Christoph Damerius, Peter Kling, Minming Li, Chenyang Xu, Ruilong Zhang
{"title":"Scheduling with a Limited Testing Budget","authors":"Christoph Damerius, Peter Kling, Minming Li, Chenyang Xu, Ruilong Zhang","doi":"10.48550/arXiv.2306.15597","DOIUrl":"https://doi.org/10.48550/arXiv.2306.15597","url":null,"abstract":"Scheduling with testing falls under the umbrella of the research on optimization with explorable uncertainty. In this model, each job has an upper limit on its processing time that can be decreased to a lower limit (possibly unknown) by some preliminary action (testing). Recently, D{\"{u}}rr et al. cite{DBLP:journals/algorithmica/DurrEMM20} has studied a setting where testing a job takes a unit time, and the goal is to minimize total completion time or makespan on a single machine. In this paper, we extend their problem to the budget setting in which each test consumes a job-specific cost, and we require that the total testing cost cannot exceed a given budget. We consider the offline variant (the lower processing time is known) and the oblivious variant (the lower processing time is unknown) and aim to minimize the total completion time or makespan on a single machine. For the total completion time objective, we show NP-hardness and derive a PTAS for the offline variant based on a novel LP rounding scheme. We give a $(4+epsilon)$-competitive algorithm for the oblivious variant based on a framework inspired by the worst-case lower-bound instance. For the makespan objective, we give an FPTAS for the offline variant and a $(2+epsilon)$-competitive algorithm for the oblivious variant. Our algorithms for the oblivious variants under both objectives run in time $O(poly(n/epsilon))$. Lastly, we show that our results are essentially optimal by providing matching lower bounds.","PeriodicalId":201778,"journal":{"name":"Embedded Systems and Applications","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114062079","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}
M. Czekanski, S. Kimmel, R. T. Witter, Inge Li, Martin Farach-Colton, S. Puglisi, Grzegorz Herman
{"title":"Robust and Space-Efficient Dual Adversary Quantum Query Algorithms","authors":"M. Czekanski, S. Kimmel, R. T. Witter, Inge Li, Martin Farach-Colton, S. Puglisi, Grzegorz Herman","doi":"10.48550/arXiv.2306.15040","DOIUrl":"https://doi.org/10.48550/arXiv.2306.15040","url":null,"abstract":"The general adversary dual is a powerful tool in quantum computing because it gives a query-optimal bounded-error quantum algorithm for deciding any Boolean function. Unfortunately, the algorithm uses linear qubits in the worst case, and only works if the constraints of the general adversary dual are exactly satisfied. The challenge of improving the algorithm is that it is brittle to arbitrarily small errors since it relies on a reflection over a span of vectors. We overcome this challenge and build a robust dual adversary algorithm that can handle approximately satisfied constraints. As one application of our robust algorithm, we prove that for any Boolean function with polynomially many 1-valued inputs (or in fact a slightly weaker condition) there is a query-optimal algorithm that uses logarithmic qubits. As another application, we prove that numerically derived, approximate solutions to the general adversary dual give a bounded-error quantum algorithm under certain conditions. Further, we show that these conditions empirically hold with reasonable iterations for Boolean functions with small domains. We also develop several tools that may be of independent interest, including a robust approximate spectral gap lemma, a method to compress a general adversary dual solution using the Johnson-Lindenstrauss lemma, and open-source code to find solutions to the general adversary dual.","PeriodicalId":201778,"journal":{"name":"Embedded Systems and Applications","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128418109","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":"An Efficient Algorithm for Power Dominating Set","authors":"Thomas Bläsius, Max Göttlicher","doi":"10.48550/arXiv.2306.09870","DOIUrl":"https://doi.org/10.48550/arXiv.2306.09870","url":null,"abstract":"The problem Power Dominating Set (PDS) is motivated by the placement of phasor measurement units to monitor electrical networks. It asks for a minimum set of vertices in a graph that observes all remaining vertices by exhaustively applying two observation rules. Our contribution is twofold. First, we determine the parameterized complexity of PDS by proving it is $W[P]$-complete when parameterized with respect to the solution size. We note that it was only known to be $W[2]$-hard before. Our second and main contribution is a new algorithm for PDS that efficiently solves practical instances. Our algorithm consists of two complementary parts. The first is a set of reduction rules for PDS that can also be used in conjunction with previously existing algorithms. The second is an algorithm for solving the remaining kernel based on the implicit hitting set approach. Our evaluation on a set of power grid instances from the literature shows that our solver outperforms previous state-of-the-art solvers for PDS by more than one order of magnitude on average. Furthermore, our algorithm can solve previously unsolved instances of continental scale within a few minutes.","PeriodicalId":201778,"journal":{"name":"Embedded Systems and Applications","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124359533","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}
Thomas Bläsius, T. Friedrich, Maximilian Katzmann, János Ruff, Ziena Zeif
{"title":"On the Giant Component of Geometric Inhomogeneous Random Graphs","authors":"Thomas Bläsius, T. Friedrich, Maximilian Katzmann, János Ruff, Ziena Zeif","doi":"10.48550/arXiv.2306.09506","DOIUrl":"https://doi.org/10.48550/arXiv.2306.09506","url":null,"abstract":"In this paper we study the threshold model of emph{geometric inhomogeneous random graphs} (GIRGs); a generative random graph model that is closely related to emph{hyperbolic random graphs} (HRGs). These models have been observed to capture complex real-world networks well with respect to the structural and algorithmic properties. Following comprehensive studies regarding their emph{connectivity}, i.e., which parts of the graphs are connected, we have a good understanding under which circumstances a emph{giant} component (containing a constant fraction of the graph) emerges. While previous results are rather technical and challenging to work with, the goal of this paper is to provide more accessible proofs. At the same time we significantly improve the previously known probabilistic guarantees, showing that GIRGs contain a giant component with probability $1 - exp(-Omega(n^{(3-tau)/2}))$ for graph size $n$ and a degree distribution with power-law exponent $tau in (2, 3)$. Based on that we additionally derive insights about the connectivity of certain induced subgraphs of GIRGs.","PeriodicalId":201778,"journal":{"name":"Embedded Systems and Applications","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122090485","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":"Improved Algorithms for Distance Selection and Related Problems","authors":"Haitao Wang, Yiming Zhao","doi":"10.48550/arXiv.2306.01073","DOIUrl":"https://doi.org/10.48550/arXiv.2306.01073","url":null,"abstract":"In this paper, we propose new techniques for solving geometric optimization problems involving interpoint distances of a point set in the plane. Given a set $P$ of $n$ points in the plane and an integer $1 leq k leq binom{n}{2}$, the distance selection problem is to find the $k$-th smallest interpoint distance among all pairs of points of $P$. The previously best deterministic algorithm solves the problem in $O(n^{4/3} log^2 n)$ time [Katz and Sharir, SIAM J. Comput. 1997 and SoCG 1993]. In this paper, we improve their algorithm to $O(n^{4/3} log n)$ time. Using similar techniques, we also give improved algorithms on both the two-sided and the one-sided discrete Fr'{e}chet distance with shortcuts problem for two point sets in the plane. For the two-sided problem (resp., one-sided problem), we improve the previous work [Avraham, Filtser, Kaplan, Katz, and Sharir, ACM Trans. Algorithms 2015 and SoCG 2014] by a factor of roughly $log^2(m+n)$ (resp., $(m+n)^{epsilon}$), where $m$ and $n$ are the sizes of the two input point sets, respectively. Other problems whose solutions can be improved by our techniques include the reverse shortest path problems for unit-disk graphs. Our techniques are quite general and we believe they will find many other applications in future.","PeriodicalId":201778,"journal":{"name":"Embedded Systems and Applications","volume":"33 49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116276471","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":"Optimal energetic paths for electric cars","authors":"D. Dorfman, Haim Kaplan, R. Tarjan, Uri Zwick","doi":"10.48550/arXiv.2305.19015","DOIUrl":"https://doi.org/10.48550/arXiv.2305.19015","url":null,"abstract":"A weighted directed graph $G=(V,A,c)$, where $Asubseteq Vtimes V$ and $c:Ato R$, describes a road network in which an electric car can roam. An arc $uv$ models a road segment connecting the two vertices $u$ and $v$. The cost $c(uv)$ of an arc $uv$ is the amount of energy the car needs to traverse the arc. This amount may be positive, zero or negative. To make the problem realistic, we assume there are no negative cycles. The car has a battery that can store up to $B$ units of energy. It can traverse an arc $uvin A$ only if it is at $u$ and the charge $b$ in its battery satisfies $bge c(uv)$. If it traverses the arc, it reaches $v$ with a charge of $min(b-c(uv),B)$. Arcs with positive costs deplete the battery, arcs with negative costs charge the battery, but not above its capacity of $B$. Given $s,tin V$, can the car travel from $s$ to $t$, starting at $s$ with an initial charge $b$, where $0le ble B$? If so, what is the maximum charge with which the car can reach $t$? Equivalently, what is the smallest $delta_{B,b}(s,t)$ such that the car can reach $t$ with a charge of $b-delta_{B,b}(s,t)$, and which path should the car follow to achieve this? We refer to $delta_{B,b}(s,t)$ as the energetic cost of traveling from $s$ to $t$. We let $delta_{B,b}(s,t)=infty$ if the car cannot travel from $s$ to $t$ starting with an initial charge of $b$. The problem of computing energetic costs is a strict generalization of the standard shortest paths problem. We show that the single-source minimum energetic paths problem can be solved using simple, but subtle, adaptations of the Bellman-Ford and Dijkstra algorithms. To make Dijkstra's algorithm work in the presence of negative arcs, but no negative cycles, we use a variant of the $A^*$ search heuristic. These results are explicit or implicit in some previous papers. We provide a simpler and unified description of these algorithms.","PeriodicalId":201778,"journal":{"name":"Embedded Systems and Applications","volume":"137 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124477131","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}
Amir Abboud, N. Fischer, Elazar Goldenberg, Ron Safier
{"title":"Can You Solve Closest String Faster than Exhaustive Search?","authors":"Amir Abboud, N. Fischer, Elazar Goldenberg, Ron Safier","doi":"10.48550/arXiv.2305.16878","DOIUrl":"https://doi.org/10.48550/arXiv.2305.16878","url":null,"abstract":"We study the fundamental problem of finding the best string to represent a given set, in the form of the Closest String problem: Given a set $X subseteq Sigma^d$ of $n$ strings, find the string $x^*$ minimizing the radius of the smallest Hamming ball around $x^*$ that encloses all the strings in $X$. In this paper, we investigate whether the Closest String problem admits algorithms that are faster than the trivial exhaustive search algorithm. We obtain the following results for the two natural versions of the problem: $bullet$ In the continuous Closest String problem, the goal is to find the solution string $x^*$ anywhere in $Sigma^d$. For binary strings, the exhaustive search algorithm runs in time $O(2^d poly(nd))$ and we prove that it cannot be improved to time $O(2^{(1-epsilon) d} poly(nd))$, for any $epsilon>0$, unless the Strong Exponential Time Hypothesis fails. $bullet$ In the discrete Closest String problem, $x^*$ is required to be in the input set $X$. While this problem is clearly in polynomial time, its fine-grained complexity has been pinpointed to be quadratic time $n^{2 pm o(1)}$ whenever the dimension is $omega(log n)<d<n^{o(1)}$. We complement this known hardness result with new algorithms, proving essentially that whenever $d$ falls out of this hard range, the discrete Closest String problem can be solved faster than exhaustive search. In the small-$d$ regime, our algorithm is based on a novel application of the inclusion-exclusion principle. Interestingly, all of our results apply (and some are even stronger) to the natural dual of the Closest String problem, called the Remotest String problem, where the task is to find a string maximizing the Hamming distance to all the strings in $X$.","PeriodicalId":201778,"journal":{"name":"Embedded Systems and Applications","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129690024","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":"Aggregating over Dominated Points by Sorting, Scanning, Zip and Flat Maps","authors":"J. Sroka, Jerzy Tyszkiewicz","doi":"10.48550/arXiv.2305.16751","DOIUrl":"https://doi.org/10.48550/arXiv.2305.16751","url":null,"abstract":"Prefix aggregation operation (also called scan), and its particular case, prefix summation, is an important parallel primitive and enjoys a lot of attention in the research literature. It is also used in many algorithms as one of the steps. Aggregation over dominated points in $mathbb{R}^m$ is a multidimensional generalisation of prefix aggregation. It is also intensively researched, both as a parallel primitive and as a practical problem, encountered in computational geometry, spatial databases and data warehouses. In this paper we show that, for a constant dimension $m$, aggregation over dominated points in $mathbb{R}^m$ can be computed by $O(1)$ basic operations that include sorting the whole dataset, zipping sorted lists of elements, computing prefix aggregations of lists of elements and flat maps, which expand the data size from initial $n$ to $nlog^{m-1}n$. Thereby we establish that prefix aggregation suffices to express aggregation over dominated points in more dimensions, even though the latter is a far-reaching generalisation of the former. Many problems known to be expressible by aggregation over dominated points become expressible by prefix aggregation, too. We rely on a small set of primitive operations which guarantee an easy transfer to various distributed architectures and some desired properties of the implementation.","PeriodicalId":201778,"journal":{"name":"Embedded Systems and Applications","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126483980","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":"On k-means for segments and polylines","authors":"Sergio Cabello, P. Giannopoulos","doi":"10.48550/arXiv.2305.10922","DOIUrl":"https://doi.org/10.48550/arXiv.2305.10922","url":null,"abstract":"We study the problem of $k$-means clustering in the space of straight-line segments in $mathbb{R}^{2}$ under the Hausdorff distance. For this problem, we give a $(1+epsilon)$-approximation algorithm that, for an input of $n$ segments, for any fixed $k$, and with constant success probability, runs in time $O(n+ epsilon^{-O(k)} + epsilon^{-O(k)}cdot log^{O(k)} (epsilon^{-1}))$. The algorithm has two main ingredients. Firstly, we express the $k$-means objective in our metric space as a sum of algebraic functions and use the optimization technique of Vigneron~cite{Vigneron14} to approximate its minimum. Secondly, we reduce the input size by computing a small size coreset using the sensitivity-based sampling framework by Feldman and Langberg~cite{Feldman11, Feldman2020}. Our results can be extended to polylines of constant complexity with a running time of $O(n+ epsilon^{-O(k)})$.","PeriodicalId":201778,"journal":{"name":"Embedded Systems and Applications","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116917598","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":"Parameterized Complexity of Equality MinCSP","authors":"George Osipov, Magnus Wahlstrom","doi":"10.48550/arXiv.2305.11131","DOIUrl":"https://doi.org/10.48550/arXiv.2305.11131","url":null,"abstract":"We study the parameterized complexity of MinCSP for so-called equality languages, i.e., for finite languages over an infinite domain such as $mathbb{N}$, where the relations are defined via first-order formulas whose only predicate is $=$. This is an important class of languages that forms the starting point of all study of infinite-domain CSPs under the commonly used approach pioneered by Bodirsky, i.e., languages defined as reducts of finitely bounded homogeneous structures. Moreover, MinCSP over equality languages forms a natural class of optimisation problems in its own right, covering such problems as Edge Multicut, Steiner Multicut and (under singleton expansion) Edge Multiway Cut. We classify MinCSP$(Gamma)$ for every finite equality language $Gamma$, under the natural parameter, as either FPT, W[1]-hard but admitting a constant-factor FPT-approximation, or not admitting a constant-factor FPT-approximation unless FPT=W[2]. In particular, we describe an FPT case that slightly generalises Multicut, and show a constant-factor FPT-approximation for Disjunctive Multicut, the generalisation of Multicut where the ``cut requests'' come as disjunctions over $d = O(1)$ individual cut requests $s_i neq t_i$. We also consider singleton expansions of equality languages, i.e., enriching an equality language with the capability for assignment constraints $(x=i)$ for either finitely or infinitely many constants $i in mathbb{N}$, and fully characterize the complexity of the resulting MinCSP.","PeriodicalId":201778,"journal":{"name":"Embedded Systems and Applications","volume":"116 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117239615","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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