Distributed Algorithms for Scheduling on Line and Tree Networks

We have a set of processors (or agents) and a set of graph networks defined over some vertex set. Each processor can access a subset of the graph networks. Each processor has a demand specified as a pair of vertices $$, along with a profit; the processor wishes to send data between $u$ and $v$. Towards that goal, the processor needs to select a graph network accessible to it and a path connecting $u$ and $v$ within the selected network. The processor requires exclusive access to the chosen path, in order to route the data. Thus, the processors are competing for routes/channels. A feasible solution selects a subset of demands and schedules each selected demand on a graph network accessible to the processor owning the demand; the solution also specifies the paths to use for this purpose. The requirement is that for any two demands scheduled on the same graph network, their chosen paths must be edge disjoint. The goal is to output a solution having the maximum aggregate profit. Prior work has addressed the above problem in a distibuted setting for the special case where all the graph networks are simply paths (i.e, line-networks). Distributed constant factor approximation algorithms are known for this case. The main contributions of this paper are twofold. First we design a distributed constant factor approximation algorithm for the more general case of tree-networks. The core component of our algorithm is a tree-decomposition technique, which may be of independent interest. Secondly, for the case of line-networks, we improve the known approximation guarantees by a factor of 5. Our algorithms can also handle the capacitated scenario, wherein the demands and edges have bandwidth requirements and capacities, respectively.Comment: Accepted to PODC 2012, full versio

Faster (1+ε)-approximation for unsplittable flow on a path via resource augmentation and back

Unsplittable flow on a path (UFP) is an important and well-studied problem. We are given a path with capacities on its edges, and a set of tasks where for each task we are given a demand, a subpath, and a weight. The goal is to select the set of tasks of maximum total weight whose total demands do not exceed the capacity on any edge. UFP admits an (1+ε)-approximation with a running time of n^{O_{ε}(poly(log n))}, i.e., a QPTAS {[}Bansal et al., STOC 2006; Batra et al., SODA 2015{]} and it is considered an important open problem to construct a PTAS. To this end, in a series of papers polynomial time approximation algorithms have been developed, which culminated in a (5/3+ε)-approximation {[}Grandoni et al., STOC 2018{]} and very recently an approximation ratio of (1+1/(e+1)+ε) < 1.269 {[}Grandoni et al., 2020{]}. In this paper, we address the search for a PTAS from a different angle: we present a faster (1+ε)-approximation with a running time of only n^{O_{ε}(log log n)}. We first give such a result in the relaxed setting of resource augmentation and then transform it to an algorithm without resource augmentation. For this, we present a framework which transforms algorithms for (a slight generalization of) UFP under resource augmentation in a black-box manner into algorithms for UFP without resource augmentation, with only negligible loss

Data transfer scheduling with advance reservation and provisioning

Over the years, scientific applications have become more complex and more data intensive. Although through the use of distributed resources the institutions and organizations gain access to the resources needed for their large-scale applications, complex middleware is required to orchestrate the use of these storage and network resources between collaborating parties, and to manage the end-to-end processing of data. We present a new data scheduling paradigm with advance reservation and provisioning. Our methodology provides a basis for provisioning end-to-end high performance data transfers which require integration between system, storage and network resources, and coordination between reservation managers and data transfer nodes. This allows researchers/users and higher level meta-schedulers to use data placement as a service where they can plan ahead and reserve time and resources for their data movement operations. We present a novel approach for evaluating time-dependent structures with bandwidth guaranteed paths. We present a practical online scheduling model using advance reservation in dynamic network with time constraints. In addition, we report a new polynomial algorithm presenting possible reservation options and alternatives for earliest completion and shortest transfer duration. We enhance the advance network reservation system by extending the underlying mechanism to provide a new service in which users submit their constraints and the system suggests possible reservation requests satisfying users\u27 requirements. We have studied scheduling data transfer operation with resource and time conflicts. We have developed a new scheduling methodology considering resource allocation in client sites and bandwidth allocation on network link connecting resources. Some other major contributions of our study include enhanced reliability, adaptability, and performance optimization of distributed data placement tasks. While designing this new data scheduling architecture, we also developed other important methodologies such as early error detection, failure awareness, job aggregation, and dynamic adaptation of distributed data placement tasks. The adaptive tuning includes dynamically setting data transfer parameters and controlling utilization of available network capacity. Our research aims to provide a middleware to improve the data bottleneck in high performance computing systems

Improved Pseudo-Polynomial-Time Approximation for Strip Packing

We study the strip packing problem, a classical packing problem which generalizes both bin packing and makespan minimization. Here we are given a set of axis-parallel rectangles in the two-dimensional plane and the goal is to pack them in a vertical strip of fixed width such that the height of the obtained packing is minimized. The packing must be non-overlapping and the rectangles cannot be rotated. A reduction from the partition problem shows that no approximation better than 3/2 is possible for strip packing in polynomial time (assuming P!=NP). Nadiradze and Wiese [SODA16] overcame this barrier by presenting a (7/5+epsilon)-approximation algorithm in pseudo-polynomial-time (PPT). As the problem is strongly NP-hard, it does not admit an exact PPT algorithm (though a PPT approximation scheme might exist). In this paper we make further progress on the PPT approximability of strip packing, by presenting a (4/3+epsilon)-approximation algorithm. Our result is based on a non-trivial repacking of some rectangles in the "empty space" left by the construction by Nadiradze and Wiese, and in some sense pushes their approach to its limit. Our PPT algorithm can be adapted to the case where we are allowed to rotate the rectangles by 90 degrees, achieving the same approximation factor and breaking the polynomial-time approximation barrier of 3/2 for the case with rotations as well

Approximation Algorithms for Demand Strip Packing

In the Demand Strip Packing problem (DSP), we are given a time interval and a collection of tasks, each characterized by a processing time and a demand for a given resource (such as electricity, computational power, etc.). A feasible solution consists of a schedule of the tasks within the mentioned time interval. Our goal is to minimize the peak resource consumption, i.e. the maximum total demand of tasks executed at any point in time. It is known that DSP is NP-hard to approximate below a factor 3/2, and standard techniques for related problems imply a (polynomial-time) 2-approximation. Our main result is a (5/3+?)-approximation algorithm for any constant ? > 0. We also achieve best-possible approximation factors for some relevant special cases

Combinatorial Optimization (hybrid meeting)

Combinatorial Optimization deals with optimization problems defined on combinatorial structures such as graphs and networks. Motivated by diverse practical problem setups, the topic has developed into a rich mathematical discipline with many connections to other fields of Mathematics (such as, e.g., Combinatorics, Convex Optimization and Geometry, and Real Algebraic Geometry). It also has strong ties to Theoretical Computer Science and Operations Research. A series of Oberwolfach Workshops have been crucial for establishing and developing the field. The workshop we report about was a particularly exciting event - due to the depth of results that were presented, the spectrum of developments that became apparent from the talks, the breadth of the connections to other mathematical fields that were explored, and last but not least because for many of the particiants it was the first opportunity to exchange ideas and to collaborate during an on-site workshop since almost two years

Fairness in Communication and Computer Network Design

In communication networks, fair sharing of resources is an important issue for one main reason. The growth of network capacity is in general not matching the rapid growth of traffic. Consequently, the resources consumed by each user have to be limited. This implies that users cannot always be assigned the end-to-end bandwidth they ask for. Instead, the limited network resources should be distributed to users in a way that assures fair end-to-end bandwidth assignment among them. Obtaining fairness between network users and at the same time assuring efficient network utilization, is a source of non-trivial network optimization problems. Complicating factors are that each user has limited access to the (limited) network resources and that different users require and consume different amounts and types of resources. In this thesis different types of optimization problems associated with fair resource sharing in communication networks are studied. Initially, the notions of max-min fairness, proportional fairness, alpha-fairness etc., are put in a formal framework of fair rational preference relations. A clear, unified definition of fairness is presented. The theory is first applied to different types of allocation problems. Focus is put on convex and non-convex max-min fair traffic allocation problems, and a difference in difficulty between the two groups of problems is demonstrated. The studies are continued by an investigation of proportionally fair dimensioning. Two different cases are studied -- a simpler, when no resilience to failures is required, and a more complicated, assuming the possibility of link failures. In the context of fair sharing of the resources of a communication network, this thesis presents several original theoretical findings as well as solution algorithms for the studied problems. The results are accompanied by numerical results, illustrating algorithm efficiency for virtually all of the studied problems

용량 제약이 없는 부보상 문제의 혼합 이진 이차 문제로의 모형화를 통한 해법

학위논문(석사) -- 서울대학교대학원 : 공과대학 산업공학과, 2022. 8. 홍성필.부보상 문제는 비순환 유향 그래프 상에서 출발, 도착 마디를 잇는 경로와 그 경로 상의 흐름을 결정하는 문제이다. 부보상은 도시 1에서 n까지 이동하면서 각 도시를 경유하 거나 지나치며, 경유하는 도시에서만 상품을 매매할 수 있지만 이동 거리와 상품량에 따른 비용 또한 지불해야 한다. 이 때, 부보상은 자신의 수익, 즉 총 상품의 판매량에서 얻는 수익과 지불 비용의 차를 최대화하고자 한다. 본 연구에서는 용량 제약이 없는 경우만을 다루며 기존 부보상 문제를 혼합이진이차문제으로 재모형화하여 분지절단법 으로 문제를 푼다. 이 때 목적 함수를 볼록화하고 연속 완화시켜 얻을 수 있는 상한을 비교하기 위해 여러 볼록화 방법들을 비교하고 비교실험한 결과 또한 제시한다.Bubosang Problem is a problem set on a directed acyclic graph path concerning both the path and multi-commodity flow decisions. A merchant travels from city 1 through n, either transiting through a city and trading products or passing by the city to the next city on his route. He wants to choose the path and trading product quantity to maximize his net profit which is defined by the difference between the total sales revenue and the traveling cost. The scope of the study considers only the uncapacitated case. In this study, we reformulate BP into a mixed binary quadratic problem to employ the branch-and-cut algorithm to solve the problem. Specifically, we compare the upper bound obtained through the continuous relaxation and convexification of the objective by studying different convexification methods. Computational results of the comparison are also provided.Chapter 1 Introduction 1 1.1 Background 1 1.2 Literature Review 3 1.3 Research Motivations 5 1.4 Organization of the Thesis 6 Chapter 2 Problem Definition and Mathematical Formulations 7 2.1 Problem Definition 7 2.2 Flow Arc Formulation 8 2.3 MBQP Formulation 11 2.3.1 MBQP 13 2.4 Branch-and-Cut Algorithm 14 2.4.1 Overall Setting 14 2.4.2 Cutset Inequality 14 2.4.3 Lower Bound 15 2.4.4 Upper Bound 18 Chapter 3 Convexification Methods 19 3.1 One Coefficient Case : Eigenvalue Method 21 3.2 Criteria for Convexification Evaluation 22 3.2.1 Criterion for Unweighted Methods 22 3.3 Two Coefficient Case : (α, β) - SDP method 23 3.4 Two Coefficient Case : (α, β) - Sum of Squares Method 24 3.5 Four Coefficient Case : (α, β, γ, δ) - method 26 3.5.1 (α, β, γ, δ) - SDP method 26 3.5.2 (α, β, γ, δ) - Sum of Squares method 28 3.6 Five Coefficient Case : (α, β, γ, δ, τ ) - Sum of Squares method 29 3.7 Weighted methods 30 3.7.1 Criterion for Weighted Methods 30 Chapter 4 Computational Experiments 32 Chapter 5 Conclusion 36 Bibliography 37 국문초록 41석

An Online Scheduling Algorithm with Advance Reservation for Large-Scale Data Transfers

Scientific applications and experimental facilities generate massive data sets that need to be transferred to remote collaborating sites for sharing, processing, and long term storage. In order to support increasingly data-intensive science, next generation research networks have been deployed to provide high-speed on-demand data access between collaborating institutions. In this paper, we present a practical model for online data scheduling in which data movement operations are scheduled in advance for end-to-end high performance transfers. In our model, data scheduler interacts with reservation managers and data transfer nodes in order to reserve available bandwidth to guarantee completion of jobs that are accepted and confirmed to satisfy preferred time constraint given by the user. Our methodology improves current systems by allowing researchers and higher level meta-schedulers to use data placement as a service where theycan plan ahead and reserve the scheduler time in advance for their data movement operations. We have implemented our algorithm and examined possible techniques for incorporation into current reservation frameworks. Performance measurements confirm that the proposed algorithm is efficient and scalable