The Complexity of the List Partition Problem for Graphs

The k-partition problem is as follows: Given a graph G and a positive integer k, partition the vertices of G into at most k parts A1, A2, . . . , Ak, where it may be specified that Ai induces a stable set, a clique, or an arbitrary subgraph, and pairs Ai, Aj (i≠j) be completely nonadjacent, completely adjacent, or arbitrarily adjacent. The list k-partition problem generalizes the k-partition problem by specifying for each vertex x, a list L(x) of parts in which it is allowed to be placed. Many well-known graph problems can be formulated as list k-partition problems: e.g., 3-colorability, clique cutset, stable cutset, homogeneous set, skew partition, and 2-clique cutset. We classify, with the exception of two polynomially equivalent problems, each list 4-partition problem as either solvable in polynomial time or NP-complete. In doing so, we provide polynomial-time algorithms for many problems whose polynomial-time solvability was open, including the list 2-clique cutset problem. This also allows us to classify each list generalized 2-clique cutset problem and list generalized skew partition problem as solvable in polynomial time or NP-complete

Perfect Graphs

This chapter is a survey on perfect graphs with an algorithmic flavor. Our emphasis is on important classes of perfect graphs for which there are fast and efficient recognition and optimization algorithms. The classes of graphs we discuss in this chapter are chordal, comparability, interval, perfectly orderable, weakly chordal, perfectly contractile, and chi-bound graphs. For each of these classes, when appropriate, we discuss the complexity of the recognition algorithm and algorithms for finding a minimum coloring, and a largest clique in the graph and its complement

Ergodicity for the 3D Stochastic Navier-Stokes Equations Perturbed by Lévy Noise

In this work we construct a Markov family of martingale solutions for 3D stochastic Navier–Stokes equations (SNSE) perturbed by Lévy noise with periodic boundary conditions. Using the Kolmogorov equations of integrodifferential type associated with the SNSE perturbed by Lévy noise, we construct a transition semigroup and establish the existence of a unique invariant measure. We also show that it is ergodic and strongly mixing. Abstract © Wiley

Conditional value-at-risk for water management in Lake Burley Griffin

Copyright © Australian Mathematical SocietyAs the centrepiece of Canberra, Lake Burley Griffin provides the setting for buildings of national importance and a venue for aquatic recreation while, as part of the Molonglo River, the lake has a role in the ecological processes of its broader setting. For the purposes of recreation and landscape a constant water level is preferred: the management plan requires the lake to be maintained at a prescribed normal level. In years of low rainfall this requirement could conflict with the water demands of other users. Episodes of high rainfall may also require compromise between competing objectives. For example, drawdown of lake levels for flood mitigation could impact on the lake's recreational and amenity values and the spill may not be a good use of water. Conditional Value at Risk, a risk measure developed by the financial industry for portfolio management, is defined as the expected loss given that some loss threshold is exceeded. Here, Conditional Value at Risk is applied as decision support for strategic planning and day-to-day operational problems in the hydraulic management of Lake Burley Griffin.R. B. Webby, J. Boland, P. G. Howlett, A. V. Metcalfe, T. Srithara

Pile Setup in Cohesive Soil. I: Experimental Investigation

Pile setup in cohesive soils has been a known phenomenon for several decades. However, a systematic field investigation to provide the needed data to develop analytical procedures and integrate pile setup into the design method rarely exists. This paper summarizes a recently completed field investigation on five fully instrumented steel H-piles embedded in cohesive soils, while a companion paper discusses the development of the pile setup method. During the field investigation, detailed soil characterization, monitoring of soil total lateral stress and pore-water pressure, collection of pile dynamic restrike data as a function of time, and vertical static load tests were completed. Restrike measurements confirm that pile setup occurs at a logarithmic rate following the end of driving, and its development correlates well with the rate of dissipation of the measured porewater pressure. Based on the field data collected, it was concluded that the skin friction component, not the end bearing, contributes predominantly to the setup, which can be accurately estimated for practical purposes using soil properties, such as coefficient of consolidation, undrained shear strength, and the standard penetration testN-value

A multi-centre randomised controlled trial comparing radiofrequency and mechanical occlusion chemically assisted ablation of varicose veins - final results of the Venefit versus Clarivein for varicose veins trial

BACKGROUND: Endovenous thermal ablation has revolutionised varicose vein treatment. New non-thermal techniques such as mechanical occlusion chemically assisted endovenous ablation (MOCA) allow treatment of entire trunks with single anaesthetic injections. Previous non-randomised work has shown reduced pain post-operatively with MOCA. This study presents a multi-centre randomised controlled trial assessing the difference in pain during truncal ablation using MOCA and radiofrequency endovenous ablation (RFA) with six months' follow-up. METHODS: Patients undergoing local anaesthetic endovenous ablation for primary varicose veins were randomised to either MOCA or RFA. Pain scores using Visual Analogue Scale and number scale (0-10) during truncal ablation were recorded. Adjunctive procedures were completed subsequently. Pain after phlebectomy was not assessed. Patients were reviewed at one and six months with clinical scores, quality of life scores and duplex ultrasound assessment of the treated leg. RESULTS: A total of 170 patients were recruited over a 21-month period from 240 screened. Patients in the MOCA group experienced significantly less maximum pain during the procedure by Visual Analogue Scale (MOCA median 15 mm (interquartile range 7-36 mm) versus RFA 34 mm (interquartile range 16-53 mm), p = 0.003) and number scale (MOCA median 3 (interquartile range 1-5) versus RFA 4 mm (interquartile range 3-6.5), p = 0.002). 'Average' pain scores were also significantly less in the MOCA group; 74% underwent simultaneous phlebectomy. Occlusion rates, clinical severity scores, disease specific and generic quality of life scores were similar between groups at one and six months. There were two deep vein thromboses, one in each group. CONCLUSION: Pain secondary to truncal ablation is less painful with MOCA than RFA with similar short-term technical, quality of life and safety outcomes