M. Balko, S. Chaplick, R. Ganian, Siddharth Gupta, M. Hoffmann, P. Valtr, A. Wolff
{"title":"Bounding and computing obstacle numbers of graphs","authors":"M. Balko, S. Chaplick, R. Ganian, Siddharth Gupta, M. Hoffmann, P. Valtr, A. Wolff","doi":"10.48550/arXiv.2206.15414","DOIUrl":"https://doi.org/10.48550/arXiv.2206.15414","url":null,"abstract":"An obstacle representation of a graph $G$ consists of a set of pairwise disjoint simply-connected closed regions and a one-to-one mapping of the vertices of $G$ to points such that two vertices are adjacent in $G$ if and only if the line segment connecting the two corresponding points does not intersect any obstacle. The obstacle number of a graph is the smallest number of obstacles in an obstacle representation of the graph in the plane such that all obstacles are simple polygons. It is known that the obstacle number of each $n$-vertex graph is $O(n log n)$ [Balko, Cibulka, and Valtr, 2018] and that there are $n$-vertex graphs whose obstacle number is $Omega(n/(loglog n)^2)$ [Dujmovi'c and Morin, 2015]. We improve this lower bound to $Omega(n/loglog n)$ for simple polygons and to $Omega(n)$ for convex polygons. To obtain these stronger bounds, we improve known estimates on the number of $n$-vertex graphs with bounded obstacle number, solving a conjecture by Dujmovi'c and Morin. We also show that if the drawing of some $n$-vertex graph is given as part of the input, then for some drawings $Omega(n^2)$ obstacles are required to turn them into an obstacle representation of the graph. Our bounds are asymptotically tight in several instances. We complement these combinatorial bounds by two complexity results. First, we show that computing the obstacle number of a graph $G$ is fixed-parameter tractable in the vertex cover number of $G$. Second, we show that, given a graph $G$ and a simple polygon $P$, it is NP-hard to decide whether $G$ admits an obstacle representation using $P$ as the only obstacle.","PeriodicalId":201778,"journal":{"name":"Embedded Systems and Applications","volume":"447 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126996613","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}
{"title":"Tight Bounds for Online Matching in Bounded-Degree Graphs with Vertex Capacities","authors":"S. Albers, S. Schubert","doi":"10.48550/arXiv.2206.15336","DOIUrl":"https://doi.org/10.48550/arXiv.2206.15336","url":null,"abstract":"We study the $b$-matching problem in bipartite graphs $G=(S,R,E)$. Each vertex $sin S$ is a server with individual capacity $b_s$. The vertices $rin R$ are requests that arrive online and must be assigned instantly to an eligible server. The goal is to maximize the size of the constructed matching. We assume that $G$ is a $(k,d)$-graph~cite{NW}, where $k$ specifies a lower bound on the degree of each server and $d$ is an upper bound on the degree of each request. This setting models matching problems in timely applications. We present tight upper and lower bounds on the performance of deterministic online algorithms. In particular, we develop a new online algorithm via a primal-dual analysis. The optimal competitive ratio tends to~1, for arbitrary $kgeq d$, as the server capacities increase. Hence, nearly optimal solutions can be computed online. Our results also hold for the vertex-weighted problem extension, and thus for AdWords and auction problems in which each bidder issues individual, equally valued bids. Our bounds improve the previous best competitive ratios. The asymptotic competitiveness of~1 is a significant improvement over the previous factor of $1-1/e^{k/d}$, for the interesting range where $k/dgeq 1$ is small. Recall that $1-1/eapprox 0.63$. Matching problems that admit a competitive ratio arbitrarily close to~1 are rare. Prior results rely on randomization or probabilistic input models.","PeriodicalId":201778,"journal":{"name":"Embedded Systems and Applications","volume":"116 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132804566","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}
Aleksander Figiel, Vincent Froese, A. Nichterlein, R. Niedermeier
{"title":"There and Back Again: On Applying Data Reduction Rules by Undoing Others","authors":"Aleksander Figiel, Vincent Froese, A. Nichterlein, R. Niedermeier","doi":"10.48550/arXiv.2206.14698","DOIUrl":"https://doi.org/10.48550/arXiv.2206.14698","url":null,"abstract":"Data reduction rules are an established method in the algorithmic toolbox for tackling computationally challenging problems. A data reduction rule is a polynomial-time algorithm that, given a problem instance as input, outputs an equivalent, typically smaller instance of the same problem. The application of data reduction rules during the preprocessing of problem instances allows in many cases to considerably shrink their size, or even solve them directly. Commonly, these data reduction rules are applied exhaustively and in some fixed order to obtain irreducible instances. It was often observed that by changing the order of the rules, different irreducible instances can be obtained. We propose to “undo” data reduction rules on irreducible instances, by which they become larger, and then subsequently apply data reduction rules again to shrink them. We show that this somewhat counter-intuitive approach can lead to significantly smaller irreducible instances. The process of undoing data reduction rules is not limited to “rolling back” data reduction rules applied to the instance during preprocessing. Instead, we formulate so-called backward rules, which essentially undo a data reduction rule, but without using any information about which data reduction rules were applied to it previously. In particular, based on the example of Vertex Cover we propose two methods applying backward rules to shrink the instances further. In our experiments we show that this way smaller irreducible instances consisting of real-world graphs from the SNAP and DIMACS datasets can be computed.","PeriodicalId":201778,"journal":{"name":"Embedded Systems and Applications","volume":"25 1 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":"131338114","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}
Davide Bilò, Gianlorenzo D'angelo, Luciano Gualà, S. Leucci, Mirko Rossi
{"title":"Sparse Temporal Spanners with Low Stretch","authors":"Davide Bilò, Gianlorenzo D'angelo, Luciano Gualà, S. Leucci, Mirko Rossi","doi":"10.48550/arXiv.2206.11113","DOIUrl":"https://doi.org/10.48550/arXiv.2206.11113","url":null,"abstract":"A temporal graph is an undirected graph G = ( V, E ) along with a function λ : E → N + that assigns a time-label to each edge in E . A path in G such that the traversed time-labels are non-decreasing is called temporal path . Accordingly, the distance from u to v is the minimum length (i.e., the number of edges) of a temporal path from u to v . A temporal α -spanner of G is a (temporal) subgraph H that preserves the distances between any pair of vertices in V , up to a multiplicative stretch factor of α . The size of H is measured as the number of its edges. In this work, we study the size-stretch trade-offs of temporal spanners. In particular we show that temporal cliques always admit a temporal (2 k − 1) − spanner with (cid:101) O ( kn 1+ 1 k ) edges, where k > 1 is an integer parameter of choice. Choosing k = (cid:98) log n (cid:99) , we obtain a temporal O (log n )-spanner with (cid:101) O ( n ) edges that has almost the same size (up to logarithmic factors) as the temporal spanner given in [Casteigts et al., JCSS 2021] which only preserves temporal connectivity. We then turn our attention to general temporal graphs. Since Ω( n 2 ) edges might be needed by any connectivity-preserving temporal subgraph [Axiotis et al., ICALP’16], we focus on approximating distances from a single source . We show that (cid:101) O ( n/ log(1 + ε )) edges suffice to obtain a stretch of (1 + ε ), for any small ε > 0. This result is essentially tight in the following sense: there are temporal graphs G for which any temporal subgraph preserving exact distances from a single-source must use Ω( n 2 ) edges. Interestingly enough, our analysis can be extended to the case of additive stretch for which we prove an upper bound of (cid:101) O ( n 2 /β ) on the size of any temporal β -additive spanner, which we show to be tight up to polylogarithmic factors. Finally, we investigate how the lifetime of G , i.e., the number of its distinct time-labels, affects the trade-off between the size and the stretch of a temporal spanner.","PeriodicalId":201778,"journal":{"name":"Embedded Systems and Applications","volume":"57 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133348564","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}
{"title":"List Colouring Trees in Logarithmic Space","authors":"H. Bodlaender, C. Groenland, Hugo Jacob","doi":"10.48550/arXiv.2206.09750","DOIUrl":"https://doi.org/10.48550/arXiv.2206.09750","url":null,"abstract":"We show that List Colouring can be solved on n -vertex trees by a deterministic Turing machine using O (log n ) bits on the worktape. Given an n -vertex graph G = ( V, E ) and a list L ( v ) ⊆ { 1 , . . . , n } of available colours for each v ∈ V , a list colouring for G is a proper colouring c such that c ( v ) ∈ L ( v ) for all v .","PeriodicalId":201778,"journal":{"name":"Embedded Systems and Applications","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124720900","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}
{"title":"The Price of Hierarchical Clustering","authors":"Anna Arutyunova, Heiko Röglin","doi":"10.48550/arXiv.2205.01417","DOIUrl":"https://doi.org/10.48550/arXiv.2205.01417","url":null,"abstract":"Hierarchical Clustering is a popular tool for understanding the hereditary properties of a data set. Such a clustering is actually a sequence of clusterings that starts with the trivial clustering in which every data point forms its own cluster and then successively merges two existing clusters until all points are in the same cluster. A hierarchical clustering achieves an approximation factor of $alpha$ if the costs of each $k$-clustering in the hierarchy are at most $alpha$ times the costs of an optimal $k$-clustering. We study as cost functions the maximum (discrete) radius of any cluster ($k$-center problem) and the maximum diameter of any cluster ($k$-diameter problem). In general, the optimal clusterings do not form a hierarchy and hence an approximation factor of $1$ cannot be achieved. We call the smallest approximation factor that can be achieved for any instance the price of hierarchy. For the $k$-diameter problem we improve the upper bound on the price of hierarchy to $3+2sqrt{2}approx 5.83$. Moreover we significantly improve the lower bounds for $k$-center and $k$-diameter, proving a price of hierarchy of exactly $4$ and $3+2sqrt{2}$, respectively.","PeriodicalId":201778,"journal":{"name":"Embedded Systems and Applications","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133915857","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}
A. D. Cunha, Francesco d’Amore, F. Giroire, Hicham Lesfari, Emanuele Natale, L. Viennot
{"title":"Revisiting the Random Subset Sum Problem","authors":"A. D. Cunha, Francesco d’Amore, F. Giroire, Hicham Lesfari, Emanuele Natale, L. Viennot","doi":"10.4230/LIPIcs.ESA.2023.37","DOIUrl":"https://doi.org/10.4230/LIPIcs.ESA.2023.37","url":null,"abstract":"The average properties of the well-known Subset Sum Problem can be studied by the means of its randomised version, where we are given a target value $z$, random variables $X_1, ldots, X_n$, and an error parameter $varepsilon>0$, and we seek a subset of the $X_i$s whose sum approximates $z$ up to error $varepsilon$. In this setup, it has been shown that, under mild assumptions on the distribution of the random variables, a sample of size $mathcal{O}(log(1/varepsilon))$ suffices to obtain, with high probability, approximations for all values in $[-1/2, 1/2]$. Recently, this result has been rediscovered outside the algorithms community, enabling meaningful progress in other fields. In this work we present an alternative proof for this theorem, with a more direct approach and resourcing to more elementary tools.","PeriodicalId":201778,"journal":{"name":"Embedded Systems and Applications","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131101427","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}
{"title":"Fast Computation of Zigzag Persistence","authors":"T. Dey, T. Hou","doi":"10.48550/arXiv.2204.11080","DOIUrl":"https://doi.org/10.48550/arXiv.2204.11080","url":null,"abstract":"Zigzag persistence is a powerful extension of the standard persistence which allows deletions of simplices besides insertions. However, computing zigzag persistence usually takes considerably more time than the standard persistence. We propose an algorithm called FastZigzag which narrows this efficiency gap. Our main result is that an input simplex-wise zigzag filtration can be converted to a cell-wise non-zigzag filtration of a $Delta$-complex with the same length, where the cells are copies of the input simplices. This conversion step in FastZigzag incurs very little cost. Furthermore, the barcode of the original filtration can be easily read from the barcode of the new cell-wise filtration because the conversion embodies a series of diamond switches known in topological data analysis. This seemingly simple observation opens up the vast possibilities for improving the computation of zigzag persistence because any efficient algorithm/software for standard persistence can now be applied to computing zigzag persistence. Our experiment shows that this indeed achieves substantial performance gain over the existing state-of-the-art softwares.","PeriodicalId":201778,"journal":{"name":"Embedded Systems and Applications","volume":"71 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131052230","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}
{"title":"Determinants from homomorphisms","authors":"Radu Curticapean","doi":"10.48550/arXiv.2204.10718","DOIUrl":"https://doi.org/10.48550/arXiv.2204.10718","url":null,"abstract":"We give a new combinatorial explanation for well-known relations between determinants and traces of matrix powers. Such relations can be used to obtain polynomial-time and poly-logarithmic space algorithms for the determinant. Our new explanation avoids linear-algebraic arguments and instead exploits a classical connection between subgraph and homomorphism counts.","PeriodicalId":201778,"journal":{"name":"Embedded Systems and Applications","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115286806","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}
{"title":"Faster Approximate Covering of Subcurves under the Fréchet Distance","authors":"Frederik Brüning, Jacobus Conradi, A. Driemel","doi":"10.48550/arXiv.2204.09949","DOIUrl":"https://doi.org/10.48550/arXiv.2204.09949","url":null,"abstract":"Subtrajectory clustering is an important variant of the trajectory clustering problem, where the start and endpoints of trajectory patterns within the collected trajectory data are not known in advance. We study this problem in the form of a set cover problem for a given polygonal curve: find the smallest number k of representative curves such that any point on the input curve is contained in a subcurve that has Fréchet distance at most a given ∆ to a representative curve. We focus on the case where the representative curves are line segments and approach this NP-hard problem with classical techniques from the area of geometric set cover: we use a variant of the multiplicative weights update method which was first suggested by Brönniman and Goodrich for set cover instances with small VC-dimension. We obtain a bicriteria-approximation algorithm that computes a set of O ( k log( k )) line segments that cover a given polygonal curve of n vertices under Fréchet distance at most O (∆). We show that the algorithm runs in e O ( k 2 n + kn 3 ) time in expectation and uses e O ( kn + n 3 ) space. For input curves that are c -packed and lie in the plane, we bound the expected running time by e O ( k 2 c 2 n ) and the space by e O ( kn + c 2 n ). In addition, we present a variant of the algorithm that uses implicit weight updates on the candidate set and thereby achieves near-linear running time in n without any assumptions on the input curve, while keeping the same approximation bounds. This comes at the expense of a small (polylogarithmic) dependency on the relative arclength.","PeriodicalId":201778,"journal":{"name":"Embedded Systems and Applications","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126703726","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}
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