2 research outputs found
Generalized Load Balancing and Clustering Problems with Norm Minimization
In many fundamental combinatorial optimization problems, a feasible solution
induces some real cost vectors as an intermediate result, and the optimization
objective is a certain function of the vectors. For example, in the problem of
makespan minimization on unrelated parallel machines, a feasible job assignment
induces a vector containing the sizes of assigned jobs for each machine, and
the goal is to minimize the norm of norms of the vectors.
Another example is fault-tolerant -center, where each client is connected to
multiple open facilities, thus having a vector of distances to these
facilities, and the goal is to minimize the norm of norms
of these vectors. In this paper, we study the maximum of norm problem. Given an
arbitrary symmetric monotone norm , the objective is defined as the maximum
( norm) of -norm values of the induced cost vectors. This
versatile formulation captures a wide variety of problems, including makespan
minimization, fault-tolerant -center and many others. We give concrete
results for load balancing on unrelated parallel machines and clustering
problems, including constant-factor approximation algorithms when belongs
with a certain rich family of norms, and -approximations when is
general and satisfies some mild assumptions. We also consider the
aforementioned problems in a generalized fairness setting. As a concrete
example, the insight is to prevent a scheduling algorithm from assigning too
many jobs consistently on any machine in a job-recurring scenario, and causing
the machine's controller to fail. Our algorithm needs to stochastically output
a feasible solution minimizing the objective function, and satisfy the given
marginal fairness constraints
Ordered -Median with Outliers and Fault-Tolerance
In this paper, we study two natural generalizations of ordered -median,
named robust ordered -median and fault-tolerant ordered -median. In
ordered -median, given a finite metric space , we seek to open
facilities which induce a service cost vector
, and minimize the ordered objective
. Here is the minimum
distance between and facilities in , is a given
non-increasing non-negative vector, and is the
non-increasingly sorted version of . The current best result is a
-approximation [CS19].
We first consider robust ordered -median, a.k.a. ordered -median with
outliers, where the input consists of an ordered -median instance and
parameter . The goal is to open facilities , select
clients and assign the nearest open facility to each . The service cost vector is and is in
. We introduce a novel yet simple objective function that enables
linear analysis of the non-linear ordered objective, apply an iterative
rounding framework [KLS18] and obtain a constant-factor approximation. We
devise the first constant-approximations for ordered matroid median and ordered
knapsack median using the same method.
We also consider fault-tolerant ordered -median, where besides the same
input as ordered -median, we are also given additional client requirements
and need to assign distinct open
facilities to each client . The service cost of is the sum of
distances to its assigned facilities, and the objective is the same. We obtain
a constant-factor approximation using a novel LP relaxation with constraints
created via a new sparsification technique