Algorithmica最新文献

{"title":"Limitations of the Impagliazzo–Nisan–Wigderson Pseudorandom Generator Against Permutation Branching Programs","authors":"William M. Hoza,&nbsp;Edward Pyne,&nbsp;Salil Vadhan","doi":"10.1007/s00453-024-01251-2","DOIUrl":"10.1007/s00453-024-01251-2","url":null,"abstract":"<div><p>The classic Impagliazzo–Nisan–Wigderson (INW) pseudorandom generator (PRG) (STOC ‘94) for space-bounded computation uses a seed of length <span>(O(log n cdot log (nw/varepsilon )+log d))</span> to fool ordered branching programs of length <i>n</i>, width <i>w</i>, and alphabet size <i>d</i> to within error <span>(varepsilon )</span>. A series of works have shown that the analysis of the INW generator can be improved for the class of <i>permutation</i> branching programs or the more general <i>regular</i> branching programs, improving the <span>(O(log ^2 n))</span> dependence on the length <i>n</i> to <span>(O(log n))</span> or <span>({tilde{O}}(log n))</span>. However, when also considering the dependence on the other parameters, these analyses still fall short of the optimal PRG seed length <span>(O(log (nwd/varepsilon )))</span>. In this paper, we prove that any “spectral analysis” of the INW generator requires seed length </p><div><div><span>$$begin{aligned} Omega left( log ncdot log log left( min {n,d}right) +log ncdot log left( w/varepsilon right) +log dright) end{aligned}$$</span></div></div><p>to fool ordered permutation branching programs of length <i>n</i>, width <i>w</i>, and alphabet size <i>d</i> to within error <span>(varepsilon )</span>. By “spectral analysis” we mean an analysis of the INW generator that relies only on the spectral expansion of the graphs used to construct the generator; this encompasses all prior analyses of the INW generator. Our lower bound matches the upper bound of Braverman–Rao–Raz–Yehudayoff (FOCS 2010, SICOMP 2014) for regular branching programs of alphabet size <span>(d=2)</span> except for a gap between their <span>(Oleft( log n cdot log log nright) )</span> term and our <span>(Omega left( log n cdot log log min {n,d}right) )</span> term. It also matches the upper bounds of Koucký–Nimbhorkar–Pudlák (STOC 2011), De (CCC 2011), and Steinke (ECCC 2012) for constant-width (<span>(w=O(1))</span>) permutation branching programs of alphabet size <span>(d=2)</span> to within a constant factor. To fool permutation branching programs in the measure of <i>spectral norm</i>, we prove that any spectral analysis of the INW generator requires a seed of length <span>(Omega left( log ncdot log log n+log ncdot log (1/varepsilon )right) )</span> when the width is at least polynomial in <i>n</i> (<span>(w=n^{Omega (1)})</span>), matching the recent upper bound of Hoza–Pyne–Vadhan (ITCS 2021) to within a constant factor.\u0000</p></div>","PeriodicalId":50824,"journal":{"name":"Algorithmica","volume":"86 10","pages":"3153 - 3185"},"PeriodicalIF":0.9,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00453-024-01251-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141868112","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Minimizing Energy Consumption for Real-Time Tasks on Heterogeneous Platforms Under Deadline and Reliability Constraints","authors":"Yiqin Gao,&nbsp;Li Han,&nbsp;Jing Liu,&nbsp;Yves Robert,&nbsp;Frédéric Vivien","doi":"10.1007/s00453-024-01253-0","DOIUrl":"10.1007/s00453-024-01253-0","url":null,"abstract":"<div><p>As real-time systems are safety critical, guaranteeing a high reliability threshold is as important as meeting all deadlines. Periodic tasks are replicated to mitigate the negative impact of transient faults, which leads to redundancy and high energy consumption. On the other hand, energy saving is widely identified as increasingly relevant issues in real-time systems. In this paper, we formalize this challenging tri-criteria optimization problem, i.e., minimizing the expected energy consumption while enforcing the reliability threshold and meeting all task deadlines, and propose several mapping and scheduling heuristics to solve it. Specifically, a novel approach is designed to (i) map an arbitrary number of replicas onto processors, (ii) schedule each replica of each task instance on its assigned processor with less temporal overlap. The platform is composed of processing units with different characteristics, including speed profile, energy cost and fault rate. The heterogeneity of the computing platform makes the problem more complicated, because different mappings achieve different levels of reliability and consume different amounts of energy. Moreover, scheduling plays an important role in energy saving, as the expected energy consumption is the average over all failure scenarios. Once a task replica is successful, the other replicas of that task instance can be canceled, which calls for minimizing the overlap between any replica pair. Finally, to quantitatively analyze our methods, we derive a theoretical lower-bound for the expected energy consumption. Comprehensive experiments are conducted on a large set of execution scenarios and parameters. The comparison results reveal that our strategies perform better than the random baseline under almost all settings, with an average gain in energy consumption of more than 40%, and our best heuristic achieves an excellent performance: its energy saving is only 2% less than the lower-bound on average.</p></div>","PeriodicalId":50824,"journal":{"name":"Algorithmica","volume":"86 10","pages":"3079 - 3114"},"PeriodicalIF":0.9,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141740782","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Tight Runtime Bounds for Static Unary Unbiased Evolutionary Algorithms on Linear Functions","authors":"Carola Doerr,&nbsp;Duri Andrea Janett,&nbsp;Johannes Lengler","doi":"10.1007/s00453-024-01258-9","DOIUrl":"10.1007/s00453-024-01258-9","url":null,"abstract":"<div><p>In a seminal paper in 2013, Witt showed that the (1+1) Evolutionary Algorithm with standard bit mutation needs time <span>((1+o(1))n ln n/p_1)</span> to find the optimum of any linear function, as long as the probability <span>(p_1)</span> to flip exactly one bit is <span>(Theta (1))</span>. In this paper we investigate how this result generalizes if standard bit mutation is replaced by an arbitrary unbiased mutation operator. This situation is notably different, since the stochastic domination argument used for the lower bound by Witt no longer holds. In particular, starting closer to the optimum is not necessarily an advantage, and OneMax is no longer the easiest function for arbitrary starting positions. Nevertheless, we show that Witt’s result carries over if <span>(p_1)</span> is not too small, with different constraints for upper and lower bounds, and if the number of flipped bits has bounded expectation <span>(chi )</span>. Notably, this includes some of the heavy-tail mutation operators used in fast genetic algorithms, but not all of them. We also give examples showing that algorithms with unbounded <span>(chi )</span> have qualitatively different trajectories close to the optimum.</p></div>","PeriodicalId":50824,"journal":{"name":"Algorithmica","volume":"86 10","pages":"3115 - 3152"},"PeriodicalIF":0.9,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141740781","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"New Partitioning Techniques and Faster Algorithms for Approximate Interval Scheduling","authors":"Spencer Compton,&nbsp;Slobodan Mitrović,&nbsp;Ronitt Rubinfeld","doi":"10.1007/s00453-024-01252-1","DOIUrl":"10.1007/s00453-024-01252-1","url":null,"abstract":"<div><p>Interval scheduling is a basic algorithmic problem and a classical task in combinatorial optimization. We develop techniques for partitioning and grouping jobs based on their starting/ending times, enabling us to view an instance of interval scheduling on <i>many</i> jobs as a union of multiple interval scheduling instances, each containing only <i>a few</i> jobs. Instantiating these techniques in a dynamic setting produces several new results. For <span>((1+varepsilon ))</span>-approximation of job scheduling of <i>n</i> jobs on a single machine, we develop a fully dynamic algorithm with <span>(O(nicefrac {log {n}}{varepsilon }))</span> update and <span>(O(log {n}))</span> query worst-case time. Our techniques are also applicable in a setting where jobs have weights. We design a fully dynamic <i>deterministic</i> algorithm whose worst-case update and query times are <span>(text {poly} (log n,frac{1}{varepsilon }))</span>. This is <i>the first</i> algorithm that maintains a <span>((1+varepsilon ))</span>-approximation of the maximum independent set of a collection of weighted intervals in <span>(text {poly} (log n,frac{1}{varepsilon }))</span> time updates/queries. This is an exponential improvement in <span>(1/varepsilon )</span> over the running time of an algorithm of Henzinger, Neumann, and Wiese  [SoCG, 2020]. Our approach also removes all dependence on the values of the jobs’ starting/ending times and weights.</p></div>","PeriodicalId":50824,"journal":{"name":"Algorithmica","volume":"86 9","pages":"2997 - 3026"},"PeriodicalIF":0.9,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142412312","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Non-crossing Hamiltonian Paths and Cycles in Output-Polynomial Time","authors":"David Eppstein","doi":"10.1007/s00453-024-01255-y","DOIUrl":"10.1007/s00453-024-01255-y","url":null,"abstract":"<div><p>We show that, for planar point sets, the number of non-crossing Hamiltonian paths is polynomially bounded in the number of non-crossing paths, and the number of non-crossing Hamiltonian cycles (polygonalizations) is polynomially bounded in the number of surrounding cycles. As a consequence, we can list the non-crossing Hamiltonian paths or the polygonalizations, in time polynomial in the output size, by filtering the output of simple backtracking algorithms for non-crossing paths or surrounding cycles respectively. We do not assume that the points are in general position. To prove these results we relate the numbers of non-crossing structures to two easily-computed parameters of the point set: the minimum number of points whose removal results in a collinear set, and the number of points interior to the convex hull. These relations also lead to polynomial-time approximation algorithms for the numbers of structures of all four types, accurate to within a constant factor of the logarithm of these numbers.\u0000</p></div>","PeriodicalId":50824,"journal":{"name":"Algorithmica","volume":"86 9","pages":"3027 - 3053"},"PeriodicalIF":0.9,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00453-024-01255-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142412335","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On a Traveling Salesman Problem for Points in the Unit Cube","authors":"József Balogh,&nbsp;Felix Christian Clemen,&nbsp;Adrian Dumitrescu","doi":"10.1007/s00453-024-01257-w","DOIUrl":"10.1007/s00453-024-01257-w","url":null,"abstract":"<div><p>Let <i>X</i> be an <i>n</i>-element point set in the <i>k</i>-dimensional unit cube <span>([0,1]^k)</span> where <span>(k ge 2)</span>. According to an old result of Bollobás and Meir (Oper Res Lett 11:19–21, 1992) , there exists a cycle (tour) <span>(x_1, x_2, ldots , x_n)</span> through the <i>n</i> points, such that <span>(left( sum _{i=1}^n |x_i - x_{i+1}|^k right) ^{1/k} le c_k)</span>, where <span>(|x-y|)</span> is the Euclidean distance between <i>x</i> and <i>y</i>, and <span>(c_k)</span> is an absolute constant that depends only on <i>k</i>, where <span>(x_{n+1} equiv x_1)</span>. From the other direction, for every <span>(k ge 2)</span> and <span>(n ge 2)</span>, there exist <i>n</i> points in <span>([0,1]^k)</span>, such that their shortest tour satisfies <span>(left( sum _{i=1}^n |x_i - x_{i+1}|^k right) ^{1/k} = 2^{1/k} cdot sqrt{k})</span>. For the plane, the best constant is <span>(c_2=2)</span> and this is the only exact value known. Bollobás and Meir showed that one can take <span>(c_k = 9 left( frac{2}{3} right) ^{1/k} cdot sqrt{k})</span> for every <span>(k ge 3)</span> and conjectured that the best constant is <span>(c_k = 2^{1/k} cdot sqrt{k})</span>, for every <span>(k ge 2)</span>. Here we significantly improve the upper bound and show that one can take <span>(c_k = 3 sqrt{5} left( frac{2}{3} right) ^{1/k} cdot sqrt{k})</span> or <span>(c_k = 2.91 sqrt{k} (1+o_k(1)))</span>. Our bounds are constructive. We also show that <span>(c_3 ge 2^{7/6})</span>, which disproves the conjecture for <span>(k=3)</span>. Connections to matching problems, power assignment problems, related problems, including algorithms, are discussed in this context. A slightly revised version of the Bollobás–Meir conjecture is proposed.</p></div>","PeriodicalId":50824,"journal":{"name":"Algorithmica","volume":"86 9","pages":"3054 - 3078"},"PeriodicalIF":0.9,"publicationDate":"2024-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00453-024-01257-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142412318","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Sublinear Algorithms in T-Interval Dynamic Networks","authors":"Irvan Jahja,&nbsp;Haifeng Yu","doi":"10.1007/s00453-024-01250-3","DOIUrl":"10.1007/s00453-024-01250-3","url":null,"abstract":"<div><p>We consider standard <i>T</i>-<i>interval dynamic networks</i>, under the synchronous timing model and the broadcast CONGEST model. In a <i>T</i>-<i>interval dynamic network</i>, the set of nodes is always fixed and there are no node failures. The edges in the network are always undirected, but the set of edges in the topology may change arbitrarily from round to round, as determined by some <i>adversary</i> and subject to the following constraint: For every <i>T</i> consecutive rounds, the topologies in those rounds must contain a common connected spanning subgraph. Let <span>(H_r)</span> to be the maximum (in terms of number of edges) such subgraph for round <i>r</i> through <span>(r+T-1)</span>. We define the <i>backbone diameter</i> <i>d</i> of a <i>T</i>-interval dynamic network to be the maximum diameter of all such <span>(H_r)</span>’s, for <span>(rge 1)</span>. We use <i>n</i> to denote the number of nodes in the network. Within such a context, we consider a range of fundamental distributed computing problems including <span>Count</span>/<span>Max</span>/<span>Median</span>/<span>Sum</span>/<span>LeaderElect</span>/<span>Consensus</span>/<span>ConfirmedFlood</span>. Existing algorithms for these problems all have time complexity of <span>(Omega (n))</span> rounds, even for <span>(T=infty )</span> and even when <i>d</i> is as small as <i>O</i>(1). This paper presents a novel approach/framework, based on the idea of <i>massively parallel aggregation</i>. Following this approach, we develop a novel deterministic <span>Count</span> algorithm with <span>(O(d^3 log ^2 n))</span> complexity, for <i>T</i>-interval dynamic networks with <span>(T ge ccdot d^2 log ^2n)</span>. Here <i>c</i> is a (sufficiently large) constant independent of <i>d</i>, <i>n</i>, and <i>T</i>. To our knowledge, our algorithm is the very first such algorithm whose complexity does not contain a <span>(Theta (n))</span> term. This paper further develops novel algorithms for solving <span>Max</span>/<span>Median</span>/<span>Sum</span>/<span>LeaderElect</span>/<span>Consensus</span>/<span>ConfirmedFlood</span>, while incurring <span>(O(d^3 text{ polylog }(n)))</span> complexity. Again, for all these problems, our algorithms are the first ones whose time complexity does not contain a <span>(Theta (n))</span> term.</p></div>","PeriodicalId":50824,"journal":{"name":"Algorithmica","volume":"86 9","pages":"2959 - 2996"},"PeriodicalIF":0.9,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141612701","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Stagnation Detection in Highly Multimodal Fitness Landscapes","authors":"Amirhossein Rajabi,&nbsp;Carsten Witt","doi":"10.1007/s00453-024-01249-w","DOIUrl":"10.1007/s00453-024-01249-w","url":null,"abstract":"<div><p>Stagnation detection has been proposed as a mechanism for randomized search heuristics to escape from local optima by automatically increasing the size of the neighborhood to find the so-called gap size, i. e., the distance to the next improvement. Its usefulness has mostly been considered in simple multimodal landscapes with few local optima that could be crossed one after another. In multimodal landscapes with a more complex location of optima of similar gap size, stagnation detection suffers from the fact that the neighborhood size is frequently reset to  1 without using gap sizes that were promising in the past. In this paper, we investigate a new mechanism called <i>radius memory</i> which can be added to stagnation detection to control the search radius more carefully by giving preference to values that were successful in the past. We implement this idea in an algorithm called SD-RLS<span>(^{text {m}})</span> and show compared to previous variants of stagnation detection that it yields speed-ups for linear functions under uniform constraints and the minimum spanning tree problem. Moreover, its running time does not significantly deteriorate on unimodal functions and a generalization of the <span>Jump</span> benchmark. Finally, we present experimental results carried out to study SD-RLS<span>(^{text {m}})</span> and compare it with other algorithms.</p></div>","PeriodicalId":50824,"journal":{"name":"Algorithmica","volume":"86 9","pages":"2929 - 2958"},"PeriodicalIF":0.9,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00453-024-01249-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141550368","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Parameterized Complexity of Streaming Diameter and Connectivity Problems","authors":"Jelle J. Oostveen,&nbsp;Erik Jan van Leeuwen","doi":"10.1007/s00453-024-01246-z","DOIUrl":"10.1007/s00453-024-01246-z","url":null,"abstract":"<div><p>We initiate the investigation of the parameterized complexity of <span>Diameter</span> and <span>Connectivity</span> in the streaming paradigm. On the positive end, we show that knowing a vertex cover of size <i>k</i> allows for algorithms in the Adjacency List (AL) streaming model whose number of passes is constant and memory is <span>(mathcal {O}(log n))</span> for any fixed <i>k</i>. Underlying these algorithms is a method to execute a breadth-first search in <span>(mathcal {O}(k))</span> passes and <span>(mathcal {O}(k log n))</span> bits of memory. On the negative end, we show that many other parameters lead to lower bounds in the AL model, where <span>(Omega (n/p))</span> bits of memory is needed for any <i>p</i>-pass algorithm even for constant parameter values. In particular, this holds for graphs with a known modulator (deletion set) of constant size to a graph that has no induced subgraph isomorphic to a fixed graph <i>H</i>, for most <i>H</i>. For some cases, we can also show one-pass, <span>(Omega (n log n))</span> bits of memory lower bounds. We also prove a much stronger <span>(Omega (n^2/p))</span> lower bound for <span>Diameter</span> on bipartite graphs. Finally, using the insights we developed into streaming parameterized graph exploration algorithms, we show a new streaming kernelization algorithm for computing a vertex cover of size <i>k</i>. This yields a kernel of 2<i>k</i> vertices (with <span>(mathcal {O}(k^2))</span> edges) produced as a stream in <span>(text {poly}(k))</span> passes and only <span>(mathcal {O}(k log n))</span> bits of memory.</p></div>","PeriodicalId":50824,"journal":{"name":"Algorithmica","volume":"86 9","pages":"2885 - 2928"},"PeriodicalIF":0.9,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00453-024-01246-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141550330","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Approximation Algorithms for the Two-Watchman Route in a Simple Polygon","authors":"Bengt J. Nilsson,&nbsp;Eli Packer","doi":"10.1007/s00453-024-01245-0","DOIUrl":"10.1007/s00453-024-01245-0","url":null,"abstract":"<div><p>The <i>two-watchman route problem</i> is that of computing a pair of closed tours in an environment so that the two tours together see the whole environment and some length measure on the two tours is minimized. Two standard measures are: the minmax measure, where we want the tours where the longest of them has smallest length, and the minsum measure, where we want the tours for which the sum of their lengths is the smallest. It is known that computing a minmax two-watchman route is NP-hard for simple rectilinear polygons and thus also for simple polygons. Also, any <i>c</i>-approximation algorithm for the minmax two-watchman route is automatically a 2<i>c</i>-approximation algorithm for the minsum two-watchman route. We exhibit two constant factor approximation algorithms for computing minmax two-watchman routes in simple polygons with approximation factors 5.969 and 11.939, having running times <span>(O(n^8))</span> and <span>(O(n^4))</span> respectively, where <i>n</i> is the number of vertices of the polygon. We also use the same techniques to obtain a 6.922-approximation for the <i>fixed two-watchman route problem</i> running in <span>(O(n^2))</span> time, i.e., when two starting points of the two tours are given as input.\u0000</p></div>","PeriodicalId":50824,"journal":{"name":"Algorithmica","volume":"86 9","pages":"2845 - 2884"},"PeriodicalIF":0.9,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00453-024-01245-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141552997","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}