An Introduction to Chordal Graphs and Clique Trees

Blair, Jean R. S.; Peyton, Barry W.

doi:10.1007/978-1-4613-8369-7_1

Cited by 302 publications

(301 citation statements)

References 34 publications

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“…, ±h m0 . Hence we can apply Theorem 3.20 and, therefore, there is a decomposition 12) where s 0 , s j are sums of squares and q ∈ J . To finish the proof we distinguish the two cases (i), (ii).…”

Section: Finite Convergencementioning

confidence: 99%

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Sums of Squares, Moment Matrices and Optimization Over Polynomials

Laurent

2008

Emerging Applications of Algebraic Geometry

614

703

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Abstract. We consider the problem of minimizing a polynomial over a semialgebraic set defined by polynomial equations and inequalities, which is NP-hard in general. Hierarchies of semidefinite relaxations have been proposed in the literature, involving positive semidefinite moment matrices and the dual theory of sums of squares of polynomials. We present these hierarchies of approximations and their main properties: asymptotic/finite convergence, optimality certificate, and extraction of global optimum solutions. We review the mathematical tools underlying these properties, in particular, some sums of squares representation results for positive polynomials, some results about moment matrices (in particular, of Curto and Fialkow), and the algebraic eigenvalue method for solving zero-dimensional systems of polynomial equations. We try whenever possible to provide detailed proofs and background.

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Section: Finite Convergencementioning

confidence: 99%

“…e.g. [12] for details about chordal graphs. The following strategy is proposed in [153] for identifying a sparsity structure like (8.2)-(8.3).…”

Section: Sums Of Squares Moments and Polynomial Optimization 69mentioning

confidence: 99%

Sums of Squares, Moment Matrices and Optimization Over Polynomials

Laurent

2008

Emerging Applications of Algebraic Geometry

614

703

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“…1 Add a universal vertex r / ∈ V (G ′ ) to the graph G ′ to obtain G; 2 Obtain a nice tree decomposition T of G as follows:; for every leaf bag of T , add {r} as a child-bag; 6 R = treedepth-rec(G, t + 1, T , X);…”

Section: Algorithm 1: Treedepthmentioning

confidence: 99%

“…First, check whether ω(G) > t and if that is the case, output that the treedepth of G is greater than t. Otherwise, ω (G) t which implies that tw(G) t. Compute a clique tree of G in linear time (cf. [2]), i.e. a tree decomposition of G in which every bag induces a clique.…”

Section: Treedepth and Chordal Graphsmentioning

confidence: 99%

A Faster Parameterized Algorithm for Treedepth

Reidl

Rossmanith

Villaamil

et al. 2014

Automata, Languages, and Programming

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The width measure treedepth, also known as vertex ranking, centered coloring and elimination tree height, is a well-established notion which has recently seen a resurgence of interest. We present an algorithm which-given as input an n-vertex graph, a tree decomposition of the graph of width w, and an integer t-decides Treedepth, i.e. whether the treedepth of the graph is at most t, in time 2 O(wt) · n. If necessary, a witness structure for the treedepth can be constructed in the same running time. In conjunction with previous results we provide a simple algorithm and a fast algorithm which decide treedepth in time 2 2 O(t) · n and 2 O(t 2 ) · n, respectively, which do not require a tree decomposition as part of their input. The former answers an open question posed by Ossona de Mendez and Nešetřil as to whether deciding Treedepth admits an algorithm with a linear running time (for every fixed t) that does not rely on Courcelle's Theorem or other heavy machinery. For chordal graphs we can prove a running time of 2 O(t log t) · n for the same algorithm. * Research funded by DFG-Project RO 927/13-1 "Pragmatic Parameterized Algorithms".

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“…In the centralized approach, Ma-STP is considered as a single STP and is solved incrementally by repeatedly applying single shoot algorithm, such as all pairs shortest path algorithms or an alternative algorithm Directed Path Consistency (DPC) [4], [5] or the L. Xu and B. Choueiry's algorithm, △ STP, in [6] that extended from Partial Path Consistency (PPC) algorithm [7] and triangulated algorithm [8], [9]. The other useful algorithm applying PPC-based algorithms is represented by Planken, Weerdt, and Krogt [10] called new algorithm P3C, which sweep the triangles process in a systematic order, resulting in an improved performance.…”

Section: Introductionmentioning

confidence: 99%

Distributed Algorithm for Incrementally Solving the Decoupled Multi-agent Simple Temporal Problem

Giap¹,

Hien²

2016

IJCTE

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Abstract-Applying temporal constraint in planning is a well-known problem, which keeps a plan is flexible until a specific schedule is generated. In this area, Decoupled Multi-agent simple temporal problem (DMaSTP) is suitably applied for planning of a multi-agents system. However, in scheduling problem, new events or temporal constraints are added regularly and force scheduler to check the consistency of exist MaSTP and retighten exist constraints. In this paper, we study a distributed scheduling algorithm for incrementally solving a DMaSTP. We have strongly considered the problem of adding a set of new constraints into a tightening consistent DMaSTP that tightened by also a distributed algorithm or set as empty. The algorithm checks whether the new adding constraints threaten the consistence of DMaSTP or not and decouple such new adding constraints when necessary and retighten DMaSTP. We have proposed a distributed algorithm, called DI-DMaSTP that solves the above problem and theoretically prove its correctness and outperformance, besides we have experienced with the variant datasets.Index Terms-Distributed algorithm, parallel, incremental, decoupled, multi-agents, simple temporal problem.

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An Introduction to Chordal Graphs and Clique Trees

Cited by 302 publications

References 34 publications

Sums of Squares, Moment Matrices and Optimization Over Polynomials

Sums of Squares, Moment Matrices and Optimization Over Polynomials

A Faster Parameterized Algorithm for Treedepth

Distributed Algorithm for Incrementally Solving the Decoupled Multi-agent Simple Temporal Problem

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