Two methods for the generation of chordal graphs

In this paper two methods for automatic generation of connected chordal graphs are proposed: the first one is based on results concerning the dynamic maintainance of chordality under edge insertions; the second is based on expansion/merging of maximal cliques. In both methods, chordality is preserved along the whole generation process

Succinct Data Structures for Chordal Graphs

We study the problem of approximate shortest path queries in chordal graphs and give a n log n + o(n log n) bit data structure to answer the approximate distance query to within an additive constant of 1 in O(1) time. We study the problem of succinctly storing a static chordal graph to answer adjacency, degree, neighbourhood and shortest path queries. Let G be a chordal graph with n vertices. We design a data structure using the information theoretic minimal n^2/4 + o(n^2) bits of space to support the queries: - whether two vertices u,v are adjacent in time f(n) for any f(n) in omega(1). - the degree of a vertex in O(1) time. - the vertices adjacent to u in (f(n))^2 time per neighbour - the length of the shortest path from u to v in O(nf(n)) tim

最小費用全域木ゲームのShapley値に対する近似アルゴリズム (高度情報化社会に向けた数理最適化の新潮流)

最小費用全域木ゲームはそれを定義するネットワークの費用関数が木距離である場合には木距離最小費用全域木ゲームと呼ばれる. 一般の最小費用全域木ゲームのShapley値の計算は#P-困難であるが, 木距離最小費用全域木ゲームのShapley値は多項式時間で計算できることが知られている. 本研究では, 木距離最小費用全域木ゲームに対する多項式時間アルゴリズムのアイデアに基づいて, 一般の最小費用全域木ゲームのShapley値に対する多項式時間近似アルゴリズムを導入した. このアルゴリズムの近似精度を評価するためにランダムに生成した費用関数を入力として数値実験を行った結果, 与えられた費用関数が2次元ユークリッド距離の場合には最大でも14%程度の相対誤差を持つということが観察された

Fast minimal triangulation algorithm using minimum degree criterion

AbstractWe propose an algorithm for minimal triangulation which, using simple and efficient strategy, subdivides the input graph in different, almost non-overlapping, subgraphs. Using the technique of matrix multiplication for saturating the minimal separators, we show that the partition of the graph can be computed in time O(nα) where nα is the time required by the binary matrix multiplication. After saturating the minimal separators, the same procedure is recursively applied on each subgraphs. We also present a variant of the algorithm in which the minimum degree criterion is used. In this way, we obtain an algorithm that uses minimum degree criterion and at the same time produces a minimal triangulation, thus shedding new light on the effectiveness of the minimum degree heuristics

Recognition of split-graphic sequences

Using different definitions of split graphs we propose quick algorithms for the recognition and extremal reconstruction of split sequences among integer, regular, and graphic sequences

Discrete parametric graphical models with Dirichlet type priors

Typically, statistical graphical models are either continuous and parametric (Gaussian, parameterized by the graph-dependent precision matrix) or discrete and non-parametric (with graph-dependent probabilities of cells). Eventually, the two types are mixed. We propose a way to break this dichotomy by introducing two discrete parametric graphical models on finite decomposable graphs: the graph negative multinomial and the graph multinomial distributions. These models interpolate between the product of univariate negative multinomial and negative multinomial distributions, and between the product of binomial and multinomial distributions, respectively. We derive their Markov decomposition and present probabilistic models leading to both. Additionally, we introduce graphical versions of the Dirichlet distribution and inverted Dirichlet distribution, which serve as conjugate priors for the two discrete graphical Markov models. We derive explicit normalizing constants for both graphical Dirichlet laws and demonstrate that their independence structure (a graphical version of neutrality) yields a strong hyper Markov property for both Bayesian models. We also provide characterization theorems for the generalized Dirichlet distributions via strong hyper Markov property. Finally, we develop a Bayesian model selection procedure for the graphical negative multinomial model with respective Dirichlet-type priors.Comment: 36 page

Succinct Data Structures for Chordal Graphs

We study the problem of approximate shortest path queries in chordal graphs and give a n log n + o(n log n) bit data structure to answer the approximate distance query to within an additive constant of 1 in O(1) time. We study the problem of succinctly storing a static chordal graph to answer adjacency, degree, neighbourhood and shortest path queries. Let G be a chordal graph with n vertices. We design a data structure using the information theoretic minimal n^2/4 + o(n^2) bits of space to support the queries: whether two vertices u,v are adjacent in time f(n) for any f(n) \in \omega(1). the degree of a vertex in O(1) time. the vertices adjacent to u in O(f(n)^2) time per neighbour the length of the shortest path from u to v in O(n f(n)) tim

Linear time algorithms for graphs close to chordal graphs.

Ho Man Lam.Thesis (M.Phil.)--Chinese University of Hong Kong, 2003.Includes bibliographical references (leaves 51-54).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.1Chapter 1.1 --- Statement of problems --- p.1Chapter 1.2 --- Notation and definitions --- p.3Chapter 1.3 --- Graph families --- p.4Chapter 1.4 --- Related work --- p.5Chapter 1.4.1 --- Graph modification problems --- p.5Chapter 1.4.2 --- Independent set --- p.6Chapter 1.5 --- Overview of the thesis --- p.7Chapter 2 --- Recognition of Nearly Chordal Graphs --- p.8Chapter 2.1 --- Critical edges not in triangles --- p.9Chapter 2.2 --- Critical edges in triangles --- p.10Chapter 2.3 --- A linear time algorithm --- p.13Chapter 3 --- Recognition of Almost Chordal Graphs --- p.15Chapter 3.1 --- Minimal separator --- p.16Chapter 3.2 --- "All chordless cycles passing through the minimal (x, z)-separator S" --- p.18Chapter 3.3 --- Algorithm for almost chordal graphs recognition --- p.22Chapter 3.4 --- Another approach to find a critical vertex if all chordless cycles pass through S --- p.26Chapter 3.5 --- A linear algorithm for all chordless cycles passing through S --- p.28Chapter 4 --- Maximum Independent Bases of Chordal Graphs --- p.32Chapter 4.1 --- Maximum independent base --- p.32Chapter 4.1.1 --- Finding a maximum independent set of a chordal graph . --- p.33Chapter 4.1.2 --- Another approach to prove the algorithm --- p.33Chapter 4.1.3 --- Maximum independent base --- p.34Chapter 4.1.4 --- Vertices in the maximum independent base --- p.36Chapter 4.1.5 --- A linear time algorithm --- p.38Chapter 4.2 --- Generating all maximum independent sets --- p.39Chapter 4.2.1 --- Relation between two maximum independent sets --- p.39Chapter 4.2.2 --- Algorithm --- p.40Chapter 4.3 --- Maximum induced split graph of a chordal graph --- p.43Chapter 4.3.1 --- Property of maximum induced split subgraph --- p.44Chapter 4.3.2 --- A linear time algorithm --- p.45Chapter 5 --- Concluding Remarks --- p.48Chapter 5.1 --- Summary of results --- p.48Chapter 5.2 --- Open problems --- p.48Bibliography --- p.5