Pub Date : 2022-06-29DOI: 10.4230/LIPIcs.SWAT.2022.8
Patrizio Angelini, M. Bekos, G. D. Lozzo, Martin Gronemann, Fabrizio Montecchiani, Alessandra Tappini
A map is a partition of the sphere into interior-disjoint regions homeomorphic to closed disks. Some regions are labeled as nations, while the remaining ones are labeled as holes. A map in which at most k nations touch at the same point is a k-map, while it is hole-free if it contains no holes. A graph is a map graph if there is a bijection between its vertices and the nations of a map, such that two nations touch if and only the corresponding vertices are connected by an edge. We present a fixed-parameter tractable algorithm for recognizing map graphs parameterized by treewidth. Its time complexity is linear in the size of the graph. It reports a certificate in the form of a so-called witness, if the input is a yes-instance. Our algorithmic framework is general enough to test, for any k, if the input graph admits a k-map or a hole-free k-map.
{"title":"Recognizing Map Graphs of Bounded Treewidth","authors":"Patrizio Angelini, M. Bekos, G. D. Lozzo, Martin Gronemann, Fabrizio Montecchiani, Alessandra Tappini","doi":"10.4230/LIPIcs.SWAT.2022.8","DOIUrl":"https://doi.org/10.4230/LIPIcs.SWAT.2022.8","url":null,"abstract":"A map is a partition of the sphere into interior-disjoint regions homeomorphic to closed disks. Some regions are labeled as nations, while the remaining ones are labeled as holes. A map in which at most k nations touch at the same point is a k-map, while it is hole-free if it contains no holes. A graph is a map graph if there is a bijection between its vertices and the nations of a map, such that two nations touch if and only the corresponding vertices are connected by an edge. We present a fixed-parameter tractable algorithm for recognizing map graphs parameterized by treewidth. Its time complexity is linear in the size of the graph. It reports a certificate in the form of a so-called witness, if the input is a yes-instance. Our algorithmic framework is general enough to test, for any k, if the input graph admits a k-map or a hole-free k-map.","PeriodicalId":447445,"journal":{"name":"Scandinavian Workshop on Algorithm Theory","volume":"160 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122041778","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-05-06DOI: 10.48550/arXiv.2205.03450
Matt Gibson-Lopez, Serge Zamarripa
Representation of Euclidean objects in a digital space has been a focus of research for over 30 years. Digital line segments are particularly important as other digital objects depend on their definition (e.g., digital convex objects or digital star-shaped objects). It may be desirable for the digital line segment systems to satisfy some nice properties that their Euclidean counterparts also satisfy. The system is a consistent digital line segment system (CDS) if it satisfies five properties, most notably the subsegment property (the intersection of any two digital line segments should be connected) and the prolongation property (any digital line segment should be able to be extended into a digital line). It is known that any CDS must have Ω(log n ) Hausdorff distance to their Euclidean counterparts, where n is the number of grid points on a segment. In fact this lower bound even applies to consistent digital rays (CDR) where for a fixed p ∈ Z 2 , we consider the digital segments from p to q for each q ∈ Z 2 . In this paper, we consider families of weak consistent digital rays (WCDR) where we maintain four of the CDR properties but exclude the prolongation property. In this paper, we give a WCDR construction that has optimal Hausdorff distance to the exact constant. That is, we give a construction whose Hausdorff distance is 1.5 under the L ∞ metric, and we show that for every ϵ > 0, it is not possible to have a WCDR with Hausdorff distance at most 1 . 5 − ϵ .
{"title":"Optimal Bounds for Weak Consistent Digital Rays in 2D","authors":"Matt Gibson-Lopez, Serge Zamarripa","doi":"10.48550/arXiv.2205.03450","DOIUrl":"https://doi.org/10.48550/arXiv.2205.03450","url":null,"abstract":"Representation of Euclidean objects in a digital space has been a focus of research for over 30 years. Digital line segments are particularly important as other digital objects depend on their definition (e.g., digital convex objects or digital star-shaped objects). It may be desirable for the digital line segment systems to satisfy some nice properties that their Euclidean counterparts also satisfy. The system is a consistent digital line segment system (CDS) if it satisfies five properties, most notably the subsegment property (the intersection of any two digital line segments should be connected) and the prolongation property (any digital line segment should be able to be extended into a digital line). It is known that any CDS must have Ω(log n ) Hausdorff distance to their Euclidean counterparts, where n is the number of grid points on a segment. In fact this lower bound even applies to consistent digital rays (CDR) where for a fixed p ∈ Z 2 , we consider the digital segments from p to q for each q ∈ Z 2 . In this paper, we consider families of weak consistent digital rays (WCDR) where we maintain four of the CDR properties but exclude the prolongation property. In this paper, we give a WCDR construction that has optimal Hausdorff distance to the exact constant. That is, we give a construction whose Hausdorff distance is 1.5 under the L ∞ metric, and we show that for every ϵ > 0, it is not possible to have a WCDR with Hausdorff distance at most 1 . 5 − ϵ .","PeriodicalId":447445,"journal":{"name":"Scandinavian Workshop on Algorithm Theory","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131332049","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-26DOI: 10.48550/arXiv.2204.12614
Max Bannach, Pamela Fleischmann, Malte Skambath
The natural generalization of the Boolean satisfiability problem to optimization problems is the task of determining the maximum number of clauses that can simultaneously be satisfied in a propositional formula in conjunctive normal form. In the weighted maximum satisfiability problem each clause has a positive weight and one seeks an assignment of maximum weight. The literature almost solely considers the case of positive weights. While the general case of the problem is only restricted slightly by this constraint, many special cases become trivial in the absence of negative weights. In this work we study the problem with negative weights and observe that the problem becomes computationally harder - which we formalize from a parameterized perspective in the sense that various variations of the problem become W[1]-hard if negative weights are present. Allowing negative weights also introduces new variants of the problem: Instead of maximizing the sum of weights of satisfied clauses, we can maximize the absolute value of that sum. This turns out to be surprisingly expressive even restricted to monotone formulas in disjunctive normal form with at most two literals per clause. In contrast to the versions without the absolute value, however, we prove that these variants are fixed-parameter tractable. As technical contribution we present a kernelization for an auxiliary problem on hypergraphs in which we seek, given an edge-weighted hypergraph, an induced subgraph that maximizes the absolute value of the sum of edge-weights.
{"title":"MaxSAT with Absolute Value Functions: A Parameterized Perspective","authors":"Max Bannach, Pamela Fleischmann, Malte Skambath","doi":"10.48550/arXiv.2204.12614","DOIUrl":"https://doi.org/10.48550/arXiv.2204.12614","url":null,"abstract":"The natural generalization of the Boolean satisfiability problem to optimization problems is the task of determining the maximum number of clauses that can simultaneously be satisfied in a propositional formula in conjunctive normal form. In the weighted maximum satisfiability problem each clause has a positive weight and one seeks an assignment of maximum weight. The literature almost solely considers the case of positive weights. While the general case of the problem is only restricted slightly by this constraint, many special cases become trivial in the absence of negative weights. In this work we study the problem with negative weights and observe that the problem becomes computationally harder - which we formalize from a parameterized perspective in the sense that various variations of the problem become W[1]-hard if negative weights are present. Allowing negative weights also introduces new variants of the problem: Instead of maximizing the sum of weights of satisfied clauses, we can maximize the absolute value of that sum. This turns out to be surprisingly expressive even restricted to monotone formulas in disjunctive normal form with at most two literals per clause. In contrast to the versions without the absolute value, however, we prove that these variants are fixed-parameter tractable. As technical contribution we present a kernelization for an auxiliary problem on hypergraphs in which we seek, given an edge-weighted hypergraph, an induced subgraph that maximizes the absolute value of the sum of edge-weights.","PeriodicalId":447445,"journal":{"name":"Scandinavian Workshop on Algorithm Theory","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121086337","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-04-19DOI: 10.48550/arXiv.2204.08992
Haitao Wang
Given a set $P$ of $n$ points in the plane, we consider the problem of computing the number of points of $P$ in a query unit disk (i.e., all query disks have the same radius). We show that the main techniques for simplex range searching in the plane can be adapted to this problem. For example, by adapting Matouv{s}ek's results, we can build a data structure of $O(n)$ space so that each query can be answered in $O(sqrt{n})$ time. Our techniques lead to improvements for several other classical problems, such as batched range searching, counting/reporting intersecting pairs of unit circles, distance selection, discrete 2-center, etc. For example, given a set of $n$ unit disks and a set of $n$ points in the plane, the batched range searching problem is to compute for each disk the number of points in it. Previous work [Katz and Sharir, 1997] solved the problem in $O(n^{4/3}log n)$ time while our new algorithm runs in $O(n^{4/3})$ time.
{"title":"Unit-Disk Range Searching and Applications","authors":"Haitao Wang","doi":"10.48550/arXiv.2204.08992","DOIUrl":"https://doi.org/10.48550/arXiv.2204.08992","url":null,"abstract":"Given a set $P$ of $n$ points in the plane, we consider the problem of computing the number of points of $P$ in a query unit disk (i.e., all query disks have the same radius). We show that the main techniques for simplex range searching in the plane can be adapted to this problem. For example, by adapting Matouv{s}ek's results, we can build a data structure of $O(n)$ space so that each query can be answered in $O(sqrt{n})$ time. Our techniques lead to improvements for several other classical problems, such as batched range searching, counting/reporting intersecting pairs of unit circles, distance selection, discrete 2-center, etc. For example, given a set of $n$ unit disks and a set of $n$ points in the plane, the batched range searching problem is to compute for each disk the number of points in it. Previous work [Katz and Sharir, 1997] solved the problem in $O(n^{4/3}log n)$ time while our new algorithm runs in $O(n^{4/3})$ time.","PeriodicalId":447445,"journal":{"name":"Scandinavian Workshop on Algorithm Theory","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115169492","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-01DOI: 10.48550/arXiv.2203.00285
J. Boyar, Lene M. Favrholdt, Kim S. Larsen
A variant of the online knapsack problem is considered in the settings of trusted and untrusted predictions. In Unit Profit Knapsack, the items have unit profit, and it is easy to find an optimal solution offline: Pack as many of the smallest items as possible into the knapsack. For Online Unit Profit Knapsack, the competitive ratio is unbounded. In contrast, previous work on online algorithms with untrusted predictions generally studied problems where an online algorithm with a constant competitive ratio is known. The prediction, possibly obtained from a machine learning source, that our algorithm uses is the average size of those smallest items that fit in the knapsack. For the prediction error in this hard online problem, we use the ratio $r=frac{a}{hat{a}}$ where $a$ is the actual value for this average size and $hat{a}$ is the prediction. The algorithm presented achieves a competitive ratio of $frac{1}{2r}$ for $rgeq 1$ and $frac{r}{2}$ for $rleq 1$. Using an adversary technique, we show that this is optimal in some sense, giving a trade-off in the competitive ratio attainable for different values of $r$. Note that the result for accurate advice, $r=1$, is only $frac{1}{2}$, but we show that no algorithm knowing the value $a$ can achieve a competitive ratio better than $frac{e-1}{e}approx 0.6321$ and present an algorithm with a matching upper bound. We also show that this latter algorithm attains a competitive ratio of $rfrac{e-1}{e}$ for $r leq 1$ and $frac{e-r}{e}$ for $1 leq r
{"title":"Online Unit Profit Knapsack with Untrusted Predictions","authors":"J. Boyar, Lene M. Favrholdt, Kim S. Larsen","doi":"10.48550/arXiv.2203.00285","DOIUrl":"https://doi.org/10.48550/arXiv.2203.00285","url":null,"abstract":"A variant of the online knapsack problem is considered in the settings of trusted and untrusted predictions. In Unit Profit Knapsack, the items have unit profit, and it is easy to find an optimal solution offline: Pack as many of the smallest items as possible into the knapsack. For Online Unit Profit Knapsack, the competitive ratio is unbounded. In contrast, previous work on online algorithms with untrusted predictions generally studied problems where an online algorithm with a constant competitive ratio is known. The prediction, possibly obtained from a machine learning source, that our algorithm uses is the average size of those smallest items that fit in the knapsack. For the prediction error in this hard online problem, we use the ratio $r=frac{a}{hat{a}}$ where $a$ is the actual value for this average size and $hat{a}$ is the prediction. The algorithm presented achieves a competitive ratio of $frac{1}{2r}$ for $rgeq 1$ and $frac{r}{2}$ for $rleq 1$. Using an adversary technique, we show that this is optimal in some sense, giving a trade-off in the competitive ratio attainable for different values of $r$. Note that the result for accurate advice, $r=1$, is only $frac{1}{2}$, but we show that no algorithm knowing the value $a$ can achieve a competitive ratio better than $frac{e-1}{e}approx 0.6321$ and present an algorithm with a matching upper bound. We also show that this latter algorithm attains a competitive ratio of $rfrac{e-1}{e}$ for $r leq 1$ and $frac{e-r}{e}$ for $1 leq r","PeriodicalId":447445,"journal":{"name":"Scandinavian Workshop on Algorithm Theory","volume":"327 1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123099709","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-17DOI: 10.4230/LIPIcs.SWAT.2022.19
P. Bose, Pat Morin, Saeed Odak
The emph{Product Structure Theorem} for planar graphs (Dujmovi'c et al. emph{JACM}, textbf{67}(4):22) states that any planar graph is contained in the strong product of a planar $3$-tree, a path, and a $3$-cycle. We give a simple linear-time algorithm for finding this decomposition as well as several related decompositions. This improves on the previous $O(nlog n)$ time algorithm (Morin. emph{Algorithmica}, textbf{85}(5):1544--1558).
平面图的emph{积结构定理}(dujmovovic et al.)emph{JACM}, textbf{67}(4):22)指出任何平面图都包含在平面$3$ -tree、路径和$3$ -cycle的强积中。我们给出了一个简单的线性时间算法来找到这个分解，以及几个相关的分解。这改进了以前的$O(nlog n)$时间算法(Morin。emph{算法}，textbf{85}(5):1544—1558。
{"title":"An Optimal Algorithm for Product Structure in Planar Graphs","authors":"P. Bose, Pat Morin, Saeed Odak","doi":"10.4230/LIPIcs.SWAT.2022.19","DOIUrl":"https://doi.org/10.4230/LIPIcs.SWAT.2022.19","url":null,"abstract":"The emph{Product Structure Theorem} for planar graphs (Dujmovi'c et al. emph{JACM}, textbf{67}(4):22) states that any planar graph is contained in the strong product of a planar $3$-tree, a path, and a $3$-cycle. We give a simple linear-time algorithm for finding this decomposition as well as several related decompositions. This improves on the previous $O(nlog n)$ time algorithm (Morin. emph{Algorithmica}, textbf{85}(5):1544--1558).","PeriodicalId":447445,"journal":{"name":"Scandinavian Workshop on Algorithm Theory","volume":"53 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125488613","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-12-06DOI: 10.4230/LIPIcs.SWAT.2022.9
A. Antoniadis, Peyman Jabbarzade Ganje, Golnoosh Shahkarami
Given the rapid rise in energy demand by data centers and computing systems in general, it is fundamental to incorporate energy considerations when designing (scheduling) algorithms. Machine learning can be a useful approach in practice by predicting the future load of the system based on, for example, historical data. However, the effectiveness of such an approach highly depends on the quality of the predictions and can be quite far from optimal when predictions are sub-par. On the other hand, while providing a worst-case guarantee, classical online algorithms can be pessimistic for large classes of inputs arising in practice. This paper, in the spirit of the new area of machine learning augmented algorithms, attempts to obtain the best of both worlds for the classical, deadline based, online speed-scaling problem: Based on the introduction of a novel prediction setup, we develop algorithms that (i) obtain provably low energy-consumption in the presence of adequate predictions, and (ii) are robust against inadequate predictions, and (iii) are smooth, i.e., their performance gradually degrades as the prediction error increases.
{"title":"A Novel Prediction Setup for Online Speed-Scaling","authors":"A. Antoniadis, Peyman Jabbarzade Ganje, Golnoosh Shahkarami","doi":"10.4230/LIPIcs.SWAT.2022.9","DOIUrl":"https://doi.org/10.4230/LIPIcs.SWAT.2022.9","url":null,"abstract":"Given the rapid rise in energy demand by data centers and computing systems in general, it is fundamental to incorporate energy considerations when designing (scheduling) algorithms. Machine learning can be a useful approach in practice by predicting the future load of the system based on, for example, historical data. However, the effectiveness of such an approach highly depends on the quality of the predictions and can be quite far from optimal when predictions are sub-par. On the other hand, while providing a worst-case guarantee, classical online algorithms can be pessimistic for large classes of inputs arising in practice. This paper, in the spirit of the new area of machine learning augmented algorithms, attempts to obtain the best of both worlds for the classical, deadline based, online speed-scaling problem: Based on the introduction of a novel prediction setup, we develop algorithms that (i) obtain provably low energy-consumption in the presence of adequate predictions, and (ii) are robust against inadequate predictions, and (iii) are smooth, i.e., their performance gradually degrades as the prediction error increases.","PeriodicalId":447445,"journal":{"name":"Scandinavian Workshop on Algorithm Theory","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126935344","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-11DOI: 10.4230/LIPIcs.SWAT.2022.28
Tanmay Inamdar, Kasturi R. Varadarajan
In the Non-Uniform k -Center (NU k C) problem, a generalization of the famous k -center clustering problem, we want to cover the given set of points in a metric space by finding a placement of balls with specified radii. In t -NU k C, we assume that the number of distinct radii is equal to t , and we are allowed to use k i balls of radius r i , for 1 ≤ i ≤ t . This problem was introduced by Chakrabarty et al. [ACM Trans. Alg. 16(4):46:1-46:19], who showed that a constant approximation for t -NU k C is not possible if t is unbounded, assuming P ̸ = NP . On the other hand, they gave a bicriteria approximation that violates the number of allowed balls as well as the given radii by a constant factor. They also conjectured that a constant approximation for t -NU k C should be possible if t is a fixed constant. Since then, there has been steady progress towards resolving this conjecture – currently, a constant approximation for 3-NU k C is known via the results of Chakrabarty and Negahbani [IPCO 2021], and Jia et al. [SOSA 2022]. We push the horizon by giving an O (1)-approximation for the Non-Uniform k -Center for 4 distinct types of radii. Our result is obtained via a novel combination of tools and techniques from the k -center literature, which also demonstrates that the different generalizations of k -center involving non-uniform radii, and multiple coverage constraints (i.e., colorful k -center ), are closely interlinked with each other. We hope that our ideas will contribute towards a deeper understanding of the t -NU k C problem, eventually bringing us closer to the resolution of the CGK conjecture.
在非均匀k中心(NU k C)问题中，一个著名的k中心聚类问题的推广，我们希望通过寻找具有指定半径的球的位置来覆盖度量空间中的给定点集。在t - k C中，我们假设不同半径的个数等于t，我们可以使用k个半径为ri的球，对于1≤i≤t。这个问题是Chakrabarty等人提出的。[al16(4):46:1-46:19]，他证明了如果t无界，假设P P = NP, t -NU k C不可能有常数近似。另一方面，他们给出了一个双标准近似值，该近似值违反了允许的球数以及给定半径的常数因子。他们还推测，如果t是固定常数，t - k C的常数近似应该是可能的。从那时起，在解决这一猜想方面取得了稳步进展——目前，通过Chakrabarty和Negahbani [IPCO 2021]和Jia等[SOSA 2022]的结果，已知了3-NU k C的恒定近似值。我们通过对4种不同类型的半径给出非均匀k中心的O(1)近似来推动视界。我们的结果是通过k -center文献中的工具和技术的新颖组合获得的，这也表明，涉及非均匀半径的k -center的不同推广，以及多个覆盖约束(即彩色k -center)，彼此密切相关。我们希望我们的想法将有助于更深入地理解t -NU k C问题，最终使我们更接近CGK猜想的解决。
{"title":"Non-Uniform k-Center and Greedy Clustering","authors":"Tanmay Inamdar, Kasturi R. Varadarajan","doi":"10.4230/LIPIcs.SWAT.2022.28","DOIUrl":"https://doi.org/10.4230/LIPIcs.SWAT.2022.28","url":null,"abstract":"In the Non-Uniform k -Center (NU k C) problem, a generalization of the famous k -center clustering problem, we want to cover the given set of points in a metric space by finding a placement of balls with specified radii. In t -NU k C, we assume that the number of distinct radii is equal to t , and we are allowed to use k i balls of radius r i , for 1 ≤ i ≤ t . This problem was introduced by Chakrabarty et al. [ACM Trans. Alg. 16(4):46:1-46:19], who showed that a constant approximation for t -NU k C is not possible if t is unbounded, assuming P ̸ = NP . On the other hand, they gave a bicriteria approximation that violates the number of allowed balls as well as the given radii by a constant factor. They also conjectured that a constant approximation for t -NU k C should be possible if t is a fixed constant. Since then, there has been steady progress towards resolving this conjecture – currently, a constant approximation for 3-NU k C is known via the results of Chakrabarty and Negahbani [IPCO 2021], and Jia et al. [SOSA 2022]. We push the horizon by giving an O (1)-approximation for the Non-Uniform k -Center for 4 distinct types of radii. Our result is obtained via a novel combination of tools and techniques from the k -center literature, which also demonstrates that the different generalizations of k -center involving non-uniform radii, and multiple coverage constraints (i.e., colorful k -center ), are closely interlinked with each other. We hope that our ideas will contribute towards a deeper understanding of the t -NU k C problem, eventually bringing us closer to the resolution of the CGK conjecture.","PeriodicalId":447445,"journal":{"name":"Scandinavian Workshop on Algorithm Theory","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126423671","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-10-25DOI: 10.4230/LIPIcs.SWAT.2022.14
Benjamin Aram Berendsohn
The caterpillar associahedron A ( G ) is a polytope arising from the rotation graph of search trees on a caterpillar tree G , generalizing the rotation graph of binary search trees (BSTs) and thus the conventional associahedron. We show that the diameter of A ( G ) is Θ( n + m · ( H + 1)), where n is the number of vertices, m is the number of leaves, and H is the entropy of the leaf distribution of G . Our proofs reveal a strong connection between caterpillar associahedra and searching in BSTs. We prove the lower bound using Wilber’s first lower bound for dynamic BSTs, and the upper bound by reducing the problem to searching in static BSTs. discussions and suggestions.
{"title":"The diameter of caterpillar associahedra","authors":"Benjamin Aram Berendsohn","doi":"10.4230/LIPIcs.SWAT.2022.14","DOIUrl":"https://doi.org/10.4230/LIPIcs.SWAT.2022.14","url":null,"abstract":"The caterpillar associahedron A ( G ) is a polytope arising from the rotation graph of search trees on a caterpillar tree G , generalizing the rotation graph of binary search trees (BSTs) and thus the conventional associahedron. We show that the diameter of A ( G ) is Θ( n + m · ( H + 1)), where n is the number of vertices, m is the number of leaves, and H is the entropy of the leaf distribution of G . Our proofs reveal a strong connection between caterpillar associahedra and searching in BSTs. We prove the lower bound using Wilber’s first lower bound for dynamic BSTs, and the upper bound by reducing the problem to searching in static BSTs. discussions and suggestions.","PeriodicalId":447445,"journal":{"name":"Scandinavian Workshop on Algorithm Theory","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116678809","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-28DOI: 10.4230/LIPIcs.SWAT.2022.13
C. Bazgan, Katrin Casel, Pierre Cazals
We consider the problem of partitioning a graph into a non-fixed number of non-overlapping subgraphs of maximum density. The density of a partition is the sum of the densities of the subgraphs, where the density of a subgraph is its average degree, that is, the ratio of its number of edges and its number of vertices. This problem, called Dense Graph Partition, is known to be NP-hard on general graphs and polynomial-time solvable on trees, and polynomial-time 2-approximable. In this paper we study the restriction of Dense Graph Partition to particular sparse and dense graph classes. In particular, we prove that it is NP-hard on dense bipartite graphs as well as on cubic graphs. On dense graphs on $n$ vertices, it is polynomial-time solvable on graphs with minimum degree $n-3$ and NP-hard on $(n-4)$-regular graphs. We prove that it is polynomial-time $4/3$-approximable on cubic graphs and admits an efficient polynomial-time approximation scheme on graphs of minimum degree $n-t$ for any constant $tgeq 4$.
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