Multiply Balanced k −Partitioning

Abstract

Abstract. The problem of partitioning an edge-capacitated graph on n vertices into k balanced parts has been amply researched. Motivated by applications such as load balancing in distributed systems and market segmentation in social networks, we propose a new variant of the problem, called Multiply Balanced k Partitioning, where the vertex-partition must be balanced under d vertex-weight functions simultaneously. We design bicriteria approximation algorithms for this problem, i.e., they partition the vertices into up to k parts that are nearly balanced simultaneously for all weight functions, and their approximation factor for the capacity of cut edges matches the bounds known for a single weight function times d. For the case where d = 2, for vertex weights that are integers bounded by a polynomial in n and any fixed > 0, we obtain a (2 + , O( √ log n log k))-bicriteria approximation, namely, we partition the graph into parts whose weight is at most 2+ times that of a perfectly balanced part (simultaneously for both weight functions), and whose cut capacity is O( √ log n log k) · OPT. For unbounded (exponential) vertex weights, we achieve approximation (3, O(log n)). Our algorithm generalizes to d weight functions as follows: For vertex weights that are integers bounded by a polynomial in n and any fixed > 0, we obtain a (2d + , O(d √ log n log k))-bicriteria approximation. For unbounded (exponential) vertex weights, we achieve approximation (2d + 1, O(d log n)).

Задача минимального неравномерного разбиения графа на части c несвязанными весами

“…Объем передаваемых данных между задачами u и v равен суммарному весу ребер между соответствующими вершинами u и v. Целью является распределение задач между k машинами (с учетом заданных ограничений на производительность) для минимизации суммарной передаваемой информации между машинами. 1 Данная задача весьма нетривиальна. В [5] отмечается, что многие из существующих подходов не применимы к этой задаче.…”

“…Требуется разбить множество V таким образом, чтобы выполнялись ограничения на размер частей для всех ресурсов. В [1] был изучен d-мерный вариант задачи (равномерного) минимального разбиения на k частей, где предполагалось, что величины ρ i = 1/k одни и те же, а r j не зависит от i. Определение 1.3 (задача минимального разбиения графа на неравные части с неравными d-мерными весами). Задаются n-вершинный граф G = (V, E) и неотрицательные числа c j (i), r j (u, i), i ∈ {1, .…”

Multi-dimensional balanced graph partitioning via projected gradient descent

“…While a large body of work exists offering practical solutions for the one-dimensional version of the problem [7,12,13,14,22,23,33,41,42], as well as on theoretical foundations of graph partitioning [4,26,32], literature on principled and scalable approaches for the multi-dimensional case is quite sparse [24,35,36,37]. In particular, if the weight functions are unrelated to each other, one can easily construct examples when no feasible solution exists that satisfies all balance constraints even for two weight functions.…”

Minimum nonuniform graph partitioning with unrelated weights

“…note that many existing techniques do not work for this problem. Particularly, it is not clear how to solve this problem on tree graphs 2 and consequently k-Partitioning was previously studied by Amir et al (2014). In their problem, all ρ i = 1/k are the same, and r j 's do not depend on i.…”