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Register a new userApproximation Schemes for Partitioning: Convex Decomposition and Surface Approximation
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Added by
Santanu Bhowmick
Publication date
2014
fields
Informatics Engineering
and research's language is
English
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We revisit two NP-hard geometric partitioning problems - convex decomposition and surface approximation. Building on recent developments in geometric separators, we present quasi-polynomial time algorithms for these problems with improved approximation guarantees.
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Given $n$ points in a $d$ dimensional Euclidean space, the Minimum Enclosing Ball (MEB) problem is to find the ball with the smallest radius which contains all $n$ points. We give a $O(ndQcal/sqrt{epsilon})$ approximation algorithm for producing an enclosing ball whose radius is at most $epsilon$ away from the optimum (where $Qcal$ is an upper bound on the norm of the points). This improves existing results using emph{coresets}, which yield a $O(nd/epsilon)$ greedy algorithm. Finding the Minimum Enclosing Convex Polytope (MECP) is a related problem wherein a convex polytope of a fixed shape is given and the aim is to find the smallest magnification of the polytope which encloses the given points. For this problem we present a $O(mndQcal/epsilon)$ approximation algorithm, where $m$ is the number of faces of the polytope. Our algorithms borrow heavily from convex duality and recently developed techniques in non-smooth optimization, and are in contrast with existing methods which rely on geometric arguments. In particular, we specialize the excessive gap framework of citet{Nesterov05a} to obtain our results.
Approximating the optimal social welfare while preserving truthfulness is a well studied problem in algorithmic mechanism design. Assuming that the social welfare of a given mechanism design problem can be optimized by an integer program whose integrality gap is at most $alpha$, Lavi and Swamy~cite{Lavi11} propose a general approach to designing a randomized $alpha$-approximation mechanism which is truthful in expectation. Their method is based on decomposing an optimal solution for the relaxed linear program into a convex combination of integer solutions. Unfortunately, Lavi and Swamys decomposition technique relies heavily on the ellipsoid method, which is notorious for its poor practical performance. To overcome this problem, we present an alternative decomposition technique which yields an $alpha(1 + epsilon)$ approximation and only requires a quadratic number of calls to an integrality gap verifier.
We provide a randomized linear time approximation scheme for a generic problem about clustering of binary vectors subject to additional constrains. The new constrained clustering problem encompasses a number of problems and by solving it, we obtain the first linear time-approximation schemes for a number of well-studied fundamental problems concerning clustering of binary vectors and low-rank approximation of binary matrices. Among the problems solvable by our approach are textsc{Low GF(2)-Rank Approximation}, textsc{Low Boolean-Rank Approximation}, and vario
In this paper, we prove that the Max-Morse Matching Problem is approximable, thus resolving an open problem posed by Joswig and Pfetsch. We describe two different approximation algorithms for the Max-Morse Matching Problem. For $D$-dimensional simplicial complexes, we obtain a $frac{(D+1)}{(D^2+D+1)}$-factor approximation ratio using a simple edge reorientation algorithm that removes cycles. Our second result is an algorithm that provides a $frac{2}{D}$-factor approximation for simplicial manifolds by processing the simplices in increasing order of dimension. One application of these algorithms is towards efficient homology computation of simplicial complexes. Experiments using a prototype implementation on several datasets indicate that the algorithm computes near optimal results.
The subspace approximation problem with outliers, for given $n$ points in $d$ dimensions $x_{1},ldots, x_{n} in R^{d}$, an integer $1 leq k leq d$, and an outlier parameter $0 leq alpha leq 1$, is to find a $k$-dimensional linear subspace of $R^{d}$ that minimizes the sum of squared distances to its nearest $(1-alpha)n$ points. More generally, the $ell_{p}$ subspace approximation problem with outliers minimizes the sum of $p$-th powers of distances instead of the sum of squared distances. Even the case of robust PCA is non-trivial, and previous work requires additional assumptions on the input. Any multiplicative approximation algorithm for the subspace approximation problem with outliers must solve the robust subspace recovery problem, a special case in which the $(1-alpha)n$ inliers in the optimal solution are promised to lie exactly on a $k$-dimensional linear subspace. However, robust subspace recovery is Small Set Expansion (SSE)-hard. We show how to extend dimension reduction techniques and bi-criteria approximations based on sampling to the problem of subspace approximation with outliers. To get around the SSE-hardness of robust subspace recovery, we assume that the squared distance error of the optimal $k$-dimensional subspace summed over the optimal $(1-alpha)n$ inliers is at least $delta$ times its squared-error summed over all $n$ points, for some $0 < delta leq 1 - alpha$. With this assumption, we give an efficient algorithm to find a subset of $poly(k/epsilon) log(1/delta) loglog(1/delta)$ points whose span contains a $k$-dimensional subspace that gives a multiplicative $(1+epsilon)$-approximation to the optimal solution. The running time of our algorithm is linear in $n$ and $d$. Interestingly, our results hold even when the fraction of outliers $alpha$ is large, as long as the obvious condition $0 < delta leq 1 - alpha$ is satisfied.
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