Efficient algorithm for the k-means problem with Must-Link and Cannot-Link constraints

Constrained clustering, such as k -means with instance-level Must-Link (ML) and Cannot-Link (CL) auxiliary information as the constraints, has been extensively studied recently, due to its broad applications in data science and AI. Despite some heuristic approaches, there has not been any algorithm providing a non-trivial approximation ratio to the constrained k -means problem. To address this issue, we propose an algorithm with a provable approximation ratio of O(logk) when only ML constraints are considered. We also empirically evaluate the performance of our algorithm on real-world datasets having artificial ML and disjoint CL constraints. The experimental results show that our algorithm outperforms the existing greedy-based heuristic methods in clustering accuracy

Forward modeling of P- and S-waves response of fractures intersected with horizontal wells in tight reservoirs

Horizontal wells play an important role in expanding the drilling volume of reservoirs and oil production area, and are widely used in unconventional reservoirs. Fractures have a positive effect on reservoir permeability, but fractures can also cause accidents such as casing deformation and inter-well frac-hit. It is of great significance to identify and evaluate fractures intersected with horizontal wells in tight reservoirs. In this paper, a three-dimensional numerical model of horizontal wells and fractures in tight reservoirs is designed. The responses of monopole P-wave and dipole S-wave to fractures with different width, dip angle and filling medium are systematically studied, by using three-dimensional finite difference algorithm. The results show that when the fracture is filled with calcite, the amplitude attenuation of monopole P-wave and dipole S-wave has a monotonic exponential increase with the increase of fracture width and the decrease of fracture dip angle. In the real data processing, the amplitude attenuation of P- and S-waves can be used to jointly evaluate the fracture filled with calcite. When the fracture is filled with water, both P- and S-waves have prominent amplitude attenuation. P wave amplitude attenuation does not have a monotonic variation with the increase of fracture width but it has a monotonic increase with the decrease of fracture dip angle. S wave amplitude attenuation has a monotonic increase with the increase of fracture width and the decrease of fracture dip angle. The amplitude attenuation of P- and S- waves rises significantly when the fracture is filled with natural gas. This study is crucial for better understanding the response of P- and S-waves to fractures intersected with borehole in tight reservoirs, and it provides useful information for the inversion of fracture parameters by using P- and S-waves

Approximation algorithms for resource allocation optimization.

Nowadays, data storage, server replicas/mirrors, virtual machines, and various kinds of services can all be regarded as different types of resources. These resources play an important role in today’s computer world because of the continuing advances in information technology. It is usual that similar resources are grouped together at the same site, and can then be allocated to geographically distributed clients. This is the resource allocation paradigm considered in this thesis. Optimizing solutions to a variety of problems arising from this paradigm remains a key challenge, since these problems are NP-hard. For all the resource allocation problems studied in this thesis, we are given a set of sites containing facilities as resources, a set of clients to access these facilities, an opening cost for each facility, and a connection cost for each allocation of a facility to a client. The general goal is to decide the number of facilities to open at each site and allocate the open facilities to clients so that the total cost incurred is minimized. This class of the problems extends the classical NP-hard facility location problems with additional abilities to capture various practical resource allocation scenarios. To cope with the NP-hardness of the resource allocation problems, the thesis focuses on the design and analysis of approximation algorithms. The main techniques we adopt are linear programming based, such as primal-dual schema, linear program rounding, and reductions via linear programs. Our developed solutions have great potential for optimizing the performances of many contemporary distributed systems such as cloud computing, content delivery networks, Web caching, and Web services provisioning.Thesis (Ph.D.) -- University of Adelaide, School of Computer Science, 201

LP-based approximation algorithms for reliable resource allocation

We initiate the study of the reliable resource allocation (RRA) problem. In this problem, we are given a set of sites ℱ each with an unconstrained number of facilities as resources. Every facility at site i ∈ ℱ has an opening cost and a service reliability pi. There is also a set of clients \u1d49e to be allocated to facilities. Every client j ∈ \u1d49e accesses a facility at i with a connection cost and reliability lij. In addition, every client j has a minimum reliability requirement (MRR) rj for accessing facilities. The objective of the problem is to decide the number of facilities to open at each site and connect these facilities to clients such that all clients’ MRRs are satisfied at a minimum total cost. The unconstrained fault-tolerant resource allocation problem studied in Liao and Shen [(2011) Unconstrained and Constrained Fault-Tolerant Resource Allocation. Proceedings of the 17th Annual International Conference on Computing and Combinatorics (COCOON), Dallas, Texas, USA, August 14–16, pp. 555–566. Springer, Berlin] is a special case of RRA. Both of these resource allocation problems are derived from the classical facility location theory. In this paper, for solving the general RRA problem, we develop two equivalent primal-dual algorithms where the second one is an acceleration of the first and runs in quasi-quadratic time. In the algorithm's ratio analysis, we first obtain a constant approximation factor of 2+2√2 and then a reduced ratio of 3.722 using a factor revealing program, when lij's are uniform on i (partially uniform) and rj's are uniform above the threshold reliability that a single access to a facility is able to provide. The analysis further elaborates and generalizes the inverse dual-fitting technique introduced in Xu and Shen [(2009) The Fault-Tolerant Facility Allocation Problem. Proceedings of the 20th International Symposium on Algorithms and Computation (ISAAC), Honolulu, HI, USA, December 16–18, pp. 689–698. Springer, Berlin]. Moreover, we formalize this technique for analyzing the minimum set cover problem. For a special case of RRA, where all rj's and lij's are uniform, we derive its approximation ratio through a novel reduction to the uncapacitated facility location problem. The reduction demonstrates some useful and generic linear programming techniques

On the shallow-light Steiner tree problem

Let G = (V, E) be a given graph with nonnegative integral edge cost and delay, S ⊆ V be a terminal set and r ∈ S be the selected root. The shallow-light Steiner tree (SLST) problem is to compute a minimum cost tree spanning the terminals of S, such that the delay between r and every other terminal is bounded by a given delay constraint D ∈ ℤ 0 + . It is known that the SLST problem is NP-hard and unless NP ⊆ DTIME(n log log n ) there exists no approximation algorithm with ratio (1, γ log2 n) for some fixed γ > 0 [12]. Nevertheless, under the same assumption it admits no approximation ratio better than (1, γ log 2 n) for some fixed γ > 0 even when D = 2 [2]. This paper first gives an exact algorithm with time complexity O(3 t nD + 2 t n 2 D 2 + n 3 D 3 ), where n and t are the numbers of vertices and terminals of the given graph respectively. This is a pseudo polynomial time parameterized algorithm with respect to the parameterization “number of terminals”. Later, this algorithm is improved to a parameterized approximation algorithm with a time complexity O(3 t n 2 /∈ + 2 t n 4 /∈ 2 + n 6 /∈ 3 ) and a bifactor approximation ratio (1 + ∈, 1). That is, for any small real number ∈ > 0, the algorithm computes a Steiner tree with delay and cost bounded by (1 + ∈)D and the optimum cost respectively

Improved approximation algorithms for constrained fault-tolerant resource allocation

In Constrained Fault-Tolerant Resource Allocation (FTRA) problem, we are given a set of sites containing facilities as resources and a set of clients accessing these resources. Each site i can open at most facilities with opening cost . Each client j requires an allocation of open facilities and connecting j to any facility at site i incurs a connection cost . The goal is to minimize the total cost of this resource allocation scenario. FTRA generalizes the Unconstrained Fault-Tolerant Resource Allocation () [1] and the classical Fault-Tolerant Facility Location (FTFL) [2] problems: for every site i, does not have the constraint , whereas FTFL sets . These problems are said to be uniform if all 's are the same, and general otherwise. For the general metric FTRA, we first give an LP-rounding algorithm achieving an approximation ratio of 4. Then we show the problem reduces to FTFL, implying the ratio of 1.7245 from [3]. For the uniform FTRA, we provide a 1.52-approximation primal–dual algorithm in time, where n is the total number of sites and clients

Fast approximation algorithms for multiple coverage with unit disks

Effective monitoring of applications in wireless sensor networks can be underpinned by the multiple coverage problem with unit disks. In the problem, we are given a set of targets T = {t 1 , t 2 , …, t n } distributed in the plane, where t i needs to be covered f(t i ) times for any positive integer f(t i ). The aim is to place a minimum number of disks, such that all the targets can be covered as desired. In the paper, we first present a 5-approximation algorithm with runtime O(n + m) for m = max i {f(t i )}. Then, we give a theoretically improved 4-approximation algorithm, albeit with an increased time complexity to O(n 2 ). In addition, we consider the online setting where targets arrive in sequence and upon each arrival the corresponding coverage disk must be placed. For this setting, we devise an online algorithm with a competitive ratio of 6 and constant update time. To verify aforementioned theoretical findings, numerical experiments are conducted to demonstrate and compare the practical performance of the proposed algorithms