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Register a new userWorst-Case Efficient Dynamic Geometric Independent Set
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John Iacono
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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We consider the problem of maintaining an approximate maximum independent set of geometric objects under insertions and deletions. We present data structures that maintain a constant-factor approximate maximum independent set for broad classes of fat objects in $d$ dimensions, where $d$ is assumed to be a constant, in sublinear textit{worst-case} update time. This gives the first results for dynamic independent set in a wide variety of geometric settings, such as disks, fat polygons, and their high-dimensional equivalents. For axis-aligned squares and hypercubes, our result improves upon all (recently announced) previous works. We obtain, in particular, a dynamic $(4+epsilon)$-approximation for squares, with $O(log^4 n)$ worst-case update time. Our result is obtained via a two-level approach. First, we develop a dynamic data structure which stores all objects and provides an approximate independent set when queried, with output-sensitive running time. We show that via standard methods such a structure can be used to obtain a dynamic algorithm with textit{amortized} update time bounds. Then, to obtain worst-case update time algorithms, we develop a generic deamortization scheme that with each insertion/deletion keeps (i) the update time bounded and (ii) the number of changes in the independent set constant. We show that such a scheme is applicable to fat objects by showing an appropriate generalization of a separator theorem. Interestingly, we show that our deamortization scheme is also necessary in order to obtain worst-case update bounds: If for a class of objects our scheme is not applicable, then no constant-factor approximation with sublinear worst-case update time is possible. We show that such a lower bound applies even for seemingly simple classes of geometric objects including axis-aligned rectangles in the plane.
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We present fully dynamic approximation algorithms for the Maximum Independent Set problem on several types of geometric objects: intervals on the real line, arbitrary axis-aligned squares in the plane and axis-aligned $d$-dimensional hypercubes. It is known that a maximum independent set of a collection of $n$ intervals can be found in $O(nlog n)$ time, while it is already textsf{NP}-hard for a set of unit squares. Moreover, the problem is inapproximable on many important graph families, but admits a textsf{PTAS} for a set of arbitrary pseudo-disks. Therefore, a fundamental question in computational geometry is whether it is possible to maintain an approximate maximum independent set in a set of dynamic geometric objects, in truly sublinear time per insertion or deletion. In this work, we answer this question in the affirmative for intervals, squares and hypercubes. First, we show that for intervals a $(1+varepsilon)$-approximate maximum independent set can be maintained with logarithmic worst-case update time. This is achieved by maintaining a locally optimal solution using a constant number of constant-size exchanges per update. We then show how our interval structure can be used to design a data structure for maintaining an expected constant factor approximate maximum independent set of axis-aligned squares in the plane, with polylogarithmic amortized update time. Our approach generalizes to $d$-dimensional hypercubes, providing a $O(4^d)$-approximation with polylogarithmic update time. Those are the first approximation algorithms for any set of dynamic arbitrary size geometric objects; previous results required bounded size ratios to obtain polylogarithmic update time. Furthermore, it is known that our results for squares (and hypercubes) cannot be improved to a $(1+varepsilon)$-approximation with the same update time.
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.
We give a polynomial-time constant-factor approximation algorithm for maximum independent set for (axis-aligned) rectangles in the plane. Using a polynomial-time algorithm, the best approximation factor previously known is $O(loglog n)$. The results are based on a new form of recursive partitioning in the plane, in which faces that are constant-complexity and orthogonally convex are recursively partitioned into a constant number of such faces.
In the dynamic minimum set cover problem, a challenge is to minimize the update time while guaranteeing close to the optimal $min(O(log n), f)$ approximation factor. (Throughout, $m$, $n$, $f$, and $C$ are parameters denoting the maximum number of sets, number of elements, frequency, and the cost range.) In the high-frequency range, when $f=Omega(log n)$, this was achieved by a deterministic $O(log n)$-approximation algorithm with $O(f log n)$ amortized update time [Gupta et al. STOC17]. In the low-frequency range, the line of work by Gupta et al. [STOC17], Abboud et al. [STOC19], and Bhattacharya et al. [ICALP15, IPCO17, FOCS19] led to a deterministic $(1+epsilon)f$-approximation algorithm with $O(f log (Cn)/epsilon^2)$ amortized update time. In this paper we improve the latter update time and provide the first bounds that subsume (and sometimes improve) the state-of-the-art dynamic vertex cover algorithms. We obtain: 1. $(1+epsilon)f$-approximation ratio in $O(flog^2 (Cn)/epsilon^3)$ worst-case update time: No non-trivial worst-case update time was previously known for dynamic set cover. Our bound subsumes and improves by a logarithmic factor the $O(log^3 n/text{poly}(epsilon))$ worst-case update time for unweighted dynamic vertex cover (i.e., when $f=2$ and $C=1$) by Bhattacharya et al. [SODA17]. 2. $(1+epsilon)f$-approximation ratio in $Oleft((f^2/epsilon^3)+(f/epsilon^2) log Cright)$ amortized update time: This result improves the previous $O(f log (Cn)/epsilon^2)$ update time bound for most values of $f$ in the low-frequency range, i.e. whenever $f=o(log n)$. It is the first that is independent of $m$ and $n$. It subsumes the constant amortized update time of Bhattacharya and Kulkarni [SODA19] for unweighted dynamic vertex cover (i.e., when $f = 2$ and $C = 1$).
We study geometric set cover problems in dynamic settings, allowing insertions and deletions of points and objects. We present the first dynamic data structure that can maintain an $O(1)$-approximation in sublinear update time for set cover for axis-aligned squares in 2D. More precisely, we obtain randomized update time $O(n^{2/3+delta})$ for an arbitrarily small constant $delta>0$. Previously, a dynamic geometric set cover data structure with sublinear update time was known only for unit squares by Agarwal, Chang, Suri, Xiao, and Xue [SoCG 2020]. If only an approximate size of the solution is needed, then we can also obtain sublinear amortized update time for disks in 2D and halfspaces in 3D. As a byproduct, our techniques for dynamic set cover also yield an optimal randomized $O(nlog n)$-time algorithm for static set cover for 2D disks and 3D halfspaces, improving our earlier $O(nlog n(loglog n)^{O(1)})$ result [SoCG 2020].
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