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One of the theoretically intriguing problems in computer-aided geometric modeling comes from the stitching of the tensor product Bezier patches. When they share an extraordinary vertex, it is not possible to obtain continuity C1 or G1 along the edges emanating from that extraordinary vertex. Unfortunately, this stitching problem cannot be solved by using higher degree or rational polynomials. In this paper, we present a modified de Casteljau subdivision algorithm that can provide a solution to this problem. Our modified de Casteljau subdivision, when combined with topological modeling, provides a framework for interactive real-time modeling of piecewise smooth manifold meshes with arbitrary topology. The main advantage of the modified subdivision is that the continuity C1 on a given boundary edge does not depend on the positions of the control points on other boundary edges. The modified subdivision allows us to obtain the desired C1 continuity along the edges emanating from the extraordinary vertices along with the desired G1 continuity in the extraordinary vertices., Comment: 24 pages

One of the key problems in dealing with color in rendering, shading, compositing, or image manipulation is that we do not have algebraic structures that support operations over colors. In this paper, we present an all-encompassing framework that can support a set of algebraic structures with associativity, commutativity, and inverse properties. To provide these three properties, we build our algebraic structures on an extension of projective space by allowing for negative and complex numbers. These properties are important for (1) manipulating colors as periodic functions, (2) solving inverse problems dealing with colors, and (3) being consistent with the wave representation of the color. Allowance of negative and complex numbers is not a problem for practical applications, since we can always convert the results into desired range for display purposes as we do in High Dynamic Range imaging. This set of algebraic structures can be considered as a generalization of the Minkowski norm Lp in projective space. These structures also provide a new version of the generalized Holder average with associativity property. Our structures provide inverses of any operation by allowing for negative and complex numbers. These structures provide all properties of the generalized Holder average by providing a continuous bridge between the classical weighted average, harmonic mean, maximum, and minimum operations using a single parameter p., Comment: 37 pages

We present streaming algorithms for the graph $k$-matching problem in both the insert-only and dynamic models. Our algorithms, with space complexity matching the best upper bounds, have optimal or near-optimal update time, significantly improving on previous results. More specifically, for the insert-only streaming model, we present a one-pass algorithm with optimal space complexity $O(k^2)$ and optimal update time $O(1)$, that with high probability computes a maximum weighted $k$-matching of a given weighted graph. The update time of our algorithm significantly improves the previous upper bound of $O(\log k)$, which was derived only for $k$-matching on unweighted graphs. For the dynamic streaming model, we present a one-pass algorithm that with high probability computes a maximum weighted $k$-matching in $O(Wk^2 \cdot \mbox{polylog}(n)$ space and with $O(\mbox{polylog}(n))$ update time, where $W$ is the number of distinct edge weights. Again the update time of our algorithm improves the previous upper bound of $O(k^2 \cdot \mbox{polylog}(n))$. This algorithm, when applied to unweighted graphs, gives a streaming algorithm on the dynamic model whose space and update time complexities are both near-optimal. Our results also imply a streaming approximation algorithm for maximum weighted $k$-matching whose space complexity matches the best known upper bound with a significantly improved update time.

We present parameterized streaming algorithms for the graph matching problem in both the dynamic and the insert-only models. For the dynamic streaming model, we present a one-pass algorithm that, with high probability, computes a maximum-weight $k$-matching of a weighted graph in $\tilde{O}(Wk^2)$ space and that has $\tilde{O}(1)$ update time, where $W$ is the number of distinct edge weights and the notation $\tilde{O}()$ hides a poly-logarithmic factor in the input size. For the insert-only streaming model, we present a one-pass algorithm that runs in $O(k^2)$ space and has $O(1)$ update time, and that, with high probability, computes a maximum-weight $k$-matching of a weighted graph. The space complexity and the update-time complexity achieved by our algorithms for unweighted $k$-matching in the dynamic model and for weighted $k$-matching in the insert-only model are optimal. A notable contribution of this paper is that the presented algorithms {\it do not} rely on the apriori knowledge/promise that the cardinality of \emph{every} maximum-weight matching of the input graph is upper bounded by the parameter $k$. This promise has been a critical condition in previous works, and lifting it required the development of new tools and techniques.

We study fundamental point-line covering problems in computational geometry, in which the input is a set $S$ of points in the plane. The first is the Rich Lines problem, which asks for the set of all lines that each covers at least $\lambda$ points from $S$, for a given integer parameter $\lambda \geq 2$; this problem subsumes the 3-Points-on-Line problem and the Exact Fitting problem, which -- the latter -- asks for a line containing the maximum number of points. The second is the NP-hard problem Line Cover, which asks for a set of $k$ lines that cover the points of $S$, for a given parameter $k \in \mathbb{N}$. Both problems have been extensively studied. In particular, the Rich Lines problem is a fundamental problem whose solution serves as a building block for several algorithms in computational geometry. For Rich Lines and Exact Fitting, we present a randomized Monte Carlo algorithm that achieves a lower running time than that of Guibas et al.'s algorithm [Computational Geometry 1996], for a wide range of the parameter $\lambda$. We derive lower-bound results showing that, for $\lambda =\Omega(\sqrt{n \log n})$, the upper bound on the running time of this randomized algorithm matches the lower bound that we derive on the time complexity of Rich Lines in the algebraic computation trees model. For Line Cover, we present two kernelization algorithms: a randomized Monte Carlo algorithm and a deterministic algorithm. Both algorithms improve the running time of existing kernelization algorithms for Line Cover. We derive lower-bound results showing that the running time of the randomized algorithm we present comes close to the lower bound we derive on the time complexity of kernelization algorithms for Line Cover in the algebraic computation trees model.

We propose a new (theoretical) computational model for the study of massive data processing with limited computational resources. Our model measures the complexity of reading the very large data sets in terms of the data size N and analyzes the computational cost in terms of a parameter k that characterizes the computational power provided by limited local computing resources. We develop new algorithmic techniques that implement algorithms for solving well-known computational problems on the proposed model. In particular, we present an algorithm that finds a k-matching in a general unweighted graph in time O(N + k^{2.5}) and an algorithm that constructs a maximum weighted k-matching in a general weighted graph in time O(N + k^3 log k). Both algorithms have their space complexity bounded by O(k^2).

Motivated by the current research in data centers and cloud computing, we study the problem of scheduling a set of two-stage jobs on multiple two-stage flowshops. A new formulation for configurations of such scheduling is proposed, which leads directly to improvements to the complexity of scheduling algorithms for the problem. Motivated by the observation that the costs of the two stages can be significantly different, we present deeper study on the structures of the problem that leads to a new approach to designing scheduling algorithms for the problem. With more thorough analysis, we show that the new approach gives very significant improved scheduling algorithms for the problem when the costs of the two stages are different significantly. Improved approximation algorithms for the problem are also presented.

We study the \emph{multiterminal cut} problem, which, given an $n$-vertex graph whose edges are integer-weighted and a set of terminals, asks for a partition of the vertex set such that each terminal is in a distinct part, and the total weight of crossing edges is at most $k$. Our weapons shall be two classical results known for decades: \emph{maximum volume minimum ($s,t$)-cuts} by [Ford and Fulkerson, \emph{Flows in Networks}, 1962] and \emph{isolating cuts} by [Dahlhaus et al., \emph{SIAM J. Comp.} 23(4):864-894, 1994]. We sharpen these old weapons with the help of submodular functions, and apply them to this problem, which enable us to design a more elaborated branching scheme on deciding whether a non-terminal vertex is with a terminal or not. This bounded search tree algorithm can be shown to run in $1.84^k\cdot n^{O(1)}$ time, thereby breaking the $2^k\cdot n^{O(1)}$ barrier. As a by-product, it gives a $1.36^k\cdot n^{O(1)}$ time algorithm for $3$-terminal cut. The preprocessing applied on non-terminal vertices might be of use for study of this problem from other aspects., Comment: To fulfill the request of the European Research Council (ERC)

The Maximum Agreement Forest problem has been extensively studied in phylogenetics. Most previous work is on two binary phylogenetic trees. In this paper, we study a generalized version of the problem: the Maximum Agreement Forest problem on multiple rooted multifurcating phylogenetic trees, from the perspective of fixed-parameter algorithms. By taking advantage of a new branch-and-bound strategy, two parameterized algorithms, with running times $O(2.42^k m^3 n^4)$ and $O(2.74^k m^3 n^5)$, respectively, are presented for the hard version and the soft version of the problem, which correspond to two different biological meanings to the polytomies in multifurcating phylogenetic trees.

In the Proper Interval Vertex Deletion problem (PIVD for short), we are given a graph $G$ and an integer parameter $k>0$, and the question is whether there are at most $k$ vertices in $G$ whose removal results in a proper interval graph. It is known that the PIVD problem is fixed-parameter tractable and admits a polynomial but "unreasonably" large kernel of $O(k^{53})$ vertices. A natural question is whether the problem admits a polynomial kernel of "reasonable" size. In this paper, we answer this question by deriving an $O(k^7)$-vertex kernel for the PIVD problem. Our kernelization is based on several new observations and a refined analysis of the kernelization., Comment: This paper is combined with another paper where a significantly improved kernel of size O(k^4) is obtained. Due to this reason, the authors withdraw this paper

We study techniques for solving the Maximum Satisfiability problem (MaxSAT). Our focus is on variables of degree 4. We identify cases for degree-4 variables and show how the resolution principle and the kernelization techniques can be nicely integrated to achieve more efficient algorithms for the MaxSAT problem. As a result, we present an algorithm of time $O^*(1.3248^k)$ for the MaxSAT problem, improving the previous best upper bound $O^*(1.358^k)$ by Ivan Bliznets and Alexander.

We consider the parameterized version of the maximum internal spanning tree problem, which, given an $n$-vertex graph and a parameter $k$, asks for a spanning tree with at least $k$ internal vertices. Fomin et al. [J. Comput. System Sci., 79:1-6] crafted a very ingenious reduction rule, and showed that a simple application of this rule is sufficient to yield a $3k$-vertex kernel. Here we propose a novel way to use the same reduction rule, resulting in an improved $2k$-vertex kernel. Our algorithm applies first a greedy procedure consisting of a sequence of local exchange operations, which ends with a local-optimal spanning tree, and then uses this special tree to find a reducible structure. As a corollary of our kernel, we obtain a deterministic algorithm for the problem running in time $4^k \cdot n^{O(1)}$.

The Maximum Agreement Forest (Maf) problem is a well-studied problem in evolutionary biology, which asks for a largest common subforest of a given collection of phylogenetic trees with identical leaf label-set. However, the previous work about the Maf problem are mainly on two binary phylogenetic trees or two general (i.e., binary and non-binary) phylogenetic trees. In this paper, we study the more general version of the problem: the Maf problem on multiple general phylogenetic trees. We present a parameterized algorithm of running time $O(3^k n^2m)$ and a 3-approximation algorithm for the Maf problem on multiple rooted general phylogenetic trees, and a parameterized algorithm of running time $O(4^k n^2m)$ and a 4-approximation algorithm for the Maf problem on multiple unrooted general phylogenetic trees. We also implement the parameterized algorithm and approximation algorithm for the Maf problem on multiple rooted general phylogenetic trees, and test them on simulated data and biological data.