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Register a new userApproximation Algorithms for Max-Morse Matching
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Added by
Abhishek Rathod
Publication date
2016
fields
Informatics Engineering
and research's language is
English
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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.
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We study the problem of minimizing the number of critical simplices from the point of view of inapproximability and parameterized complexity. We first show inapproximability of Min-Morse Matching within a factor of $2^{log^{(1-epsilon)}n}$. Our second result shows that Min-Morse Matching is ${bf W{[P]}}$-hard with respect to the standard parameter. Next, we show that Min-Morse Matching with standard parameterization has no FPT approximation algorithm for any approximation factor $rho$. The above hardness results are applicable to complexes of dimension $ge 2$. On the positive side, we provide a factor $O(frac{n}{log n})$ approximation algorithm for Min-Morse Matching on $2$-complexes, noting that no such algorithm is known for higher dimensional complexes. Finally, we devise discrete gradients with very few critical simplices for typical instances drawn from a fairly wide range of parameter values of the Costa-Farber model of random complexes.
Given two sets $S$ and $T$ of points in the plane, of total size $n$, a {many-to-many} matching between $S$ and $T$ is a set of pairs $(p,q)$ such that $pin S$, $qin T$ and for each $rin Scup T$, $r$ appears in at least one such pair. The {cost of a pair} $(p,q)$ is the (Euclidean) distance between $p$ and $q$. In the {minimum-cost many-to-many matching} problem, the goal is to compute a many-to-many matching such that the sum of the costs of the pairs is minimized. This problem is a restricted version of minimum-weight edge cover in a bipartite graph, and hence can be solved in $O(n^3)$ time. In a more restricted setting where all the points are on a line, the problem can be solved in $O(nlog n)$ time [Colannino, Damian, Hurtado, Langerman, Meijer, Ramaswami, Souvaine, Toussaint; Graphs Comb., 2007]. However, no progress has been made in the general planar case in improving the cubic time bound. In this paper, we obtain an $O(n^2cdot poly(log n))$ time exact algorithm and an $O( n^{3/2}cdot poly(log n))$ time $(1+epsilon)$-approximation in the planar case. Our results affirmatively address an open problem posed in [Colannino et al., Graphs Comb., 2007].
We improve the running times of $O(1)$-approximation algorithms for the set cover problem in geometric settings, specifically, covering points by disks in the plane, or covering points by halfspaces in three dimensions. In the unweighted case, Agarwal and Pan [SoCG 2014] gave a randomized $O(nlog^4 n)$-time, $O(1)$-approximation algorithm, by using variants of the multiplicative weight update (MWU) method combined with geometric data structures. We simplify the data structure requirement in one of their methods and obtain a deterministic $O(nlog^3 nloglog n)$-time algorithm. With further new ideas, we obtain a still faster randomized $O(nlog n(loglog n)^{O(1)})$-time algorithm. For the weighted problem, we also give a randomized $O(nlog^4nloglog n)$-time, $O(1)$-approximation algorithm, by simple modifications to the MWU method and the quasi-uniform sampling technique.
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.
In the Euclidean TSP with neighborhoods (TSPN), we are given a collection of n regions (neighborhoods) and we seek a shortest tour that visits each region. As a generalization of the classical Euclidean TSP, TSPN is also NP-hard. In this paper, we present new approximation results for the TSPN, including (1) a constant-factor approximation algorithm for the case of arbitrary connected neighborhoods having comparable diameters; and (2) a PTAS for the important special case of disjoint unit disk neighborhoods (or nearly disjoint, nearly-unit disks). Our methods also yield improved approximation ratios for various special classes of neighborhoods, which have previously been studied. Further, we give a linear-time O(1)-approximation algorithm for the case of neighborhoods that are (infinite) straight lines.
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