Capacitated max-Batching with Interval Graph Compatibilities

We consider the problem of partitioning interval graphs into cliques of bounded size. Each interval has a weight, and the cost of a clique is the maximum weight of any interval in the clique. This natural graph problem can be interpreted as a batch scheduling problem. Solving an open question from [7, 4, 5], we show NP-hardness, even if the bound on the clique sizes is constant. Moreover, we give a PTAS based on a novel dynamic programming technique for this case.

A Constant Factor Approximation Algorithm for Unsplittable Flow on Paths

In the unsplittable flow problem on a path, we are given a capacitated path $P$ and $n$ tasks, each task having a demand, a profit, and start and end vertices. The goal is to compute a maximum profit set of tasks, such that for each edge $e$ of $P$, the total demand of selected tasks that use $e$ does not exceed the capacity of $e$. This is a well-studied problem that has been studied under alternative names, such as resource allocation, bandwidth allocation, resource constrained scheduling, temporal knapsack and interval packing. We present a polynomial time constant-factor approximation algorithm for this problem. This improves on the previous best known approximation ratio of $O(\log n)$. The approximation ratio of our algorithm is $7+\epsilon$ for any $\epsilon>0$. We introduce several novel algorithmic techniques, which might be of independent interest: a framework which reduces the problem to instances with a bounded range of capacities, and a new geometrically inspired dynamic program which solves a special case of the maximum weight independent set of rectangles problem to optimality. In the setting of resource augmentation, wherein the capacities can be slightly violated, we give a $(2+\epsilon)$-approximation algorithm. In addition, we show that the problem is strongly NP-hard even if all edge capacities are equal and all demands are either~1,~2, or~3.Comment: 37 pages, 5 figures Version 2 contains the same results as version 1, but the presentation has been greatly revised and improved. References have been adde

Combinatorial Algorithms for the Active Time and Busy Time Problems

In this thesis, we consider the problem of scheduling jobs in such a way that we minimize the energy consumption of the machines they are scheduled on. Job scheduling itself has a long and rich history in computer science both from theoretical and applied perspectives. A multitude of different objectives to optimize have been considered such as weighted completion time, penalties for missed deadlines, etc. However, traditional objective functions such as these do not capture or model the energy consumption of the machines these jobs run on. Energy consumption is an important facet of job scheduling to consider not only because of its relationship with the financial costs of scheduling (such as those related to cooling and the cost of powering the machines) but also due to its impact on the environment. This is especially true in the context of data centers as more and more processing is pushed to the cloud. We study two problems related to these issues - the active time problem and the busy time problem. First, we give a purely combinatorial algorithm for the active time problem which matches its best known approximation ratio (the existing algorithm is based on a rather involved LP rounding scheme). Second, we describe a local search based heuristic for the problem and also consider an experimental evaluation of these algorithms on artificially generated data. Finally, we describe two very simple algorithms which match the current best upper bounds for the busy time problem when all job lengths are equal

Heuristic algorithms for wireless mesh network planning

x, 131 leaves : ill. ; 29 cmTechnologies like IEEE 802.16j wireless mesh networks are drawing increasing attention of the research community. Mesh networks are economically viable and may extend services such as Internet to remote locations. This thesis takes interest into a planning problem in IEEE 802.16j networks, where we need to establish minimum cost relay and base stations to cover the bandwidth demand of wireless clients. A special feature of this planning problem is that any node in this network can send data to at most one node towards the next hop, thus traffic flow is unsplittable from source to destination. We study different integer programming formulations of the problem. We propose four types of heuristic algorithms that uses greedy, local search, variable neighborhood search and Lagrangian relaxation based approaches for the problem. We evaluate the algorithms on database of network instances of 500-5000 nodes, some of which are randomly generated network instances, while other network instances are generated over geometric distribution. Our experiments show that the proposed algorithms produce satisfactory result compared to benchmarks produced by generalized optimization problem solver software

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum