Breaching the 2-Approximation Barrier for Connectivity Augmentation:A Reduction to Steiner Tree

The basic goal of survivable network design is to build a cheap network that maintains the connectivity between given sets of nodes despite the failure of a few edges/nodes. The connectivity augmentation problem (CAP) is arguably one of the most basic problems in this area: given a k(-edge)-connected graph G and a set of extra edges (links), select a minimum cardinality subset A of links such that adding A to G increases its edge connectivity to k+1. Intuitively, one wants to make an existing network more reliable by augmenting it with extra edges. The best known approximation factor for this NP-hard problem is 2, and this can be achieved with multiple approaches (the first such result is in [G. N. Frederickson and J\'aj\' a, SIAM J. Comput., 10 (1981), pp. 270-283]. It is known [E. A. Dinitz, A. V. Karzanov, and M. V. Lomonosov, Studies in Discrete Optimization, Nauka, Moscow, 1976, pp. 290-306] that CAP can be reduced to the case k = 1, also known as the tree augmentation problem (TAP) for odd k, and to the case k = 2, also known as the cactus augmentation problem (CacAP) for even k. Prior to the conference version of this paper [J. Byrka, F. Grandoni, and A. Jabal Ameli, STOC'20, ACM, New York, 2020, pp. 815-825], several better than 2 approximation algorithms were known for TAP, culminating with a recent 1.458 approximation [F. Grandoni, C. Kalaitzis, and R. Zenklusen, STOC'18, ACM, New York, 1918, pp. 632-645]. However, for CacAP the best known approximation was 2. In this paper we breach the 2 approximation barrier for CacAP, hence, for CAP, by presenting a polynomial-time 2 ln(4) - 1120 967 + \varepsilon &lt; 1.91 approximation. From a technical point of view, our approach deviates quite substantially from previous work. In particular, the better-than-2 approximation algorithms for TAP either exploit greedy-style algorithms or are based on rounding carefully designed LPs. We instead use a reduction to the Steiner tree problem which was previously used in parameterized algorithms [Basavaraju et al., ICALP'14, Springer, Berlin, 2014, pp. 800-811]. This reduction is not approximation preserving, and using the current best approximation factor for a Steiner tree [Byrka et al., J. ACM, 60 (2013), 6] as a black box would not be good enough to improve on 2. To achieve the latter goal, we ``open the box"" and exploit the specific properties of the instances of a Steiner tree arising from CacAP. In our opinion this connection between approximation algorithms for survivable network design and Steiner-type problems is interesting, and might lead to other results in the area.</p

Breaching the 2-Approximation Barrier for Connectivity Augmentation: a Reduction to Steiner Tree

The basic goal of survivable network design is to build a cheap network that maintains the connectivity between given sets of nodes despite the failure of a few edges/nodes. The Connectivity Augmentation Problem (CAP) is arguably one of the most basic problems in this area: given a $k$(-edge)-connected graph $G$ and a set of extra edges (links), select a minimum cardinality subset $A$ of links such that adding $A$ to $G$ increases its edge connectivity to $k+1$. Intuitively, one wants to make an existing network more reliable by augmenting it with extra edges. The best known approximation factor for this NP-hard problem is $2$, and this can be achieved with multiple approaches (the first such result is in [Frederickson and J\'aj\'a'81]). It is known [Dinitz et al.'76] that CAP can be reduced to the case $k=1$, a.k.a. the Tree Augmentation Problem (TAP), for odd $k$, and to the case $k=2$, a.k.a. the Cactus Augmentation Problem (CacAP), for even $k$. Several better than $2$ approximation algorithms are known for TAP, culminating with a recent $1.458$ approximation [Grandoni et al.'18]. However, for CacAP the best known approximation is $2$. In this paper we breach the $2$ approximation barrier for CacAP, hence for CAP, by presenting a polynomial-time $2\ln(4)-\frac{967}{1120}+\epsilon<1.91$ approximation. Previous approaches exploit properties of TAP that do not seem to generalize to CacAP. We instead use a reduction to the Steiner tree problem which was previously used in parameterized algorithms [Basavaraju et al.'14]. This reduction is not approximation preserving, and using the current best approximation factor for Steiner tree [Byrka et al.'13] as a black-box would not be good enough to improve on $2$. To achieve the latter goal, we ``open the box'' and exploit the specific properties of the instances of Steiner tree arising from CacAP.Comment: Corrected a typo in the abstract (in metadata

Approximation Algorithms for Demand Strip Packing

In the Demand Strip Packing problem (DSP), we are given a time interval and a collection of tasks, each characterized by a processing time and a demand for a given resource (such as electricity, computational power, etc.). A feasible solution consists of a schedule of the tasks within the mentioned time interval. Our goal is to minimize the peak resource consumption, i.e. the maximum total demand of tasks executed at any point in time. It is known that DSP is NP-hard to approximate below a factor 3/2, and standard techniques for related problems imply a (polynomial-time) 2-approximation. Our main result is a (5/3+?)-approximation algorithm for any constant ? > 0. We also achieve best-possible approximation factors for some relevant special cases

A 4/3 Approximation for 2-Vertex-Connectivity

The 2-Vertex-Connected Spanning Subgraph problem (2VCSS) is among the most basic NP-hard (Survivable) Network Design problems: we are given an (unweighted) undirected graph G. Our goal is to find a subgraph S of G with the minimum number of edges which is 2-vertex-connected, namely S remains connected after the deletion of an arbitrary node. 2VCSS is well-studied in terms of approximation algorithms, and the current best (polynomial-time) approximation factor is 10/7 by Heeger and Vygen [SIDMA\u2717] (improving on earlier results by Khuller and Vishkin [STOC\u2792] and Garg, Vempala and Singla [SODA\u2793]). Here we present an improved 4/3 approximation. Our main technical ingredient is an approximation preserving reduction to a conveniently structured subset of instances which are "almost" 3-vertex-connected. The latter reduction might be helpful in future work

A PTAS for Triangle-Free 2-Matching

In the Triangle-Free (Simple) 2-Matching problem we are given an undirected graph $G=(V,E)$. Our goal is to compute a maximum-cardinality $M\subseteq E$ satisfying the following properties: (1) at most two edges of $M$ are incident on each node (i.e., $M$ is a 2-matching) and (2) $M$ does not induce any triangle. In his Ph.D. thesis from 1984, Harvitgsen presents a complex polynomial-time algorithm for this problem, with a very complex analysis. This result was never published in a journal nor reproved in a different way, to the best of our knowledge. In this paper we have a fresh look at this problem and present a simple PTAS for it based on local search. Our PTAS exploits the fact that, as long as the current solution is far enough from the optimum, there exists a short augmenting trail (similar to the maximum matching case).Comment: 27 pages, 18 figure

Performance of the First ANTARES Detector Line

In this paper we report on the data recorded with the first Antares detector line. The line was deployed on the 14th of February 2006 and was connected to the readout two weeks later. Environmental data for one and a half years of running are shown. Measurements of atmospheric muons from data taken from selected runs during the first six months of operation are presented. Performance figures in terms of time residuals and angular resolution are given. Finally the angular distribution of atmospheric muons is presented and from this the depth profile of the muon intensity is derived.Comment: 14 pages, 9 figure

Measurement of neutrino oscillation parameters with the first six detection units of KM3NeT/ORCA

Abstract: KM3NeT/ORCA is a water Cherenkov neutrino detector under construction and anchored at the bottom of the Mediterranean Sea. The detector is designed to study oscillations of atmospheric neutrinos and determine the neutrino mass ordering. This paper focuses on an initial configuration of ORCA, referred to as ORCA6, which comprises six out of the foreseen 115 detection units of photo-sensors. A high-purity neutrino sample was extracted, corresponding to an exposure of 433 kton-years. The sample of 5828 neutrino candidates is analysed following a binned log-likelihood method in the reconstructed energy and cosine of the zenith angle. The atmospheric oscillation parameters are measured to be sin2 ¿23 = 0.51+0.04 -0.05, and ¿m2 31 = 2.18+0.25 -0.35 × 10-3 eV2 ¿ {-2.25,-1.76} × 10-3 eV2 at 68% CL. The inverted neutrino mass ordering hypothesis is disfavoured with a p-value of 0.25.The authors acknowledge the financial support of the funding agencies: Funds for Scientific Research (FRS-FNRS), Francqui foundation, BAEF foundation. Czech Science Foundation (GAČR 24-12702S); Agence Nationale de la Recherche (contract ANR-15-CE31-0020), Centre National de la Recherche Scientifique (CNRS), Commission Européenne (FEDER fund and Marie Curie Program), LabEx UnivEarthS (ANR-10-LABX-0023 and ANR-18-IDEX-0001), Paris Île-de-France Region, Normandy Region (Alpha, Blue-waves and Neptune), France; Shota Rustaveli National Science Foundation of Georgia (SRNSFG, FR-22-13708), Georgia; This work is part of the MuSES project which has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 Research and Innovation Programme (grant agreement No 101142396). The General Secretariat of Research and Innovation (GSRI), Greece; Istituto Nazionale di Fisica Nucleare (INFN) and Ministero dell’Università e della Ricerca (MUR), through PRIN 2022 program (Grant PANTHEON 2022E2J4RK, Next Generation EU) and PON R&I program (Avviso n. 424 del 28 febbraio 2018, Progetto PACK-PIR01 00021), Italy; IDMAR project Po-Fesr Sicilian Region az. 1.5.1; A. De Benedittis, W. Idrissi Ibnsalih, M. Bendahman, A. Nayerhoda, G. Papalashvili, I. C. Rea, A. Simonelli have been supported by the Italian Ministero dell’Università e della Ricerca (MUR), Progetto CIR01 00021 (Avviso n. 2595 del 24 dicembre 2019); KM3NeT4RR MUR Project National Recovery and Resilience Plan (NRRP), Mission 4 Component 2 Investment 3.1, Funded by the European Union – NextGenerationEU,CUP I57G21000040001, Concession Decree MUR No. n. Prot. 123 del 21/06/2022; Ministry of Higher Education, Scientific Research and Innovation, Morocco, and the Arab Fund for Economic and Social Development, Kuwait; Nederlandse organisatie voor Wetenschappelijk Onderzoek (NWO), the Netherlands; Ministry of Research, Innovation and Digitalisation, Romania; Slovak Research and Development Agency under Contract No. APVV-22-0413; Ministry of Education, Research, Development and Youth of the Slovak Republic; MCIN for PID2021-124591NB-C41, -C42, -C43 and PDC2023-145913-I00 funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”, for ASFAE/2022/014 and ASFAE/2022 /023 with funding from the EU NextGenerationEU (PRTR-C17.I01) and Generalitat Valenciana, for Grant AST22_6.2 with funding from Consejería de Universidad, Investigación e Innovación and Gobierno de España and European Union - NextGenerationEU, for CSIC-INFRA23013 and for CNS2023-144099, Generalitat Valenciana for CIDEGENT/2018/034, /2019/043, /2020/049, /2021/23, for CIDEIG/2023/20, for CIPROM/2023/51 and for GRISOLIAP/2021/192 and EU for MSC/101025085, Spain; Khalifa University internal grants (ESIG-2023-008 and RIG-2023-070), United Arab Emirates; The European Union’s Horizon 2020 Research and Innovation Programme (ChETEC-INFRA - Project no. 101008324).Peer ReviewedPostprint (published version

The Calibration Units of KM3NeT

KM3NeT is a deep-sea infrastructure composed of two neutrino telescopes being deployed in the Mediterranean Sea : ARCA, near Sicily in Italy, designed for neutrino astronomy and ORCA, near Toulon in France, designed for neutrino oscillations. These two telescopes are 3D arrays of optical modules used to detect the Cherenkov radiation, which is a signature of charged particles going faster than light in the sea water. To achieve the best performance for the events reconstruction in the telescopes, the exact location of the optical modules, affected by sea current, must be known at any time and the timing resolution between optical modules must reach the nanosecond. Moreover, the properties of the environment in which the telescopes are deployed, such as temperature and salinity, must be continuously monitored because they affect the timing and positioning calibration. KM3NeT is going to deploy several dedicated Calibration Units to meet these calibration goals. The Calibration Base will host several instruments : a Laser Beacon for time calibration and an acoustic emitter and a hydrophone for positioning of the optical modules. To complete the positioning calibration, some of these Calibration Units will be equipped with an Instrumentation Unit hosting environmental monitoring instruments. Because of the difference in size between ARCA and ORCA, the design of the Calibration Unit is not the same for the two sites. This proceeding describes all the devices, features and purposes of the Calibration Units with a focus on ORCA Calibration Unit and its status.Article signat per 297 autors/es: M.Ageron, S. Aiello, A. Albert, M. Alshamsi, S. Alves Garre, Z. Aly, A. Ambrosone, F. Ameli, M. Andre, G. Androulakis, M. Anghinolfi, M. Anguita, G. Anton, M. Ardid, S. Ardid, W. Assal, J. Aublin, C. Bagatelas, B. Baret, S. Basegmez du Pree, M. Bendahman, F. Benfenati, E. Berbee, A. M. van den Berg, V. Bertin, S. Beurthey, V. van Beveren, S. Biagi, M. Billault, M. Bissinger, M. Boettcher, M. Bou Cabo, J. Boumaaza, M. Bouta, C. Boutonnet, G. Bouvet, M. Bouwhuis, C. Bozza, H.Brânzas, R. Bruijn, J. Brunner, R. Bruno, E. Buis, R. Buompane, J. Busto, B. Caiffi, L. Caillat, D. Calvo, S. Campion, A. Capone, H. Carduner, V. Carretero, P. Castaldi, S. Celli;, R. Cereseto, M. Chabab, C. Champion, N. Chau, A. Chen, S. Cherubini, V. Chiarella, T. Chiarusi, M. Circella, R. Cocimano, J. A. B. Coelho, A. Coleiro, M. Colomer Molla, S. Colonges, R. Coniglione, A. Cosquer, P. Coyle, M. Cresta, A. Creuso, A. Cruz, G. Cuttone, A. D’Amico, R. Dallier, B. De Martino, M. De Palma, I. Di Palma, A. F. Díaz, D. Diego- Tortosa, C. Distefano, A. Domi, C. Donzaud, D. Dornic, M. Dörr, D. Drouhin, T. Eberl, A. Eddyamoui, T. van Eeden, D. van Eijk, I. El Bojaddaini, H. Eljarrari, D. Elsaesser, A. Enzenhöfer, V. Espinosa, P. Fermani, G. Ferrara, M. D. Filipovic, F. Filippini, J. Fransen, L. A. Fusco, D. Gajanana, T. Gal, J. García Méndez, A. Garcia Soto, E. Garçon, F. Garufi, C. Gatius, N. Geißelbrecht, L. Gialanella, E. Giorgio, S. R. Gozzini, R. Gracia, K. Graf, G. Grella, D. Guderian, C. Guidi, B. Guillon, M. Gutiérrez, J. Haefner, S. Hallmann, H. Hamdaoui, H. van Haren, A. Heijboer, A. Hekalo, L. Hennig, S. Henry, J. J. Hernández-Rey, J. Hofestädt, F. Huang,W. Idrissi Ibnsalih, A. Ilioni, G. Illuminati, C.W. James, D. Janezashvili, P. Jansweijer, M. de Jong, P. de Jong, B. J. Jung, M. Kadler, P. Kalaczynski, O. Kalekin,U. F. Katz, F. Kayzel, P.Keller, N. R. Khan Chowdhury, G. Kistauri, F. van der Knaap, P. Kooijman, A. Kouchner, M. Kreter, V. Kulikovskiy, M. Labalme, P. Lagier, R. Lahmann, P. Lamare, M. Lamoureux, G. Larosa, C. Lastoria, J. Laurence, A. Lazo, R. Le Breton, E. Le Guirriec, S. Le Stum, G. Lehaut, O. Leonardi, F. Leone, E. Leonora, C. Lerouvillois, J. Lesrel, N. Lessing, G. Levi, M. Lincetto, M. Lindsey Clark, T. Lipreau, C. LLorens Alvarez, A. Lonardo, F. Longhitano, D. Lopez-Coto, N. Lumb, L. Maderer, J. Majumdar, J. Manczak, A. Margiotta, A. Marinelli, A. Marini, C. Markou, L. Martin, J. A. Martínez-Mora, A. Martini, F. Marzaioli, S. Mastroianni, K.W. Melis, G. Miele, P. Migliozzi, E. Migneco, P. Mijakowski, L. S. Miranda, C. M. Mollo, M. Mongelli, A. Moussa, R. Muller, P. Musico, M. Musumeci, L. Nauta, S. Navas, C. A. Nicolau, B. Nkosi, B. Ó Fearraigh, M. O’Sullivan, A. Orlando, G. Ottonello, S. Ottonello, J. Palacios González5, G. Papalashvili, R. Papaleo, C. Pastore, A. M. Paun, G. E. Pavalas, G. Pellegrini, C. Pellegrino, M. Perrin-Terrin, V. Pestel, P. Piattelli, C. Pieterse, O. Pisanti, C. Poirè, V. Popa, T. Pradier, F. Pratolongo, I. Probst, G. Pühlhofer, S. Pulvirenti, G. Quéméner, N. Randazzo, A. Rapicavoli, S. Razzaque, D. Real, S. Reck, G. Riccobene, L. Rigalleau, A. Romanov, A. Rovelli, J. Royon, F. Salesa Greus, D. F. E. Samtleben, A. Sánchez Losa, M. Sanguineti, A. Santangelo, D. Santonocito, P. Sapienza, J. Schmelling, J. Schnabel, M. F. Schneider, J. Schumann, H. M. Schutte, J. Seneca, I. Sgura, R. Shanidze, A. Sharma, A. Sinopoulou, B. Spisso, M. Spurio, D. Stavropoulos, J. Steijger, S. M. Stellacci, M. Taiuti, F. Tatone, Y. Tayalati, E. Tenllado, D. Tézier, T. Thakore, S. Theraube, H. Thiersen, P. Timmer, S. Tingay, S. Tsagkli, V. Tsourapis, E. Tzamariudaki, D. Tzanetatos, C. Valieri, V. Van Elewyck, G. Vasileiadis, F. Versari, S. Viola, D. Vivolo, G. de Wasseige, J.Wilms, R.Wojaczynski, E. deWolf, T. Yousfi, S. Zavatarelli, A. Zegarelli, D. Zito, J. D. Zornoza, J. Zúñiga, N. Zywucka.Postprint (published version

Nanobeacon: a time calibration device for the KM3NeT neutrino telescope

The KM3NeT Collaboration is currently constructing a multi-site high-energy neutrino telescope in the Mediterranean Sea consisting of matrices of pressure-resistant glass spheres, each holding a set of 31 small-area photomultipliers. The main goals of the telescope are the observation of neutrino sources in the Universe and the measurement of the neutrino oscillation parameters with atmospheric neutrinos. Both extraterrestrial and atmospheric neutrinos are detected through the Cherenkov light induced in seawater by charged particles produced in neutrino interactions in the surrounding medium. A relative time synchronization between photomultipliers of the order of 1 ns is needed to guarantee the required angular resolution of the detector. Due to the large detector volumes to be instrumented by KM3NeT, a cost reduction of the different systems is a priority. To this end, the inexpensive Nanobeacon has been designed and developed by the KM3NeT Collaboration to be used for detector time-calibration studies. At present, more than 600 Nanobeacons have been already produced. The characterization of the optical pulse and the wavelength emission profile of the devices are critical for the time calibration. In this paper, the main features of the Nanobeacon design, production and operation, together with the main properties of the light pulse generated are describedPeer ReviewedPostprint (author's final draft