An optimal local approximation algorithm for max-min linear programs

In a max-min LP, the objective is to maximise ω subject to Ax ≤ 1, Cx ≥ ω1, and x ≥ 0 for nonnegative matrices A and C. We present a local algorithm (constant-time distributed algorithm) for approximating max-min LPs. The approximation ratio of our algorithm is the best possible for any local algorithm; there is a matching unconditional lower bound.

Local approximability of max-min and min-max linear programs

In a max-min LP, the objective is to maximise ω subject to Ax ≤ 1, Cx ≥ ω1, and x ≥ 0. In a min-max LP, the objective is to minimise ρ subject to Ax ≤ ρ1, Cx ≥ 1, and x ≥ 0. The matrices A and C are nonnegative and sparse: each row ai of A has at most ΔI positive elements, and each row ck of C has at most ΔK positive elements. We study the approximability of max-min LPs and min-max LPs in a distributed setting; in particular, we focus on local algorithms (constant-time distributed algorithms). We show that for any ΔI ≥ 2, ΔK ≥ 2, and ε > 0 there exists a local algorithm that achieves the approximation ratio ΔI (1 − 1/ΔK) + ε. We also show that this result is the best possible: no local algorithm can achieve the approximation ratio ΔI (1 − 1/ΔK) for any ΔI ≥ 2 and ΔK ≥ 2.

Plant species diversity of buffer zones in agricultural landscapes: in search of determinants from the local to regional scale

Buffer zones are vegetated strip-edges of agricultural fields along watercourses. As linear habitats in agricultural ecosystems, buffer strips dominate and play a leading ecological role in many areas. This thesis focuses on the plant species diversity of the buffer zones in a Finnish agricultural landscape. The main objective of the present study is to identify the determinants of floral species diversity in arable buffer zones from local to regional levels. This study was conducted in a watershed area of a farmland landscape of southern Finland. The study area, Lepsämänjoki, is situated in the Nurmijärvi commune 30 km to the north of Helsinki, Finland. The biotope mosaics were mapped in GIS. A total of 59 buffer zones were surveyed, of which 29 buffer strips surveyed were also sampled by plot. Firstly, two diversity components (species richness and evenness) were investigated to determine whether the relationship between the two is equal and predictable. I found no correlation between species richness and evenness. The relationship between richness and evenness is unpredictable in a small-scale human-shaped ecosystem. Ordination and correlation analyses show that richness and evenness may result from different ecological processes, and thus should be considered separately. Species richness correlated negatively with phosphorus content, and species evenness correlated negatively with the ratio of organic carbon to total nitrogen in soil. The lack of a consistent pattern in the relationship between these two components may be due to site-specific variation in resource utilization by plant species. Within-habitat configuration (width, length, and area) were investigated to determine which is more effective for predicting species richness. More species per unit area increment could be obtained from widening the buffer strip than from lengthening it. The width of the strips is an effective determinant of plant species richness. The increase in species diversity with an increase in the width of buffer strips may be due to cross-sectional habitat gradients within the linear patches. This result can serve as a reference for policy makers, and has application value in agricultural management. In the framework of metacommunity theory, I found that both mass effect(connectivity) and species sorting (resource heterogeneity) were likely to explain species composition and diversity on a local and regional scale. The local and regional processes were interactively dominated by the degree to which dispersal perturbs local communities. In the lowly and intermediately connected regions, species sorting was of primary importance to explain species diversity, while the mass effect surpassed species sorting in the highly connected region. Increasing connectivity in communities containing high habitat heterogeneity can lead to the homogenization of local communities, and consequently, to lower regional diversity, while local species richness was unrelated to the habitat connectivity. Of all species found, Anthriscus sylvestris, Phalaris arundinacea, and Phleum pretense significantly responded to connectivity, and showed high abundance in the highly connected region. We suggest that these species may play a role in switching the force from local resources to regional connectivity shaping the community structure. On the landscape context level, the different responses of local species richness and evenness to landscape context were investigated. Seven landscape structural parameters served to indicate landscape context on five scales. On all scales but the smallest scales, the Shannon-Wiener diversity of land covers (H') correlated positively with the local richness. The factor (H') showed the highest correlation coefficients in species richness on the second largest scale. The edge density of arable field was the only predictor that correlated with species evenness on all scales, which showed the highest predictive power on the second smallest scale. The different predictive power of the factors on different scales showed a scaledependent relationship between the landscape context and local plant species diversity, and indicated that different ecological processes determine species richness and evenness. The local richness of species depends on a regional process on large scales, which may relate to the regional species pool, while species evenness depends on a fine- or coarse-grained farming system, which may relate to the patch quality of the habitats of field edges near the buffer strips. My results suggested some guidelines of species diversity conservation in the agricultural ecosystem. To maintain a high level of species diversity in the strips, a high level of phosphorus in strip soil should be avoided. Widening the strips is the most effective mean to improve species richness. Habitat connectivity is not always favorable to species diversity because increasing connectivity in communities containing high habitat heterogeneity can lead to the homogenization of local communities (beta diversity) and, consequently, to lower regional diversity. Overall, a synthesis of local and regional factors emerged as the model that best explain variations in plant species diversity. The studies also suggest that the effects of determinants on species diversity have a complex relationship with scale.

On linear order and computation : The expressiveness of interactive computations on linear orders and computations indexed by ordinals

We solve the Dynamic Ehrenfeucht-Fra\"iss\'e Game on linear orders for both players, yielding a normal form for quantifier-rank equivalence classes of linear orders in first-order logic, infinitary logic, and generalized-infinitary logics with linearly ordered clocks. We show that Scott Sentences can be manipulated quickly, classified into local information, and consistency can be decided effectively in the length of the Scott Sentence. We describe a finite set of linked automata moving continuously on a linear order. Running them on ordinals, we compute the ordinal truth predicate and compute truth in the constructible universe of set-theory. Among the corollaries are a study of semi-models as efficient database of both model-theoretic and formulaic information, and a new proof of the atomicity of the Boolean algebra of sentences consistent with the theory of linear order -- i.e., that the finitely axiomatized theories of linear order are dense.

Optimisation Problems in Wireless Sensor Networks : Local Algorithms and Local Graphs

This thesis studies optimisation problems related to modern large-scale distributed systems, such as wireless sensor networks and wireless ad-hoc networks. The concrete tasks that we use as motivating examples are the following: (i) maximising the lifetime of a battery-powered wireless sensor network, (ii) maximising the capacity of a wireless communication network, and (iii) minimising the number of sensors in a surveillance application. A sensor node consumes energy both when it is transmitting or forwarding data, and when it is performing measurements. Hence task (i), lifetime maximisation, can be approached from two different perspectives. First, we can seek for optimal data flows that make the most out of the energy resources available in the network; such optimisation problems are examples of so-called max-min linear programs. Second, we can conserve energy by putting redundant sensors into sleep mode; we arrive at the sleep scheduling problem, in which the objective is to find an optimal schedule that determines when each sensor node is asleep and when it is awake. In a wireless network simultaneous radio transmissions may interfere with each other. Task (ii), capacity maximisation, therefore gives rise to another scheduling problem, the activity scheduling problem, in which the objective is to find a minimum-length conflict-free schedule that satisfies the data transmission requirements of all wireless communication links. Task (iii), minimising the number of sensors, is related to the classical graph problem of finding a minimum dominating set. However, if we are not only interested in detecting an intruder but also locating the intruder, it is not sufficient to solve the dominating set problem; formulations such as minimum-size identifying codes and locating dominating codes are more appropriate. This thesis presents approximation algorithms for each of these optimisation problems, i.e., for max-min linear programs, sleep scheduling, activity scheduling, identifying codes, and locating dominating codes. Two complementary approaches are taken. The main focus is on local algorithms, which are constant-time distributed algorithms. The contributions include local approximation algorithms for max-min linear programs, sleep scheduling, and activity scheduling. In the case of max-min linear programs, tight upper and lower bounds are proved for the best possible approximation ratio that can be achieved by any local algorithm. The second approach is the study of centralised polynomial-time algorithms in local graphs these are geometric graphs whose structure exhibits spatial locality. Among other contributions, it is shown that while identifying codes and locating dominating codes are hard to approximate in general graphs, they admit a polynomial-time approximation scheme in local graphs.