Casting Light on the Hidden Bilevel Combinatorial Structure of the Capacitated Vertex Separator Problem

Given an undirected graph, we study the capacitated vertex separator problem that asks to find a subset of vertices of minimum cardinality, the removal of which induces a graph having a bounded number of pairwise disconnected shores (subsets of vertices) of limited cardinality. The problem is of great importance in the analysis and protection of communication or social networks against possible viral attacks and for matrix decomposition algorithms. In this article, we provide a new bilevel interpretation of the problem and model it as a two-player Stackelberg game in which the leader interdicts the vertices (i.e., decides on the subset of vertices to remove), and the follower solves a combinatorial optimization problem on the resulting graph. This approach allows us to develop a computational framework based on an integer programming formulation in the natural space of the variables. Thanks to this bilevel interpretation, we derive three different families of strengthening inequalities and show that they can be separated in polynomial time. We also show how to extend these results to a min-max version of the problem. Our extensive computational study conducted on available benchmark instances from the literature reveals that our new exact method is competitive against the state-of-the-art algorithms for the capacitated vertex separator problem and is able to improve the best-known results for several difficult classes of instances. The ideas exploited in our framework can also be extended to other vertex/edge deletion/ insertion problems or graph partitioning problems by modeling them as two-player Stackel- berg games and solving them through bilevel optimization

Tensile Strength of Geological Discontinuities Including Incipient Bedding, Rock Joints and Mineral Veins

Geological discontinuities have a controlling influence for many rock-engineering projects in terms of strength, deformability and permeability, but their characterisation is often very difficult. Whilst discontinuities are often modelled as lacking any strength, in many rock masses visible rock discontinuities are only incipient and have tensile strength that may approach and can even exceed that of the parent rock. This fact is of high importance for realistic rock mass characterisation but is generally ignored. It is argued that current ISRM and other standards for rock mass characterisation, as well as rock mass classification schemes such as RMR and Q, do not allow adequately for the incipient nature of many rock fractures or their geological variability and need to be revised, at least conceptually. This paper addresses the issue of the tensile strength of incipient discontinuities in rock and presents results from a laboratory test programme to quantify this parameter. Rock samples containing visible, natural incipient discontinuities including joints, bedding, and mineral veins have been tested in direct tension. It has been confirmed that such discontinuities can have high tensile strength, approaching that of the parent rock. Others are, of course, far weaker. The tested geological discontinuities all exhibited brittle failure at axial strain less than 0.5 % under direct tension conditions. Three factors contributing to the tensile strength of incipient rock discontinuities have been investigated and characterised. A distinction is made between sections of discontinuity that are only partially developed, sections of discontinuity that have been locally weathered leaving localised residual rock bridges and sections that have been ‘healed’ through secondary cementation. Tests on bedding surfaces within sandstone showed that tensile strength of adjacent incipient bedding can vary considerably. In this particular series of tests, values of tensile strength for bedding planes ranged from 32 to 88 % of the parent rock strength (intact without visible discontinuities), and this variability could be attributed to geological factors. Tests on incipient mineral veins also showed considerable scatter, the strength depending upon the geological nature of vein development as well as the presence of rock bridges. As might be anticipated, tensile strength of incipient rock joints decreases with degree of weathering as expressed in colour changes adjacent to rock bridges. Tensile strengths of rock bridges (lacking marked discolouration) were found to be similar to that of the parent rock. It is concluded that the degree of incipiency of rock discontinuities needs to be differentiated in the process of rock mass classification and engineering design and that this can best be done with reference to the tensile strength relative to that of the parent rock. It is argued that the science of rock mass characterisation may be advanced through better appreciation of geological history at a site thereby improving the process of prediction and extrapolating properties

New, simplified and improved interpretation of the Vaiont landslide mechanics

Both the occurrence and behaviour of the Vaiont landslide have not been satisfactorily explained previously because of difficulties arising from the assumption that the failure surface was ‘chair’ shaped. It is now known that there was no ‘chair’, which means that the 1963 landslide could not have been a reactivated ancient landslide because the residual strength of the clay interbeds would have been insufficient for stability prior to 1963. Furthermore, the moderately translational geometry reduces the influence of reservoir-induced groundwater and hence of submergence. Standard stability analyses now show that prior to 1960, the average shear strength must have significantly exceeded the peak shear strength of the clay interbeds known to have formed the majority of the failure surface. Three-dimensional stability analyses confirm these results and show that at the time of the first significant movements in 1960, the rising reservoir level had a negligible effect on the Factor of Safety. According to these results, the Vaiont landslide was most likely initiated by pore water pressures associated with transient rainfall-induced ‘perched’ groundwater above the clay layers, in combination with a smaller than hitherto assumed effect of reservoir impounding, then developed by brittle crack propagation within the clay beds, thus displaying progressive failure. Further, very heavy rainfall accelerated the process, possibly due to reservoir-induced groundwater impeding drainage of the rainwater, until the limestone beds at the northeast margin failed. With the shear strength suddenly reduced to residual throughout, the entire mass was released and was able to accelerate as observed

UCS field estimation of intact rock using the Schmidt hammer: A new empirical approach

In the present work we discuss the results of a number of Schmidt hammer tests (total number of impacts N > 2,400) that were performed in situ on rock outcrops of different lithology (marl, calcareous marl, limestone, sandstone, quartz sandstone and rhyolite) that occur in Italy. Firstly, a specific field procedure to choose the reference value of the rebound index adopted to calculate UCS of intact rock is suggested. A relationship between L and N hammer rebound index values (RL and RN, respectively) is subsequently assessed. Considering the experimental datasets provided by a Schmidt hammer construction company and other research available in literature, a new exponential equation for the correlation between RL and UCS of intact rock has been derived. Considering the here-proposed RL-RN relationship, a new exponential correlation between RN and UCS has also been defined. The newly proposed procedure and relationships were successfully utilised to determine the intact rock strength of different rocks. The calculated UCS values are very similar when using both types of Schmidt hammer (L and N) and are generally in line with previous determinations from experimental data available in literature