Coloring tournaments with few colors: Algorithms and complexity

A $k$-coloring of a tournament is a partition of its vertices into $k$ acyclic sets. Deciding if a tournament is 2-colorable is NP-hard. A natural problem, akin to that of coloring a 3-colorable graph with few colors, is to color a 2-colorable tournament with few colors. This problem does not seem to have been addressed before, although it is a special case of coloring a 2-colorable 3-uniform hypergraph with few colors, which is a well-studied problem with super-constant lower bounds. We present an efficient decomposition lemma for tournaments and show that it can be used to design polynomial-time algorithms to color various classes of tournaments with few colors, including an algorithm to color a 2-colorable tournament with ten colors. For the classes of tournaments considered, we complement our upper bounds with strengthened lower bounds, painting a comprehensive picture of the algorithmic and complexity aspects of coloring tournaments

Coloring Tournaments with Few Colors: Algorithms and Complexity

A k-coloring of a tournament is a partition of its vertices into k acyclic sets. Deciding if a tournament is 2-colorable is NP-hard. A natural problem, akin to that of coloring a 3-colorable graph with few colors, is to color a 2-colorable tournament with few colors. This problem does not seem to have been addressed before, although it is a special case of coloring a 2-colorable 3-uniform hypergraph with few colors, which is a well-studied problem with super-constant lower bounds. We present an efficient decomposition lemma for tournaments and show that it can be used to design polynomial-time algorithms to color various classes of tournaments with few colors, including an algorithm to color a 2-colorable tournament with ten colors. For the classes of tournaments considered, we complement our upper bounds with strengthened lower bounds, painting a comprehensive picture of the algorithmic and complexity aspects of coloring tournaments

Bounding the chromatic number of tournaments by arc neighborhoods

The chromatic number of a directed graph is the minimum number of induced acyclic subdigraphs that cover its vertex set, and accordingly, the chromatic number of a tournament is the minimum number of transitive subtournaments that cover its vertex set. The neighborhood of an arc $uv$ in a tournament $T$ is the set of vertices that form a directed triangle with arc $uv$. We show that if the neighborhood of every arc in a tournament has bounded chromatic number, then the whole tournament has bounded chromatic number. We show that this holds more generally for oriented graphs with bounded independence number, which we use to prove the equivalence of a conjecture of El-Zahar and Erd\H{o}s and a recent conjecture of Nguyen, Scott and Seymour relating the structure of graphs and tournaments with high chromatic number

Fiber Optic Monitoring of Active Faults at the Seafloor: the FOCUS project

Laser reflectometry (BOTDR), commonly used for structural health monitoring (bridges, dams, etc.), will for the first time be applied to study movements of an active fault on the seafloor 25 km offshore Catania Sicily. The goal of the European funded FOCUS project (ERC Advanced Grant) is to connect a 6-km long strain cable to the EMSO seafloor observatory in 2100 m water depth. Laser observations will be calibrated by seafloor geodetic instruments and seismological stations. A long-term goal is the development of dual-use telecom cables with industry partners

Applying laser reflectometry to study active submarine faults: the FOCUS project (FOCUS = Fiber Optic Cable Use for Seafloor studies of earthquake hazard and deformation)

Laser reflectometry (BOTDR), commonly used for structural health monitoring (bridges, dams, etc.), will for the first time be applied to study movements of an active fault on the seafloor, 25 km offshore Catania Sicily (an urban area of 1 million people). This technique can measure and locate micro-strains (< 1 mm) across very large distances (10 - 200 km). The goal of the European funded FOCUS project (ERC Advanced Grant) is to connect a dedicated 6-km long strain cable to the EMSO (European Multidisciplinary water-column and Seafloor Observatory) seafloor observatory in 2100 m water depth. Here, in May 2017, between the onshore fault system on the SE flank of Mount Etna and the deeper offshore Alfeo fault system, 4 cm of dextral strike-slip movement was documented as a slow slip event by seafloor acoustic ranging. For the planned seafloor operations, a detailed site survey of the seafloor will first be performed to determine the best path for deployment of the new strain cable. The next step will be to connect this 6-km long fiber optic cable to the EMSO station TSS (Test Site South) using a deep-water cable-laying system with an integrated plow to bury the cable 20 cm in the soft sediments in order to increase coupling between the cable and the seafloor. The targeted track for the cable will cross the North Alfeo Fault at three locations. Laser reflectometry measurements will be calibrated by a three-year deployment of seafloor geodetic instruments to quantify relative displacement across the fault. During the implementation of the laser reflectometry, a passive seismological experiment is planned to record regional seismicity. This will involve deployment of a temporary network of OBS (Ocean Bottom Seismometers) on the seafloor and seismic stations on land, supplemented by INGV permanent land stations. The simultaneous use of laser reflectometry, seafloor geodetic stations as well as seismological land and sea stations will provide an integrated system for monitoring a wide range of types of slipping events along the North Alfeo Fault. A long-term goal is the development of dual-use telecom cables with industry partners

Coloring Tournaments with Few Colors: Algorithms and Complexity

International audienceA k-coloring of a tournament is a partition of its vertices into k acyclic sets. Deciding if a tournament is 2-colorable is NP-hard. A natural problem, akin to that of coloring a 3-colorable graph with few colors, is to color a 2-colorable tournament with few colors. This problem does not seem to have been addressed before, although it is a special case of coloring a 2-colorable 3-uniform hypergraph with few colors, which is a well-studied problem with super-constant lower bounds. We present an efficient decomposition lemma for tournaments and show that it can be used to design polynomial-time algorithms to color various classes of tournaments with few colors, including an algorithm to color a 2-colorable tournament with ten colors. For the classes of tournaments considered, we complement our upper bounds with strengthened lower bounds, painting a comprehensive picture of the algorithmic and complexity aspects of coloring tournaments

Detecting strain with a fiber optic cable on the seafloor offshore Mount Etna, Southern Italy

Highlights • A fiber optic strain cable is used to monitor a fault offshore Catania, Sicily. • Brillouin laser reflectometry detects 2.5 cm of cable elongation on the seafloor. • The cable elongation may be caused by fault slip or by seabottom currents. • Submarine telecom cables are likely suitable to detect deformation on the seafloor. Abstract Oceans cover more than 70 percent of the Earth's surface making it difficult and costly to deploy modern seismological instruments here. The rapidly expanding global network of submarine telecom cables offers tremendous possibilities for seismological monitoring using laser light. Recent pioneer studies have demonstrated earthquake detection using lasers in onland and submarine fiber optic cables. However, permanent strain at the seafloor has never before been measured directly as it happens. With this aim, we deployed a dedicated 6-km-long fiber optic strain cable, offshore Catania Sicily, in 2000 m water depth, and connected it to a 29-km long electro-optical cable for science use. We report here that deformation of the cable equivalent to a total elongation of 2.5 cm was observed over a 21-month period (from Oct. 2020 to Jul. 2022). Brillouin laser reflectometry observations over the first 10 months indicate significant strain (+25 to +40 microstrain) at two locations where the cable crosses an active strike-slip fault on the seafloor, with most of the change occurring between 19 and 21 Nov. 2020. The cause of the strain could be fault slip or seabottom currents. During the following 11 months, the strain amplitude increased to +45 to +55 microstrain, affecting a longer portion of the cable up to 500 m to either side of the first fault crossing. A sandbag experiment performed on the distal portion of the cable (3.2–6.0 km) starting Sept. 2021 demonstrates how the fiber optic cable deforms in response to an applied load and how the deformation signal partially dissipates over time due to the elastic properties of the cable. These preliminary results are highly encouraging for the use of BOTDR (Brillouin Optical Time Domain Reflectometry) laser reflectometry as a technique to detect strain at the seafloor in near real time and to monitor the structural health of submarine cables

A novel approach for studying submarine faults: the FOCUS project (FOCUS = Fiber Optic Cable Use for Seafloor studies of earthquake hazard and deformation)

Two-thirds of the earth’s surface is covered by water and thus largely inaccessible to modern networks of seismological instruments. A novel use of fiber optic cables could help improve hazard assessment and increase early warning capability. Laser reflectometry using BOTDR (Brillouin Optical Time Domain Reflectometry), commonly used for structural health monitoring of large-scale engineering structures (e.g. - bridges, dams, pipelines, etc.) can measure very small strains (< 1 mm) at very large distances (10 - 200 km). This technique has never been used to monitor deformation caused by active faults on the seafloor. The objective of the FOCUS project is to demonstrate that this technique can measure small (1 - 2 cm) displacements on a primary test site offshore Sicily where the recently mapped North Alfeo Fault crosses the Catania EMSO seafloor observatory, 28 km long fiber optic cable. Two other EMSO test sites with fiber optic cables, the 100 km long Capo Passero (SE Sicily) and the 2 km long cable off Molene Island (W France) will also be studied. Initial reflectometry tests were performed on these three cables using a Febus BOTDR interrogator in June and July 2017. Unexpectedly high dynamic noise levels (corresponding to strains of 200 - 500 mm/m) were observed on the Molene cable, likely due to the high-energy, shallow water, open ocean environment. The tests on the EMSO infrastructure in Sicily indicated low experimental noise levels (20 - 30 mm/m) out to a distance of 15 km. BOTDR observations will have to be calibrated by other independent measurements. Therefore, targeted marine geophysical surveys of the seafloor along the trace of the cable and faults are planned, with the use of seafloor geodetic instruments to quantify fault displacement. Once the BOTDR fault-monitoring technique has been tested, demonstrated and calibrated offshore Eastern Sicily, the goal is to expand it to other fiber optic cable networks, either existing research networks in earthquake hazard zones (Japan, Cascadia) or to the Mediterranean region through access to retired (decommissioned) telecommunication cables or development of dual-use cables (two of the anticipated outcomes of the FOCUS project). This represents a potentially tremendous breakthrough in seismology, tectonics and natural hazard early warning capability

The FOCUS experiment 2020 (Fiber Optic Cable Use for Seafloor studies of earthquake hazard and deformation)

Laser reflectometry (BOTDR), commonly used for structural health monitoring (bridges, dams, etc.), for the first time is being tested to study movements of an active fault on the seafloor, 25 km offshore Catania Sicily (an urban area of 1 million people). Under ideal conditions, this technique can measure small strains (10E-6), across very large distances (10 - 200 km) and locate these strains with a spatial resolution of 10 - 50 m. As the first experiment of the European funded FOCUS project (ERC Advanced Grant), in late April 2020 we aimed to connect and deploy a dedicated 6-km long strain cable to the TSS (Test Site South) seafloor observatory in 2100 m water depth operated by INFN-LNS (Italian National Physics Institute). The work plan for the marine expedition FocusX1 onboard the research vessel PourquoiPas? is described here. First, microbathymetric mapping and a video camera survey are performed by the ROV Victor6000. Then, several intermediate junction frames and short connector cables (umbilicals) are connected. A cable-end module and 6-km long fiber-optic strain cable (manufactured by Nexans Norway) is then connected to the new junction box. Next, we use a deep-water cable-laying system with an integrated plow (updated Deep Sea Net design Ifremer, Toulon) to bury the cable 20 cm in the soft sediments in order to increase coupling between the cable and the seafloor. The targeted track for the cable crosses the North Alfeo Fault at three locations. Laser reflectometry measurements began April 2020 and will be calibrated by a three-year deployment of seafloor geodetic instruments (Canopus acoustic beacons manufactured by iXblue) also started April 2020, to quantify relative displacement across the fault. During a future marine expedition, tentatively scheduled for 2021 (FocusX2) a passive seismological experiment is planned to record regional seismicity. This will involve deployment of a temporary network of OBS (Ocean Bottom Seismometers) on the seafloor and seismic stations on land, supplemented by INGV permanent land stations. The simultaneous use of laser reflectometry, seafloor geodetic stations as well as seismological land and sea stations will provide an integrated system for monitoring a wide range of types of slipping events along the North Alfeo Fault (e.g. - creep, slow-slip, rupture). A long-term goal is the development of dual-use telecom cables with industry partners