions of it solving the DDGP both in protein modeling and in sensor networks localization frameworks. BP is an exact and exhaustive combinatorial algorithm that examines all the valid embeddings of a given weighted graph G=(V,E,d), under the hypothesis of existence of a given order on V. By comparing the two version of BP to well-known algorithms we are able to prove the e fficiency of BP in both contexts, provided that the order imposed on V is maintained

The bidimensionality theory and its algorithmic applications

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005.Includes bibliographical references (p. 201-219).Our newly developing theory of bidimensional graph problems provides general techniques for designing efficient fixed-parameter algorithms and approximation algorithms for NP- hard graph problems in broad classes of graphs. This theory applies to graph problems that are bidimensional in the sense that (1) the solution value for the k x k grid graph (and similar graphs) grows with k, typically as Q(k²), and (2) the solution value goes down when contracting edges and optionally when deleting edges. Examples of such problems include feedback vertex set, vertex cover, minimum maximal matching, face cover, a series of vertex- removal parameters, dominating set, edge dominating set, r-dominating set, connected dominating set, connected edge dominating set, connected r-dominating set, and unweighted TSP tour (a walk in the graph visiting all vertices). Bidimensional problems have many structural properties; for example, any graph embeddable in a surface of bounded genus has treewidth bounded above by the square root of the problem's solution value. These properties lead to efficient-often subexponential-fixed-parameter algorithms, as well as polynomial-time approximation schemes, for many minor-closed graph classes. One type of minor-closed graph class of particular relevance has bounded local treewidth, in the sense that the treewidth of a graph is bounded above in terms of the diameter; indeed, we show that such a bound is always at most linear. The bidimensionality theory unifies and improves several previous results.(cont.) The theory is based on algorithmic and combinatorial extensions to parts of the Robertson-Seymour Graph Minor Theory, in particular initiating a parallel theory of graph contractions. The foundation of this work is the topological theory of drawings of graphs on surfaces and our results regarding the relation (the linearity) of the size of the largest grid minor in terms of treewidth in bounded-genus graphs and more generally in graphs excluding a fixed graph H as a minor. In this thesis, we also develop the algorithmic theory of vertex separators, and its relation to the embeddings of certain metric spaces. Unlike in the edge case, we show that embeddings into L₁ (and even Euclidean embeddings) are insufficient, but that the additional structure provided by many embedding theorems does suffice for our purposes. We obtain an O[sq. root( log n)] approximation for min-ratio vertex cuts in general graphs, based on a new semidefinite relaxation of the problem, and a tight analysis of the integrality gap which is shown to be [theta][sq. root(log n)]. We also prove various approximate max-flow/min-vertex- cut theorems, which in particular give a constant-factor approximation for min-ratio vertex cuts in any excluded-minor family of graphs. Previously, this was known only for planar graphs, and for general excluded-minor families the best-known ratio was O(log n). These results have a number of applications. We exhibit an O[sq. root (log n)] pseudo-approximation for finding balanced vertex separators in general graphs.(cont.) Furthermore, we obtain improved approximation ratios for treewidth: In any graph of treewidth k, we show how to find a tree decomposition of width at most O(k[sq. root(log k)]), whereas previous algorithms yielded O(k log k). For graphs excluding a fixed graph as a minor, we give a constant-factor approximation for the treewidth; this via the bidimensionality theory can be used to obtain the first polynomial-time approximation schemes for problems like minimum feedback vertex set and minimum connected dominating set in such graphs.by MohammadTaghi Hajiaghayi.Ph.D

Solving hard subgraph problems in parallel

This thesis improves the state of the art in exact, practical algorithms for finding subgraphs. We study maximum clique, subgraph isomorphism, and maximum common subgraph problems. These are widely applicable: within computing science, subgraph problems arise in document clustering, computer vision, the design of communication protocols, model checking, compiler code generation, malware detection, cryptography, and robotics; beyond, applications occur in biochemistry, electrical engineering, mathematics, law enforcement, fraud detection, fault diagnosis, manufacturing, and sociology. We therefore consider both the ``pure'' forms of these problems, and variants with labels and other domain-specific constraints. Although subgraph-finding should theoretically be hard, the constraint-based search algorithms we discuss can easily solve real-world instances involving graphs with thousands of vertices, and millions of edges. We therefore ask: is it possible to generate ``really hard'' instances for these problems, and if so, what can we learn? By extending research into combinatorial phase transition phenomena, we develop a better understanding of branching heuristics, as well as highlighting a serious flaw in the design of graph database systems. This thesis also demonstrates how to exploit two of the kinds of parallelism offered by current computer hardware. Bit parallelism allows us to carry out operations on whole sets of vertices in a single instruction---this is largely routine. Thread parallelism, to make use of the multiple cores offered by all modern processors, is more complex. We suggest three desirable performance characteristics that we would like when introducing thread parallelism: lack of risk (parallel cannot be exponentially slower than sequential), scalability (adding more processing cores cannot make runtimes worse), and reproducibility (the same instance on the same hardware will take roughly the same time every time it is run). We then detail the difficulties in guaranteeing these characteristics when using modern algorithmic techniques. Besides ensuring that parallelism cannot make things worse, we also increase the likelihood of it making things better. We compare randomised work stealing to new tailored strategies, and perform experiments to identify the factors contributing to good speedups. We show that whilst load balancing is difficult, the primary factor influencing the results is the interaction between branching heuristics and parallelism. By using parallelism to explicitly offset the commitment made to weak early branching choices, we obtain parallel subgraph solvers which are substantially and consistently better than the best sequential algorithms

Distances and Domination in Graphs

This book presents a compendium of the 10 articles published in the recent Special Issue “Distance and Domination in Graphs”. The works appearing herein deal with several topics on graph theory that relate to the metric and dominating properties of graphs. The topics of the gathered publications deal with some new open lines of investigations that cover not only graphs, but also digraphs. Different variations in dominating sets or resolving sets are appearing, and a review on some networks’ curvatures is also present

Congress UPV Proceedings of the 21ST International Conference on Science and Technology Indicators

This is the book of proceedings of the 21st Science and Technology Indicators Conference that took place in València (Spain) from 14th to 16th of September 2016. The conference theme for this year, ‘Peripheries, frontiers and beyond’ aimed to study the development and use of Science, Technology and Innovation indicators in spaces that have not been the focus of current indicator development, for example, in the Global South, or the Social Sciences and Humanities. The exploration to the margins and beyond proposed by the theme has brought to the STI Conference an interesting array of new contributors from a variety of fields and geographies. This year’s conference had a record 382 registered participants from 40 different countries, including 23 European, 9 American, 4 Asia-Pacific, 4 Africa and Near East. About 26% of participants came from outside of Europe. There were also many participants (17%) from organisations outside academia including governments (8%), businesses (5%), foundations (2%) and international organisations (2%). This is particularly important in a field that is practice-oriented. The chapters of the proceedings attest to the breadth of issues discussed. Infrastructure, benchmarking and use of innovation indicators, societal impact and mission oriented-research, mobility and careers, social sciences and the humanities, participation and culture, gender, and altmetrics, among others. We hope that the diversity of this Conference has fostered productive dialogues and synergistic ideas and made a contribution, small as it may be, to the development and use of indicators that, being more inclusive, will foster a more inclusive and fair world