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Register a new userConnecting a Set of Circles with Minimum Sum of Radii
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Added by
Sandor P. Fekete
Publication date
2011
fields
Informatics Engineering
and research's language is
English
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We consider the problem of assigning radii to a given set of points in the plane, such that the resulting set of circles is connected, and the sum of radii is minimized. We show that the problem is polynomially solvable if a connectivity tree is given. If the connectivity tree is unknown, the problem is NP-hard if there are upper bounds on the radii and open otherwise. We give approximation guarantees for a variety of polynomial-time algorithms, describe upper and lower bounds (which are matching in some of the cases), provide polynomial-time approximation schemes, and conclude with experimental results and open problems.
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The Split Packing algorithm cite{splitpacking_ws, splitpackingsoda, splitpacking} is an offline algorithm that packs a set of circles into triangles and squares up to critical density. In this paper, we develop an online alternative to Split Packing to handle an online sequence of insertions and deletions, where the algorithm is allowed to reallocate circles into new positions at a cost proportional to their areas. The algorithm can be used to pack circles into squares and right angled triangles. If only insertions are considered, our algorithm is also able to pack to critical density, with an amortised reallocation cost of $O(clog frac{1}{c})$ for squares, and $O(c(1+s^2)log_{1+s^2}frac{1}{c})$ for right angled triangles, where $s$ is the ratio of the lengths of the second shortest side to the shortest side of the triangle, when inserting a circle of area $c$. When insertions and deletions are considered, we achieve a packing density of $(1-epsilon)$ of the critical density, where $epsilon>0$ can be made arbitrarily small, with an amortised reallocation cost of $O(c(1+s^2)log_{1+s^2}frac{1}{c} + cfrac{1}{epsilon})$.
We provide exact and approximation methods for solving a geometric relaxation of the Traveling Salesman Problem (TSP) that occurs in curve reconstruction: for a given set of vertices in the plane, the problem Minimum Perimeter Polygon (MPP) asks for a (not necessarily simply connected) polygon with shortest possible boundary length. Even though the closely related problem of finding a minimum cycle cover is polynomially solvable by matching techniques, we prove how the topological structure of a polygon leads to NP-hardness of the MPP. On the positive side, we show how to achieve a constant-factor approximation. When trying to solve MPP instances to provable optimality by means of integer programming, an additional difficulty compared to the TSP is the fact that only a subset of subtour constraints is valid, depending not on combinatorics, but on geometry. We overcome this difficulty by establishing and exploiting additional geometric properties. This allows us to reliably solve a wide range of benchmark instances with up to 600 vertices within reasonable time on a standard machine. We also show that using a natural geometry-based sparsification yields results that are on average within 0.5% of the optimum.
We study the problem of finding a minimum homology basis, that is, a shortest set of cycles that generates the $1$-dimensional homology classes with $mathbb{Z}_2$ coefficients in a given simplicial complex $K$. This problem has been extensively studied in the last few years. For general complexes, the current best deterministic algorithm, by Dey et al., runs in $O(N^omega + N^2 g)$ time, where $N$ denotes the number of simplices in $K$, $g$ denotes the rank of the $1$-homology group of $K$, and $omega$ denotes the exponent of matrix multiplication. In this paper, we present two conceptually simple randomized algorithms that compute a minimum homology basis of a general simplicial complex $K$. The first algorithm runs in $tilde{O}(m^omega)$ time, where $m$ denotes the number of edges in $K$, whereas the second algorithm runs in $O(m^omega + N m^{omega-1})$ time. We also study the problem of finding a minimum cycle basis in an undirected graph $G$ with $n$ vertices and $m$ edges. The best known algorithm for this problem runs in $O(m^omega)$ time. Our algorithm, which has a simpler high-level description, but is slightly more expensive, runs in $tilde{O}(m^omega)$ time.
Number Decision Diagrams (NDD) provide a natural finite symbolic representation for regular set of integer vectors encoded as strings of digit vectors (least or most significant digit first). The convex hull of the set of vectors represented by a NDD is proved to be an effectively computable convex polyhedron.
Given a set of $n$ terminals, which are points in $d$-dimensional Euclidean space, the minimum Manhattan network problem (MMN) asks for a minimum-length rectilinear network that connects each pair of terminals by a Manhattan path, that is, a path consisting of axis-parallel segments whose total length equals the pairs Manhattan distance. Even for $d=2$, the problem is NP-hard, but constant-factor approximations are known. For $d ge 3$, the problem is APX-hard; it is known to admit, for any $eps > 0$, an $O(n^eps)$-approximation. In the generalized minimum Manhattan network problem (GMMN), we are given a set $R$ of $n$ terminal pairs, and the goal is to find a minimum-length rectilinear network such that each pair in $R$ is connected by a Manhattan path. GMMN is a generalization of both MMN and the well-known rectilinear Steiner arborescence problem (RSA). So far, only special cases of GMMN have been considered. We present an $O(log^{d+1} n)$-approximation algorithm for GMMN (and, hence, MMN) in $d ge 2$ dimensions and an $O(log n)$-approximation algorithm for 2D. We show that an existing $O(log n)$-approximation algorithm for RSA in 2D generalizes easily to $d>2$ dimensions.
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