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Register a new userMoving intervals for packing and covering
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Minghui Jiang
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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We study several problems on geometric packing and covering with movement. Given a family $mathcal{I}$ of $n$ intervals of $kappa$ distinct lengths, and another interval $B$, can we pack the intervals in $mathcal{I}$ inside $B$ (respectively, cover $B$ by the intervals in $mathcal{I}$) by moving $tau$ intervals and keeping the other $sigma = n - tau$ intervals unmoved? We show that both packing and covering are W[1]-hard with any one of $kappa$, $tau$, and $sigma$ as single parameter, but are FPT with combined parameters $kappa$ and $tau$. We also obtain improved polynomial-time algorithms for packing and covering, including an $O(nlog^2 n)$ time algorithm for covering, when all intervals in $mathcal{I}$ have the same length.
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In the MINIMUM CONVEX COVER (MCC) problem, we are given a simple polygon $mathcal P$ and an integer $k$, and the question is if there exist $k$ convex polygons whose union is $mathcal P$. It is known that MCC is $mathsf{NP}$-hard [Culberson & Reckhow: Covering polygons is hard, FOCS 1988/Journal of Algorithms 1994] and in $existsmathbb{R}$ [ORourke: The complexity of computing minimum convex covers for polygons, Allerton 1982]. We prove that MCC is $existsmathbb{R}$-hard, and the problem is thus $existsmathbb{R}$-complete. In other words, the problem is equivalent to deciding whether a system of polynomial equations and inequalities with integer coefficients has a real solution. If a cover for our constructed polygon exists, then so does a cover consisting entirely of triangles. As a byproduct, we therefore also establish that it is $existsmathbb{R}$-complete to decide whether $k$ triangles cover a given polygon. The issue that it was not known if finding a minimum cover is in $mathsf{NP}$ has repeatedly been raised in the literature, and it was mentioned as a long-standing open question already in 2001 [Eidenbenz & Widmayer: An approximation algorithm for minimum convex cover with logarithmic performance guarantee, ESA 2001/SIAM Journal on Computing 2003]. We prove that assuming the widespread belief that $mathsf{NP} eqexistsmathbb{R}$, the problem is not in $mathsf{NP}$. An implication of the result is that many natural approaches to finding small covers are bound to give suboptimal solutions in some cases, since irrational coordinates of arbitrarily high algebraic degree can be needed for the corners of the pieces in an optimal solution.
We show that many natural two-dimensional packing problems are algorithmically equivalent to finding real roots of multivariate polynomials. A two-dimensional packing problem is defined by the type of pieces, containers, and motions that are allowed. The aim is to decide if a given set of pieces can be placed inside a given container. The pieces must be placed so that they are pairwise interior-disjoint, and only motions of the allowed type can be used to move them there. We establish a framework which enables us to show that for many combinations of allowed pieces, containers, and motions, the resulting problem is $existsmathbb R$-complete. This means that the problem is equivalent (under polynomial time reductions) to deciding whether a given system of polynomial equations and inequalities with integer coefficients has a real solution. We consider packing problems where only translations are allowed as the motions, and problems where arbitrary rigid motions are allowed, i.e., both translations and rotations. When rotations are allowed, we show that the following combinations of allowed pieces and containers are $existsmathbb R$-complete: $bullet$ simple polygons, each of which has at most 8 corners, into a square, $bullet$ convex objects bounded by line segments and hyperbolic curves into a square, $bullet$ convex polygons into a container bounded by line segments and hyperbolic curves. Restricted to translations, we show that the following combinations are $existsmathbb R$-complete: $bullet$ objects bounded by segments and hyperbolic curves into a square, $bullet$ convex polygons into a container bounded by segments and hyperbolic curves.
We study the $c$-approximate near neighbor problem under the continuous Frechet distance: Given a set of $n$ polygonal curves with $m$ vertices, a radius $delta > 0$, and a parameter $k leq m$, we want to preprocess the curves into a data structure that, given a query curve $q$ with $k$ vertices, either returns an input curve with Frechet distance at most $ccdot delta$ to $q$, or returns that there exists no input curve with Frechet distance at most $delta$ to $q$. We focus on the case where the input and the queries are one-dimensional polygonal curves -- also called time series -- and we give a comprehensive analysis for this case. We obtain new upper bounds that provide different tradeoffs between approximation factor, preprocessing time, and query time. Our data structures improve upon the state of the art in several ways. We show that for any $0 < varepsilon leq 1$ an approximation factor of $(1+varepsilon)$ can be achieved within the same asymptotic time bounds as the previously best result for $(2+varepsilon)$. Moreover, we show that an approximation factor of $(2+varepsilon)$ can be obtained by using preprocessing time and space $O(nm)$, which is linear in the input size, and query time in $O(frac{1}{varepsilon})^{k+2}$, where the previously best result used preprocessing time in $n cdot O(frac{m}{varepsilon k})^k$ and query time in $O(1)^k$. We complement our upper bounds with matching conditional lower bounds based on the Orthogonal Vectors Hypothesis. Interestingly, some of our lower bounds already hold for any super-constant value of $k$. This is achieved by proving hardness of a one-sided sparse version of the Orthogonal Vectors problem as an intermediate problem, which we believe to be of independent interest.
We study the complexity of Maximum Clique in intersection graphs of convex objects in the plane. On the algorithmic side, we extend the polynomial-time algorithm for unit disks [Clark 90, Raghavan and Spinrad 03] to translates of any fixed convex set. We also generalize the efficient polynomial-time approximation scheme (EPTAS) and subexponential algorithm for disks [Bonnet et al. 18, Bonamy et al. 18] to homothets of a fixed centrally symmetric convex set. The main open question on that topic is the complexity of Maximum Clique in disk graphs. It is not known whether this problem is NP-hard. We observe that, so far, all the hardness proofs for Maximum Clique in intersection graph classes $mathcal I$ follow the same road. They show that, for every graph $G$ of a large-enough class $mathcal C$, the complement of an even subdivision of $G$ belongs to the intersection class $mathcal I$. Then they conclude invoking the hardness of Maximum Independent Set on the class $mathcal C$, and the fact that the even subdivision preserves that hardness. However there is a strong evidence that this approach cannot work for disk graphs [Bonnet et al. 18]. We suggest a new approach, based on a problem that we dub Max Interval Permutation Avoidance, which we prove unlikely to have a subexponential-time approximation scheme. We transfer that hardness to Maximum Clique in intersection graphs of objects which can be either half-planes (or unit disks) or axis-parallel rectangles. That problem is not amenable to the previous approach. We hope that a scaled down (merely NP-hard) variant of Max Interval Permutation Avoidance could help making progress on the disk case, for instance by showing the NP-hardness for (convex) pseudo-disks.
A well-known conjecture of Tuza asserts that if a graph has at most $t$ pairwise edge-disjoint triangles, then it can be made triangle-free by removing at most $2t$ edges. If true, the factor 2 would be best possible. In the directed setting, also asked by Tuza, the analogous statement has recently been proven, however, the factor 2 is not optimal. In this paper, we show that if an $n$-vertex directed graph has at most $t$ pairwise arc-disjoint directed triangles, then there exists a set of at most $1.8t+o(n^2)$ arcs that meets all directed triangles. We complement our result by presenting two constructions of large directed graphs with $tinOmega(n^2)$ whose smallest such set has $1.5t-o(n^2)$ arcs.
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