Network Design with Edge-Connectivity and Degree Constraints

We consider the following network design problem; Given a vertex set V with a metric cost c on V, an integer k≥1, and a degree specification b, find a minimum cost k-edge-connected multigraph on V under the constraint that the degree of each vertex v∈V is equal to b(v). This problem generalizes metric TSP. In this paper, we show that the problem admits a ρ-approximation algorithm if b(v)≥2, v∈V, where ρ=2.5 if k is even, and ρ=2.5+1.5/k if k is odd. We also prove that the digraph version of this problem admits a 2.5-approximation algorithm and discuss some generalization of metric TSP.     

Fixed-Parameter Enumerability of Cluster Editing and Related Problems

Cluster Editing is transforming a graph by at most k edge insertions or deletions into a disjoint union of cliques. This problem is fixed-parameter tractable (FPT). Here we compute concise enumerations of all minimal solutions in O(2.27 ^k +k ² n+m) time. Such enumerations support efficient inference procedures, but also the optimization of further objectives such as minimizing the number of clusters. In an extended problem version, target graphs may have a limited number of overlaps of cliques, measured by the number t of edges that remain when the twin vertices are merged. This problem is still in FPT, with respect to the combined parameter k and t. The result is based on a property of twin-free graphs. We also give FPT results for problem versions avoiding certain artificial clusterings. Furthermore, we prove that all solutions with minimal edit sequences differ on a so-called full kernel with at most k ²/4+O(k) vertices, that can be found in polynomial time. The size bound is tight. We also get a bound for the number of edges in the full kernel, which is optimal up to a (large) constant factor. Numerous open problems are mentioned.     

Faster Fixed-Parameter Tractable Algorithms for Matching and Packing Problems

We obtain faster algorithms for problems such as r-dimensional matching and r-set packing when the size k of the solution is considered a parameter. We first establish a general framework for finding and exploiting small problem kernels (of size polynomial in k). This technique lets us combine Alon, Yuster and Zwick’s color-coding technique with dynamic programming to obtain faster fixed-parameter algorithms for these problems. Our algorithms run in time O(n+2 ^O(k)), an improvement over previous algorithms for some of these problems running in time O(n+k ^O(k)). The flexibility of our approach allows tuning of algorithms to obtain smaller constants in the exponent. Research initiated at the International Workshop on Fixed Parameter Tractability in Computational Geometry and Games, Bellairs Research Institute of McGill University, Holetown, Barbados, Feb. 7–13, 2004, organized by S. Whitesides. D.M. Thilikos supported by the EU within the 6th Framework Programme under contract 001907 (DELIS) and by the Spanish CICYT project TIC-2002-04498-C05-03 (TRACER).     

A Hybridizable and Superconvergent Discontinuous Galerkin Method for Biharmonic Problems

In this paper, we introduce and analyze a new discontinuous Galerkin method for solving the biharmonic problem Δ² u=f. The method has two main, distinctive features, namely, it is amenable to an efficient implementation, and it displays new superconvergence properties. Indeed, although the method uses as separate unknowns u,? u,Δu and ?Δu, the only globally coupled degrees of freedom are those of the approximations to u and Δu on the faces of the elements. This is why we say it can be efficiently implemented. We also prove that, when polynomials of degree at most k≥1 are used on all the variables, approximations of optimal convergence rates are obtained for both u and ? u; the approximations to Δu and ?Δu converge with order k+1/2 and k?1/2, respectively. Moreover, both the approximation of u as well as its numerical trace superconverge in L ²-like norms, to suitably chosen projections of u with order k+2 for k≥2. This allows the element-by-element construction of another approximation to u converging with order k+2 for k≥2. For k=0, we show that the approximation to u converges with order one, up to a logarithmic factor. Numerical experiments are provided which confirm the sharpness of our theoretical estimates.     

Approximation Algorithms for Bounded Degree Phylogenetic Roots

The Degree- Δ Closest Phylogenetic k th Root Problem (ΔCPR _k ) is the problem of finding a (phylogenetic) tree T from a given graph G=(V,E) such that (1) the degree of each internal node in T is at least 3 and at most Δ, (2) the external nodes (i.e. leaves) of T are exactly the elements of V, and (3) the number of disagreements, i.e., |E ⊕{{u,v} : u,v are leaves of T and d _T (u,v)≤k}|, is minimized, where d _T (u,v) denotes the distance between u and v in tree T. This problem arises from theoretical studies in evolutionary biology and generalizes several important combinatorial optimization problems such as the maximum matching problem. Unfortunately, it is known to be NP-hard for all fixed constants Δ,k such that either both Δ≥3 and k≥3, or Δ>3 and k=2. This paper presents a polynomial-time 8-approximation algorithm for Δ CPR ₂ for any fixed Δ>3, a quadratic-time 12-approximation algorithm for 3CPR ₃, and a polynomial-time approximation scheme for the maximization version of Δ CPR _k for any fixed Δ and k.     

Connectivity and synchronization of Vicsek model

The collective behavior of multi-agent systems is an important studying point for the investigation of complex systems, and a basic model of multi-agent systems is the so called Vicsek model, which possesses some key features of complex systems, such as dynamic behavior, local interaction, changing neighborhood, etc. This model looks simple, but the nonlinearly coupled relationship makes the theoretical analysis quite complicated. Jadbabaie et al. analyzed the linearized heading equations in this model and showed that all agents will synchronize eventually, provided that the neighbor graphs associated with the agents＇ positions satisfy a certain connectivity condition. Much subsequent research effort has been devoted to the analysis of the Vicsek model since the publication of Jadbabaie＇s work. However, an unresolved key problem is when such a connectivity is satisfied. This paper given a sufficient condition to guarantee the synchronization of the Vicsek model, which is imposed on the model parameters only. Moreover, some counterexamples are given to show that the connectivity of the neighbor graphs is not sufficient for synchronization of the Vicsek model if the initial headings are allowed to be in [0,2π）, which reveals some fundamental differences between the Vicsek model and its linearized version.     

Short Cycles Make W-hard Problems Hard: FPT Algorithms for W-hard Problems in Graphs with no Short Cycles

We show that several problems that are hard for various parameterized complexity classes on general graphs, become fixed parameter tractable on graphs with no small cycles. More specifically, we give fixed parameter tractable algorithms for Dominating Set, t -Vertex Cover (where we need to cover at least t edges) and several of their variants on graphs with girth at least five. These problems are known to be W[i]-hard for some i≥1 in general graphs. We also show that the Dominating Set problem is W[2]-hard for bipartite graphs and hence for triangle free graphs. In the case of Independent Set and several of its variants, we show these problems to be fixed parameter tractable even in triangle free graphs. In contrast, we show that the Dense Subgraph problem where one is interested in finding an induced subgraph on k vertices having at least l edges, parameterized by k, is W[1]-hard even on graphs with girth at least six. Finally, we give an O(log p) ratio approximation algorithm for the Dominating Set problem for graphs with girth at least 5, where p is the size of an optimum dominating set of the graph. This improves the previous O(log n) factor approximation algorithm for the problem, where n is the number of vertices of the input graph. A preliminary version of this paper appeared in the Proceedings of 10th Scandinavian Workshop on Algorithm Theory (SWAT), Lecture Notes in Computer Science, vol. 4059, pp. 304–315, 2006.     

Stabilization by Local Projection for Convection–Diffusion and Incompressible Flow Problems

We give a survey on recent developments of stabilization methods based on local projection type. The considered class of problems covers scalar convection–diffusion equations, the Stokes problem and the linearized Navier–Stokes equations. A new link of local projection to the streamline diffusion method is shown. Numerical tests for different type of boundary layers arising in convection–diffusion problems illustrate the stabilizing properties of the method.     

The Complexity of König Subgraph Problems and Above-Guarantee Vertex Cover

A graph is König-Egerváry if the size of a minimum vertex cover equals that of a maximum matching in the graph. These graphs have been studied extensively from a graph-theoretic point of view. In this paper, we introduce and study the algorithmic complexity of finding König-Egerváry subgraphs of a given graph. In particular, given a graph G and a nonnegative integer k, we are interested in the following questions:

1. 1.
   does there exist a set of k vertices (edges) whose deletion makes the graph König-Egerváry?
    
2. 2.
   does there exist a set of k vertices (edges) that induce a König-Egerváry subgraph?
    

We show that these problems are NP-complete and study their complexity from the points of view of approximation and parameterized complexity. Towards this end, we first study the algorithmic complexity of Above Guarantee Vertex Cover, where one is interested in minimizing the additional number of vertices needed beyond the maximum matching size for the vertex cover. Further, while studying the parameterized complexity of the problem of deleting k vertices to obtain a König-Egerváry graph, we show a number of interesting structural results on matchings and vertex covers which could be useful in other contexts.

     

Fractional Path Coloring in Bounded Degree Trees with Applications

This paper studies the natural linear programming relaxation of the path coloring problem. We prove constructively that finding an optimal fractional path coloring is Fixed Parameter Tractable (FPT), with the degree of the tree as parameter: the fractional coloring of paths in a bounded degree trees can be done in a time which is linear in the size of the tree, quadratic in the load of the set of paths, while exponential in the degree of the tree. We give an algorithm based on the generation of an efficient polynomial size linear program. Our algorithm is able to explore in polynomial time the exponential number of different fractional colorings, thanks to the notion of trace of a coloring that we introduce. We further give an upper bound on the cost of such a coloring in binary trees and extend this algorithm to bounded degree graphs with bounded treewidth. Finally, we also show some relationships between the integral and fractional problems, and derive a 1+5/3e≈1.61—approximation algorithm for the path coloring problem in bounded degree trees, improving on existing results. This classic combinatorial problem finds applications in the minimization of the number of wavelengths in wavelength division multiplexing (wdm) optical networks.     

Computation of Minimum Energy Paths for Quasi-Linear Problems

We investigate minimum energy paths of the quasi-linear problem with the p-Laplacian operator and a double-well potential. We adapt the String method of E, Ren, and Vanden-Eijnden (J. Chem. Phys. 126, 2007) to locate saddle-type solutions. In one-dimension, the String method is shown to find a minimum energy path that can align along one-dimensional “ridges” of saddle-continua. We then apply the same method to locate saddle solutions and transition paths of the two-dimensional quasi-linear problem. The method developed is applicable to a general class of quasi-linear PDEs.     

Two-Grid Discontinuous Galerkin Method for Quasi-Linear Elliptic Problems

In this paper, we consider the symmetric interior penalty discontinuous Galerkin (SIPG) method with piecewise polynomials of degree r≥1 for a class of quasi-linear elliptic problems in Ω⊂ℝ². We propose a two-grid approximation for the SIPG method which can be thought of as a type of linearization of the nonlinear system using a solution from a coarse finite element space. With this technique, solving a quasi-linear elliptic problem on the fine finite element space is reduced into solving a linear problem on the fine finite element space and solving the quasi-linear elliptic problem on a coarse space. Convergence estimates in a broken H ¹-norm are derived to justify the efficiency of the proposed two-grid algorithm. Numerical experiments are provided to confirm our theoretical findings. As a byproduct of the technique used in the analysis, we derive the optimal pointwise error estimates of the SIPG method for the quasi-linear elliptic problems in ℝ ^d ,d=2,3 and use it to establish the convergence of the two-grid method for problems in Ω⊂ℝ³.     

Characteristic Tailored Finite Point Method for Convection-Dominated Convection-Diffusion-Reaction Problems

In this paper, we propose a characteristic tailored finite point method (CTFPM) for solving the convection-diffusion-reaction equation with variable coefficients. We develop an algorithm to construct a streamline-aligned grid for the CTFPM. Our numerical tests show for small diffusion coefficient the CTFPM solution resolves the internal and boundary layers regardless the mesh size, and depicts that CTFPM method with a streamline grid has excellent performance compared with the tailored finite point method and a streamline upwind finite element method when ε is small.     

On Hierarchical Diameter-Clustering and the Supplier Problem

Given a data set in a metric space, we study the problem of hierarchical clustering to minimize the maximum cluster diameter, and the hierarchical k-supplier problem with customers arriving online. We prove that two previously known algorithms for hierarchical clustering, one (offline) due to Dasgupta and Long and the other (online) due to Charikar, Chekuri, Feder and Motwani, output essentially the same result when points are considered in the same order. We show that the analyses of both algorithms are tight and exhibit a new lower bound for hierarchical clustering. Finally we present the first constant factor approximation algorithm for the online hierarchical k-supplier problem.     

Vesicles and Amoebae: On Globally Constrained Shape Deformation

Modeling the deformation of shapes under constraints on both perimeter and area is a challenging task due to the highly nontrivial interaction between the need for flexible local rules for manipulating the boundary and the global constraints. We propose several methods to address this problem and generate “random walks” in the space of shapes obeying quite general possibly time varying constraints on their perimeter and area. Design of perimeter and area preserving deformations are an interesting and useful special case of this problem. The resulting deformation models are employed in annealing processes that evolve original shapes toward shapes that are optimal in terms of boundary bending-energy or other functionals. Furthermore, such models may find applications in the analysis of sequences of real images of deforming objects obeying global constraints as building blocks for registration and tracking algorithms.     

Interacting and Annealing Particle Filters: Mathematics and a Recipe for Applications

Interacting and annealing are two powerful strategies that are applied in different areas of stochastic modelling and data analysis. Interacting particle systems approximate a distribution of interest by a finite number of particles where the particles interact between the time steps. In computer vision, they are commonly known as particle filters. Simulated annealing, on the other hand, is a global optimization method derived from statistical mechanics. A recent heuristic approach to fuse these two techniques for motion capturing has become known as annealed particle filter. In order to analyze these techniques, we rigorously derive in this paper two algorithms with annealing properties based on the mathematical theory of interacting particle systems. Convergence results and sufficient parameter restrictions enable us to point out limitations of the annealed particle filter. Moreover, we evaluate the impact of the parameters on the performance in various experiments, including the tracking of articulated bodies from noisy measurements. Our results provide a general guidance on suitable parameter choices for different applications.

Exact Minkowksi Sums of Polyhedra and Exact and Efficient Decomposition of Polyhedra into Convex Pieces

We present the first exact and robust implementation of the 3D Minkowski sum of two non-convex polyhedra. Our implementation decomposes the two polyhedra into convex pieces, performs pairwise Minkowski sums on the convex pieces, and constructs their union. We achieve exactness and the handling of all degeneracies by building upon 3D Nef polyhedra as provided by Cgal. The implementation also supports open and closed polyhedra. This allows the handling of degenerate scenarios like the tight passage problem in robot motion planning. The bottleneck of our approach is the union step. We address efficiency by optimizing this step by two means: we implement an efficient decomposition that yields a small number of convex pieces, and develop, test and optimize multiple strategies for uniting the partial sums by consecutive binary union operations. The decomposition that we implemented as part of the Minkowski sum is interesting in its own right. It is the first robust implementation of a decomposition of polyhedra into convex pieces that yields at most O(r ²) pieces, where r is the number of edges whose adjacent facets comprise an angle of more than 180 degrees with respect to the interior of the polyhedron. This work was partially supported by the Netherlands’ Organisation for Scientific Research (NWO) under project no. 639.023.301.     

Lower Bounds for Kernelizations and Other Preprocessing Procedures

We first present a method to rule out the existence of parameter non-increasing polynomial kernelizations of parameterized problems under the hypothesis P≠NP. This method is applicable, for example, to the problem Sat parameterized by the number of variables of the input formula. Then we obtain further improvements of corresponding results in (Bodlaender et al. in Lecture Notes in Computer Science, vol. 5125, pp. 563–574, Springer, Berlin, 2008; Fortnow and Santhanam in Proceedings of the 40th ACM Symposium on the Theory of Computing (STOC’08), ACM, New York, pp. 133–142, 2008) by refining the central lemma of their proof method, a lemma due to Fortnow and Santhanam. In particular, assuming that the polynomial hierarchy does not collapse to its third level, we show that every parameterized problem with a “linear OR” and with NP-hard underlying classical problem does not have polynomial self-reductions that assign to every instance x with parameter k an instance y with |y|=k ^O(1)⋅|x|^1−ε (here ε is any given real number greater than zero). We give various applications of these results. On the structural side we prove several results clarifying the relationship between the different notions of preprocessing procedures, namely the various notions of kernelizations, self-reductions and compressions.     

Computer Assisted Proofs of Bifurcating Solutions for Nonlinear Heat Convection Problems

In previous works (Nakao et al., Reliab. Comput., 9(5):359–372, 2003; Watanabe et al., J. Math. Fluid Mech., 6(1):1–20, 2004), the authors considered the numerical verification method of solutions for two-dimensional heat convection problems known as Rayleigh-Bénard problem. In the present paper, to make the arguments self-contained, we first summarize these results including the basic formulation of the problem with numerical examples. Next, we will give a method to verify the bifurcation point itself, which should be an important information to clarify the global bifurcation structure, and show a numerical example. Finally, an extension to the three dimensional case will be described.