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This is the second paper in a series of two. The goal of the series is to give a polynomial-time algorithm for the 4-coloring problem and the 4-precoloring extension problem restricted to the class of graphs with no induced six-vertex path, thus proving a conjecture of Huang. Combined with previously known results, this completes the classification of the complexity of the 4-coloring problem for graphs with a connected forbidden induced subgraph. In this paper we give a polynomial time-algorithm that starts with a 4-precoloring of a graph with no induced six-vertex path and outputs a polynomial-sized collection of so-called excellent precolorings. Excellent precolorings are easier to handle than general ones, and, in addition, in order to determine whether the initial precoloring can be extended to the whole graph, it is enough to answer the same question for each of the excellent precolorings in the collection. The first paper in the series deals with excellent precolorings, thus providing a complete solution to the problem. [ABSTRACT FROM AUTHOR]

This is the first paper in a series whose goal is to give a polynomial-time algorithm for the 4-coloring problem and the 4-precoloring extension problem restricted to the class of graphs with no induced six-vertex path, thus proving a conjecture of Huang. Combined with previously known results, this completes the classification of the complexity of the 4-coloring problem for graphs with a connected forbidden induced subgraph. In this paper we give a polynomial-time algorithm that determines if a special kind of precoloring of a P6-free graph has a precoloring extension, and constructs such an extension if one exists. Combined with the main result of the second paper of the series, this gives a complete solution to the problem. [ABSTRACT FROM AUTHOR]

A systematic mathematical framework for the study of numerical algorithms would allow comparisons, facilitate conjugacy arguments, and enable the discovery of improved, accelerated, data-driven algorithms. Over the course of the past century, the Koopman operator has provided a mathematical framework for the study of dynamical systems which facilitates conjugacy arguments and can provide efficient reduced descriptions. More recently, numerical approximations of the operator have enabled the analysis of a large number of deterministic and stochastic dynamical systems in a completely data-driven, essentially equation-free pipeline. Discrete- or continuous-time numerical algorithms (integrators, nonlinear equation solvers, optimization algorithms are themselves dynamical systems. In this paper, we use this insight to leverage the Koopman operator framework in the data-driven study of such algorithms and discuss benefits for analysis and acceleration of numerical computation. For algorithms acting on high-dimensional spaces by quickly contracting them toward low-dimensional manifolds, we demonstrate how basis functions adapted to the data help to construct efficient reduced representations of the operator. Our illustrative examples include the gradient descent and Nesterov optimization algorithms as well as the Newton-Raphson algorithm. [ABSTRACT FROM AUTHOR]

Given a multigraph G=(V, E) with a positive rational weight w(e) on each edge e, the weighted density problem (WDP) is to find a subset $U$ of V, with |U| ≥ 3 and odd, that maximizes 2w(U)/(|U|-1), where w(U) is the total weight of all edges with both ends in U, and the weighted fractional edge-coloring problem can be formulated as the following linear program: minimize 1Tx subject to Ax = w, x ≥ 0, where $A$ is the edge-matching incidence matrix of $G$. These two problems are closely related to the celebrated Goldberg-Seymour conjecture on edge-colorings of multigraphs, and are interesting in their own right. Even when w(e) = 1 for all edges e, determining whether WDP can be solved in polynomial time was posed by Jensen and Toft [Topics in Chromatic Graph Theory, Cambridge University Press, Cambridge, 2015, pp. 327-357] and by Stiebitz et al. [Graph Edge Colouring: Vizing's Theorem and Goldberg's Conjecture, John Wiley, New York, 2012] as an open problem. In this paper we present strongly polynomial-time algorithms for solving them exactly, and develop a novel matching removal technique for multigraph edge-coloring. [ABSTRACT FROM AUTHOR]

Given a multigraph G = (V;E) with a positive rational weight w(e) on each edge e, the weighted density problem (WDP) is to find a subset U of V , with |U|≥3 and odd, that maximizes 2w(U)=(|U| - 1), where w(U) is the total weight of all edges with both ends in U, and the weighted fractional edge-coloring problem can be formulated as the following linear program: minimize 1T x subject to Ax = w, x ≥ 0, where A is the edge-matching incidence matrix of G. These two problems are closely related to the celebrated Goldberg{Seymour conjecture on edge-colorings of multigraphs, and are interesting in their own right. Even when w(e) = 1 for all edges e, determining whether WDP can be solved in polynomial time was posed by Jensen and Toft [Topics in Chromatic Graph Theory, Cambridge University Press, Cambridge, 2015, pp. 327-357] and by Stiebitz et al. [Graph Edge Colouring: Vizing's Theorem and Goldberg's Conjecture, John Wiley, New York, 2012] as an open problem. In this paper we present strongly polynomial-time algorithms for solving them exactly, and develop a novel matching removal technique for multigraph edge-coloring. [ABSTRACT FROM AUTHOR]

Let g(t) be the minimum number such that every graph G with average degree d(G) ≥ g(t)containsa Kt-minor. Such a function is known to exist, as originally shown by Mader. Kostochka and Thomason independently proved that g(t) ∈ Θ(t √log t). This paper shows that for all fixed ε > 0 and fixed sufficiently large t ≥ t(ε), if d(G) ≥ (2 + ε)g(t), then we can find this Kt-minor in linear time. This improves a previous result by Reed and Wood who gave a linear-time algorithm when d(G) ≥ 2t-2. [ABSTRACT FROM AUTHOR]

Facility location, arising in a rich variety of applications, has been studied extensively in the fields of operations research and computer science. In this paper we consider the classical uncapacitated facility location problem and its "prize-collecting" variant introduced by Baïou and Barahona, and we show that the linear systems associated with these problems are totally dual integral if and only if the input graphs do not contain a certain type of odd cycles. As corollaries, we get structural characterizations of two min-max relations on facility location. Our results strengthen the integrality theorems on facility location polytopes proved by Baïou and Barahona; our proofs lead to combinatorial polynomial-time algorithms for the facility location problems that we consider. [ABSTRACT FROM AUTHOR]

Numerical algorithms have two kinds of costs: arithmetic and communication, by which we mean either moving data between levels of a memory hierarchy (in the sequential case) or over a network connecting processors (in the parallel case). Communication costs often dominate arithmetic costs, so it is of interest to design algorithms minimizing communication. In this paper we first extend known lower bounds on the communication cost (both for bandwidth and for latency) of conventional (O(n3)) matrix multiplication to Cholesky factorization, which is used for solving dense symmetric positive definite linear systems. Second, we compare the costs of various Cholesky decomposition implementations to these lower bounds and identify the algorithms and data structures that attain them. In the sequential case, we consider both the two-level and hierarchical memory models. Combined with prior results in [J. Demmel et al., Communication-optimal Parallel and Sequential QR and LU Factorizations, Technical report EECS-2008-89, University of California, Berkeley, CA, 2008], [J. Demmel et al., Implementing Communication-optimal Parallel and Sequential QR and LU Factorizations, SIAM. J. Sci. Comp., submitted], and [J. Demmel, L. Grigori, and H. Xiang, Communication-avoiding Gaussian Elimination, Proceedings of the 2008 ACM/IEEE Conference on Supercomputing, 2008] this gives a set of communication-optimal algorithms for O(n³) implementations of the three basic factorizations of dense linear algebra: LU with pivoting, QR, and Cholesky. But it goes beyond this prior work on sequential LU by optimizing communication for any number of levels of memory hierarchy. [ABSTRACT FROM AUTHOR]

A graph is an interval graph if it is the intersection graph of intervals on a line. Interval graphs are known to be the intersection of chordal graphs and asteroidal triple-free graphs, two families where the well-known lexicographic breadth first search (LBFS) plays an important algorithmic and structural role. In this paper we show that interval graphs have a very rich LBFS structure and that by exploiting this structure one can design a linear time, easily implementable, interval graph recognition algorithm. [ABSTRACT FROM AUTHOR]

This paper considers the problem of determining whether a given set T of rooted triplets can be merged without conflicts into a galled phylogenetic network and, if so, constructing such a network. When the input T is dense, we solve the problem in O(∣T∣) time, which is optimal since the size of the input is ϴ(∣T∣). In comparison, the previously fastest algorithm for this problem runs in O(∣T∣²) time. We also develop an optimal O(∣T∣)-time algorithm for enumerating all simple phylogenetic networks leaf-labeled by L that are consistent with T , where L is the set of leaf labels in T , which is used by our main algorithm. Next, we prove that the problem becomes NP-hard if extended to nondense inputs, even for the special case of simple phylogenetic networks. We also show that for every positive integer n, there exists some set T of rooted triplets on n leaves such that any galled network can be consistent with at most 0.4883. ∣T∣ of the rooted triplets in T. On the other hand, we provide a polynomial-time approximation algorithm that always outputs a galled network consistent with at least a factor of 5 ⁄12 (> 0.4166) of the rooted triplets in T . [ABSTRACT FROM AUTHOR]

Motivated by a multitree approach to the design of reliable communication protocols, Itai and Rodeh gave a linear time algorithm for finding two independent spanning trees in a 2-connected graph. Cheriyan and Maheshwari gave an O(∣V∣²) algorithm for finding three independent spanning trees in a 3-connected graph. In this paper we present an O(∣V∣³) algorithm for finding four independent spanning trees in a 4-connected graph. We make use of chain decompositions of 4-connected graphs. [ABSTRACT FROM AUTHOR]

In this paper, we describe an O (|V (G)|| E (G)|) algorithm for finding a nonseparating planar chain in a 4-connected graph G, which will be used to decompose an arbitrary 4-connected graph into planar chains. This work was motivated by the study of a multitree approach to reliability in distributed networks, as well as the study of nonseparating induced paths in highly connected graphs. [ABSTRACT FROM AUTHOR]

For an arbitrary fixed surface S, a linear time algorithm is presented that for a given graph G either finds an embedding of G in S or identifies a subgraph of G that is homeomorphic to a minimal forbidden subgraph for embeddability in S. A side result of the proof of the algorithm is that minimal forbidden subgraphs for embeddability in S cannot be arbitrarily large. This yields a constructive proof of the result of Robertson and Seymour that for each closed surface there are only finitely many minimal forbidden subgraphs. The results and methods of this paper can be used to solve more general embedding extension problems. [ABSTRACT FROM AUTHOR]

Checking regularity (or singularity) of interval matrices is a known NP-hard problem. In this paper a general algorithm for checking regularity/singularity is presented which is not a priori exponential. The algorithm is based on a theoretical result according to which regularity may be judged from any single component of the solution set of an associated system of linear interval equations. Numerical experiments (with interval matrices up to the size n = 50) confirm that this approach brings an essential decrease in the amount of operations needed. [ABSTRACT FROM AUTHOR]

In applications, it is well known that high smoothness, small support, and high vanishing moments are the three most important properties of a biorthogonal wavelet. In this paper, we shall investigate the mutual relations among these three properties. A characterization of Lp (1 ≤ p ≤ 1) smoothness of multivariate refinable functions is presented. It is well known that there is a close relation between a fundamental refinable function and a biorthogonal wavelet. We shall demonstrate that any fundamental refinable function, whose mask is supported on [1 - 2r; 2r - 1]s for some positive integer r and satisfies the sum rules of optimal order 2r, has Lp smoothness not exceeding that of the univariate fundamental refinable function with the mask br. Here the sequence br on Z is the unique univariate interpolatory refinement mask which is supported on [1 - 2r; 2r - 1] and satisfies the sum rules of order 2r. Based on a similar idea, we shall prove that any orthogonal scaling function, whose mask is supported on [0; 2r - 1]s for some positive integer r and satisfies the sum rules of optimal order r, has Lp smoothness not exceeding that of the univariate Daubechies orthogonal scaling function whose mask is supported on [0; 2r - 1]. We also demonstrate that a similar result holds true for biorthogonal wavelets. Examples are provided to illustrate the general theory. Finally, a general CBC (cosets by cosets) algorithm is presented to construct all the dual refinement masks of any given interpolatory refinement mask with the dual masks satisfying arbitrary order of sum rules. Thus, for any scaling function which is fundamental, this algorithm can be employed to generate a dual scaling function with arbitrary approximation order. This CBC algorithm can be easily implemented. As a particular application of the general CBC algorithm, a TCBC (triangle cosets by cosets) algorithm is proposed. For any... [ABSTRACT FROM AUTHOR]

In this paper, we present an (n) time worst-case deterministic algorithm for finding the constrained Delaunay triangulation and constrained Voronoi diagram of a simple n-sided polygon in the plane. Up to now, only an O(n log n) worst-case deterministic and an O(n) expected time bound have been shown, leaving an O(n) deterministic solution open to conjecture. [ABSTRACT FROM AUTHOR]