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Added by
Sandor P. Fekete
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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We consider the online problem of packing circles into a square container. A sequence of circles has to be packed one at a time, without knowledge of the following incoming circles and without moving previously packed circles. We present an algorithm that packs any online sequence of circles with a combined area not larger than 0.350389 0.350389 of the squares area, improving the previous best value of {pi}/10 approx 0.31416; even in an offline setting, there is an upper bound of {pi}/(3 + 2 sqrt{2}) approx 0.5390. If only circles with radii of at least 0.026622 are considered, our algorithm achieves the higher value 0.375898. As a byproduct, we give an online algorithm for packing circles into a 1times b rectangle with b geq 1. This algorithm is worst case-optimal for b geq 2.36.
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We give an asymptotic approximation scheme (APTAS) for the problem of packing a set of circles into a minimum number of unit square bins. To obtain rational solutions, we use augmented bins of height $1+gamma$, for some arbitrarily small number $gamma > 0$. Our algorithm is polynomial on $log 1/gamma$, and thus $gamma$ is part of the problem input. For the special case that $gamma$ is constant, we give a (one dimensional) resource augmentation scheme, that is, we obtain a packing into bins of unit width and height $1+gamma$ using no more than the number of bins in an optimal packing. Additionally, we obtain an APTAS for the circle strip packing problem, whose goal is to pack a set of circles into a strip of unit width and minimum height. These are the first approximation and resource augmentation schemes for these problems. Our algorithm is based on novel ideas of iteratively separating small and large items, and may be extended to a wide range of packing problems that satisfy certain conditions. These extensions comprise problems with different kinds of items, such as regular polygons, or with bins of different shapes, such as circles and spheres. As an example, we obtain APTASs for the problems of packing d-dimensional spheres into hypercubes under the $L_p$-norm.
Given a set R of red points and a set B of blue points in the plane, the Red-Blue point separation problem asks if there are at most k lines that separate R from B, that is, each cell induced by the lines of the solution is either empty or monochromatic (containing points of only one color). A common variant of the problem is when the lines are required to be axis-parallel. The problem is known to be NP-complete for both scenarios, and W[1]-hard parameterized by k in the former setting and FPT in the latter. We demonstrate a polynomial-time algorithm for the special case when the points lie on a circle. Further, we also demonstrate the W-hardness of a related problem in the axis-parallel setting, where the question is if there are p horizontal and q vertical lines that separate R from B. The hardness here is shown in the parameter p.
We study emph{parallel} online algorithms: For some fixed integer $k$, a collective of $k$ parallel processes that perform online decisions on the same sequence of events forms a $k$-emph{copy algorithm}. For any given time and input sequence, the overall performance is determined by the best of the $k$ individual total results. Problems of this type have been considered for online makespan minimization; they are also related to optimization with emph{advice} on future events, i.e., a number of bits available in advance. We develop textsc{Predictive Harmonic}$_3$ (PH3), a relatively simple family of $k$-copy algorithms for the online Bin Packing Problem, whose joint competitive factor converges to 1.5 for increasing $k$. In particular, we show that $k=6$ suffices to guarantee a factor of $1.5714$ for PH3, which is better than $1.57829$, the performance of the best known 1-copy algorithm textsc{Advanced Harmonic}, while $k=11$ suffices to achieve a factor of $1.5406$, beating the known lower bound of $1.54278$ for a single online algorithm. In the context of online optimization with advice, our approach implies that 4 bits suffice to achieve a factor better than this bound of $1.54278$, which is considerably less than the previous bound of 15 bits.
We give algorithms with running time $2^{O({sqrt{k}log{k}})} cdot n^{O(1)}$ for the following problems. Given an $n$-vertex unit disk graph $G$ and an integer $k$, decide whether $G$ contains (1) a path on exactly/at least $k$ vertices, (2) a cycle on exactly $k$ vertices, (3) a cycle on at least $k$ vertices, (4) a feedback vertex set of size at most $k$, and (5) a set of $k$ pairwise vertex-disjoint cycles. For the first three problems, no subexponential time parameterized algorithms were previously known. For the remaining two problems, our algorithms significantly outperform the previously best known parameterized algorithms that run in time $2^{O(k^{0.75}log{k})} cdot n^{O(1)}$. Our algorithms are based on a new kind of tree decompositions of unit disk graphs where the separators can have size up to $k^{O(1)}$ and there exists a solution that crosses every separator at most $O(sqrt{k})$ times. The running times of our algorithms are optimal up to the $log{k}$ factor in the exponent, assuming the Exponential Time Hypothesis.
In the stochastic online vector balancing problem, vectors $v_1,v_2,ldots,v_T$ chosen independently from an arbitrary distribution in $mathbb{R}^n$ arrive one-by-one and must be immediately given a $pm$ sign. The goal is to keep the norm of the discrepancy vector, i.e., the signed prefix-sum, as small as possible for a given target norm. We consider some of the most well-known problems in discrepancy theory in the above online stochastic setting, and give algorithms that match the known offline bounds up to $mathsf{polylog}(nT)$ factors. This substantially generalizes and improves upon the previous results of Bansal, Jiang, Singla, and Sinha (STOC 20). In particular, for the Koml{o}s problem where $|v_t|_2leq 1$ for each $t$, our algorithm achieves $tilde{O}(1)$ discrepancy with high probability, improving upon the previous $tilde{O}(n^{3/2})$ bound. For Tusn{a}dys problem of minimizing the discrepancy of axis-aligned boxes, we obtain an $O(log^{d+4} T)$ bound for arbitrary distribution over points. Previous techniques only worked for product distributions and gave a weaker $O(log^{2d+1} T)$ bound. We also consider the Banaszczyk setting, where given a symmetric convex body $K$ with Gaussian measure at least $1/2$, our algorithm achieves $tilde{O}(1)$ discrepancy with respect to the norm given by $K$ for input distributions with sub-exponential tails. Our key idea is to introduce a potential that also enforces constraints on how the discrepancy vector evolves, allowing us to maintain certain anti-concentration properties. For the Banaszczyk setting, we further enhance this potential by combining it with ideas from generic chaining. Finally, we also extend these results to the setting of online multi-color discrepancy.
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