High-precision tephrostratigraphy : tracking the time-varying eruption pulse of Mt. Taranaki, North Island, New Zealand : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Earth Science, Massey University, Palmerston North, New Zealand

In this research it was proposed that a more robust record of volcanic activity for Mt. Taranaki (New Zealand) could be derived from tephras (pyroclastic fall deposits) within cores from several lakes and peatlands across a 120o arc, NE-SE of the volcano, covering a range of prevailing down-wind directions. These data were integrated with previous tephrochronology studies to construct one of the longest and most complete volcanic eruption history records ever developed for an andesitic stratovolcano. Using 44 new radiocarbon dates, electron microprobe analysis of glass shard and titanomagnetite chemical composition, along with whole-rock chemistry, a chrono- and chemostratigraphy was established. The new record identifies at least 272 tephraproducing eruptions over the last 30 cal ka BP. Six chemo-stratigraphic groups were identified: A (0.5 – 3 cal ka BP), B (3 – 4 cal ka BP), C (4 – 9.5 cal ka BP), D (9.5 – 14 cal ka BP), E (14 – 17.5 cal ka BP), and F (23.5 – 30 cal ka BP). These were used to resolve previous stratigraphic uncertainties at upper-flank (proximal) and ring-plain (medial) sites. Several well-known “marker tephras” are now recognized as being ~2000 years older than previously determined (e.g., Waipuku, Tariki, and Mangatoki Tephra units) with the prominent Korito Tephra stratigraphically positioned above the Taupo-derived Stent Tephra. Further, new markers were identified, including the Kokowai Tephra unit (~4.7 cal ka BP), at a beach-cliff exposure, 40-km north-east of the volcano. Once age-models were established for each tephra, units were matched between sites using statistical methods. Initial statistical integration showed that the immediate past high-resolution tephrochronological record suffered from a distinctive “old-carbon” effect on its ages (Lake Rotokare). This had biased the most recent probabilistic forecasting and generated artificially high probability estimates (52-59% eruption chance over the next 50 years). Once the Rotokare record was excluded and chemostratigraphy constraints were applied, a reliable multi-site tephra record could be built only for the last ~14 ka BP. The new data confirms a highly skewed distribution of mainly (98% of cases) short intervals between eruptions (mode of ~9 years and average interval ~65 years). Long intervals (up to 580 years) as seen in earlier records were reduced to 2% of the record, but can now be considered real, rather than missing data. The new data confirm a cyclic pattern of varying eruption frequency (with a five-fold range in annual frequency) on a period of ~1000-1500 years. The new time-varying frequency estimates suggest a lower probability for a new eruption at Mt. Taranaki over the next 50 years of 33-42%. The newly established chemostratigraphy was further used to investigate time-related compositional changes. Whole-lapilli analyses highlighted that a specific very evolved Ca-rich and Fe-poor composition was only found within the easterly and south-easterly depositional sites. This was explained by eruption of a stratified magma reservoir, which holds greater modal proportions of plagioclase and lower proportions of pyroxene within low-density, gas-rich upper conduit regions. During the most explosive phases of eruptions, when plumes reach the stratospheric jetstream, the lowest-density pumice is thus dispersed by high-level stable westerly winds. Further, two distinct evolutional trends were seen in the long and new tephrochronological record; from 17.5 to 3 cal ka BP and <3 cal ka BP; with wholelapilli, glass, and titanomagnetite compositions overall evolving over time. The former compositional trend indicates a crystallising and cooling magma source in the deep crust, with multiple, spatially separated magma source regions forming, each generating magmas (i.e., magma batches) with unique titanomagnetite compositions. This trend is interrupted by a distinct shift towards less-evolved compositions and the initiation of a second parasitic vent (Fanthams Peak at the southern flank of Mt. Taranaki)

Optimal partial clique edge covering guided by potential energy minimization

For given integers\ua0k,\ua0n,\ua0r\ua0we aim at families of\ua0k\ua0sub-cliques called blocks, of a clique with\ua0n\ua0vertices, such that every block has\ua0r\ua0vertices, and the blocks together cover a maximum number of edges. We demonstrate a combinatorial optimization method that generates such optimal partial clique edge coverings. It takes certain packages of columns (corresponding to vertices) in the incidence matrix of the blocks, considers the number of uncovered edges as an energy term that has to be minimized by transforming these packages. As a proof of concept we can completely solve the above maximization problem in the case of\ua0k≤4k≤4\ua0blocks and obtain optimal coverings for all integers\ua0n\ua0and\ua0r\ua0with\ua0r/n≥5/9r/n≥5/9. This generalizes known results for total coverings to partial coverings. The method as such is not restricted to\ua0k≤4k≤4\ua0blocks, but a challenge for further research (also on total coverings) is to limit the case distinctions when more blocks are involved

Distance-Based Solution of Patrolling Problems with Individual Waiting Times

In patrolling problems, robots (or other vehicles) must perpetually visit certain points without exceeding given individual waiting times. Some obvious applications are monitoring, maintenance, and periodic fetching of resources. We propose a new generic formulation of the problem. As its main advantage, it enables a reduction of the multi-robot case to the one-robot case in a certain graph/hypergraph pair, which also relates the problem to some classic path problems in graphs: NP-hardness is shown by a reduction from the Hamiltonian cycle problem, and on the positive side, the formulation allows solution heuristics using distances in the mentioned graph. We demonstrate this approach for the case of two robots patrolling on a line, a problem whose complexity status is open, apart from approximation results. Specifically, we solve all instances with up to 6 equidistant points, and we find some surprising effects, e.g., critical problem instances (which are feasible instances that become infeasible when any waiting time is diminished) may contain rather large individual waiting times

Enumerating grid layouts of graphs

We study algorithms that generate layouts of graphs with n vertices in a square grid with ν points, where adjacent vertices in the graph are also close in the grid. The problem is motivated by graph drawing and factory layout planning. In the latter application, vertices represent machines, and edges join machines that should be placed next to each other. Graphs admitting a grid layout where all edges have unit length are known as partial grid graphs. Their recognition is NP-hard already in very restricted cases. However, the moderate number of machines in practical instances suggests the use of exact algorithms that may even enumerate the possible layouts to choose from. We start with an elementary nO(√n)\ua0time algorithm, but then we argue that even simpler exponential branching algorithms are more usable for practical sizes n, although being asymptotically worse. One algorithm interpolates between obvious O∗(3n) time and O∗(4ν) time for graphs with many small connected components. It can be modified in order to accommodate also a limited number of edges that can exceed unit length. Next we show that connected graphs have at most 2.9241n\ua0grid layouts that can also be efficiently enumerated. An O∗(2.6458n) time branching algorithm solves the recognition problem, or yields a succinct enumeration of layouts with some surcharge on the time bound. In terms of the grid size we get a slightly better O∗(2.6208ν) time bound. Moreover, if we can identify a subgraph that is rigid, i.e., admits only one layout up to congruence, then all possible layouts of the entire graph are extensions of this unique layout, such that the combinatorial explosion is then confined to the rest of the graph. Therefore we also propose heuristic methods for finding certain types of large rigid subgraphs. The formulations of these results is more technical, however, the proposed method iteratively generates certain rigid subgraphs from smaller ones

Dividing splittable goods evenly and with limited fragmentation

A splittable good provided in n pieces shall be divided as evenly as possible among m agents, where every agent can take shares from at most F pieces. We call F the fragmentation and mainly restrict attention to the cases F= 1 and F= 2. For F= 1 , the max–min and min–max problems are solvable in linear time. The case F= 2 has neat formulations and structural characterizations in terms of weighted graphs. First we focus on perfectly balanced solutions. While the problem is strongly NP-hard in general, it can be solved in linear time if m≥ n- 1 , and a solution always exists in this case, in contrast to F= 1. Moreover, the problem is fixed-parameter tractable in the parameter 2 m- n. (Note that this parameter measures the number of agents above the trivial threshold m= n/ 2.) The structural results suggest another related problem where unsplittable items shall be assigned to subsets so as to balance the average sizes (rather than the total sizes) in these subsets. We give an approximation-preserving reduction from our original splitting problem with fragmentation F= 2 to this averaging problem, and some approximation results in cases when m is close to either n or n\ua0/\ua02

Computing and counting longest paths on circular-arc graphs in polynomial time.

The longest path problem asks for a path with the largest number of vertices in a given graph. The first polynomial time algorithm (with running time O(n4)) has been recently developed for interval graphs. Even though interval and circular-arc graphs look superficially similar, they differ substantially, as circular-arc graphs are not perfect. In this paper, we prove that for every path P of a circular-arc graph G, we can appropriately “cut” the circle, such that the obtained (not induced) interval subgraph G′ of G admits a path P′ on the same vertices as P. This non-trivial result is of independent interest, as it suggests a generic reduction of a number of path problems on circular-arc graphs to the case of interval graphs with a multiplicative linear time overhead of O(n). As an application of this reduction, we present the first polynomial algorithm for the longest path problem on circular-arc graphs, which turns out to have the same running time O(n4) with the one on interval graphs, as we manage to get rid of the linear overhead of the reduction. This algorithm computes in the same time an n-approximation of the number of different vertex sets that provide a longest path; in the case where G is an interval graph, we compute the exact number. Moreover, our algorithm can be directly extended with the same running time to the case where every vertex has an arbitrary positive weight

The parallel solution of domination problems on chordal and strongly chordal graphs

AbstractWe present efficient parallel algorithms for the domination problem on strongly chordal graphs and related problems, such as the set cover problem for α-acyclic hypergraphs and the dominating clique problem for strongly chordal graphs. Moreover, we present an efficient parallel algorithm which checks, for any chordal graph, whether it has a dominating clique

Paradigms for Parameterized Enumeration

The aim of the paper is to examine the computational complexity and algorithmics of enumeration, the task to output all solutions of a given problem, from the point of view of parameterized complexity. First we define formally different notions of efficient enumeration in the context of parameterized complexity. Second we show how different algorithmic paradigms can be used in order to get parameter-efficient enumeration algorithms in a number of examples. These paradigms use well-known principles from the design of parameterized decision as well as enumeration techniques, like for instance kernelization and self-reducibility. The concept of kernelization, in particular, leads to a characterization of fixed-parameter tractable enumeration problems.Comment: Accepted for MFCS 2013; long version of the pape

Letter graphs and geometric grid classes of permutations: characterization and recognition

In this paper, we reveal an intriguing relationship between two seemingly unrelated notions: letter graphs and geometric grid classes of permutations. An important property common for both of them is well-quasi-orderability, implying, in a non-constructive way, a polynomial-time recognition of geometric grid classes of permutations and $k$-letter graphs for a fixed $k$. However, constructive algorithms are available only for $k=2$. In this paper, we present the first constructive polynomial-time algorithm for the recognition of $3$-letter graphs. It is based on a structural characterization of graphs in this class.Comment: arXiv admin note: text overlap with arXiv:1108.6319 by other author