17 research outputs found
Improved approximation bounds for Vector Bin Packing
In this paper we propose an improved approximation scheme for the Vector Bin
Packing problem (VBP), based on the combination of (near-)optimal solution of
the Linear Programming (LP) relaxation and a greedy (modified first-fit)
heuristic. The Vector Bin Packing problem of higher dimension (d \geq 2) is not
known to have asymptotic polynomial-time approximation schemes (unless P = NP).
Our algorithm improves over the previously-known guarantee of (ln d + 1 +
epsilon) by Bansal et al. [1] for higher dimensions (d > 2). We provide a
{\theta}(1) approximation scheme for certain set of inputs for any dimension d.
More precisely, we provide a 2-OPT algorithm, a result which is irrespective of
the number of dimensions d.Comment: 15 pages, 3 algorithm
AFPTAS results for common variants of bin packing: A new method to handle the small items
We consider two well-known natural variants of bin packing, and show that
these packing problems admit asymptotic fully polynomial time approximation
schemes (AFPTAS). In bin packing problems, a set of one-dimensional items of
size at most 1 is to be assigned (packed) to subsets of sum at most 1 (bins).
It has been known for a while that the most basic problem admits an AFPTAS. In
this paper, we develop methods that allow to extend this result to other
variants of bin packing. Specifically, the problems which we study in this
paper, for which we design asymptotic fully polynomial time approximation
schemes, are the following. The first problem is "Bin packing with cardinality
constraints", where a parameter k is given, such that a bin may contain up to k
items. The goal is to minimize the number of bins used. The second problem is
"Bin packing with rejection", where every item has a rejection penalty
associated with it. An item needs to be either packed to a bin or rejected, and
the goal is to minimize the number of used bins plus the total rejection
penalty of unpacked items. This resolves the complexity of two important
variants of the bin packing problem. Our approximation schemes use a novel
method for packing the small items. This new method is the core of the improved
running times of our schemes over the running times of the previous results,
which are only asymptotic polynomial time approximation schemes (APTAS)
Selfish Bin Covering
In this paper, we address the selfish bin covering problem, which is greatly
related both to the bin covering problem, and to the weighted majority game.
What we mainly concern is how much the lack of coordination harms the social
welfare. Besides the standard PoA and PoS, which are based on Nash equilibrium,
we also take into account the strong Nash equilibrium, and several other new
equilibria. For each equilibrium, the corresponding PoA and PoS are given, and
the problems of computing an arbitrary equilibrium, as well as approximating
the best one, are also considered.Comment: 16 page
Packing Sporadic Real-Time Tasks on Identical Multiprocessor Systems
In real-time systems, in addition to the functional correctness recurrent
tasks must fulfill timing constraints to ensure the correct behavior of the
system. Partitioned scheduling is widely used in real-time systems, i.e., the
tasks are statically assigned onto processors while ensuring that all timing
constraints are met. The decision version of the problem, which is to check
whether the deadline constraints of tasks can be satisfied on a given number of
identical processors, has been known -complete in the strong sense.
Several studies on this problem are based on approximations involving resource
augmentation, i.e., speeding up individual processors. This paper studies
another type of resource augmentation by allocating additional processors, a
topic that has not been explored until recently. We provide polynomial-time
algorithms and analysis, in which the approximation factors are dependent upon
the input instances. Specifically, the factors are related to the maximum ratio
of the period to the relative deadline of a task in the given task set. We also
show that these algorithms unfortunately cannot achieve a constant
approximation factor for general cases. Furthermore, we prove that the problem
does not admit any asymptotic polynomial-time approximation scheme (APTAS)
unless when the task set has constrained deadlines, i.e.,
the relative deadline of a task is no more than the period of the task.Comment: Accepted and to appear in ISAAC 2018, Yi-Lan, Taiwa
Packing sporadic real-time tasks on identical multiprocessor systems
In real-time systems, in addition to the functional correctness recurrent tasks must fulfill timing constraints to ensure the correct behavior of the system. Partitioned scheduling is widely used in real-time systems, i.e., the tasks are statically assigned onto processors while ensuring that all timing constraints are met. The decision version of the problem, which is to check whether the deadline constraints of tasks can be satisfied on a given number of identical processors, has been known NP-complet
A Robust AFPTAS for Online Bin Packing with Polynomial Migration
In this paper we develop general LP and ILP techniques to find an approximate
solution with improved objective value close to an existing solution. The task
of improving an approximate solution is closely related to a classical theorem
of Cook et al. in the sensitivity analysis for LPs and ILPs. This result is
often applied in designing robust algorithms for online problems. We apply our
new techniques to the online bin packing problem, where it is allowed to
reassign a certain number of items, measured by the migration factor. The
migration factor is defined by the total size of reassigned items divided by
the size of the arriving item. We obtain a robust asymptotic fully polynomial
time approximation scheme (AFPTAS) for the online bin packing problem with
migration factor bounded by a polynomial in . This answers
an open question stated by Epstein and Levin in the affirmative. As a byproduct
we prove an approximate variant of the sensitivity theorem by Cook at el. for
linear programs
A Tight (1.5+?)-Approximation for Unsplittable Capacitated Vehicle Routing on Trees
In the unsplittable capacitated vehicle routing problem (UCVRP) on trees, we are given a rooted tree with edge weights and a subset of vertices of the tree called terminals. Each terminal is associated with a positive demand between 0 and 1. The goal is to find a minimum length collection of tours starting and ending at the root of the tree such that the demand of each terminal is covered by a single tour (i.e., the demand cannot be split), and the total demand of the terminals in each tour does not exceed the capacity of 1.
For the special case when all terminals have equal demands, a long line of research culminated in a quasi-polynomial time approximation scheme [Jayaprakash and Salavatipour, TALG 2023] and a polynomial time approximation scheme [Mathieu and Zhou, TALG 2023].
In this work, we study the general case when the terminals have arbitrary demands. Our main contribution is a polynomial time (1.5+?)-approximation algorithm for the UCVRP on trees. This is the first improvement upon the 2-approximation algorithm more than 30 years ago. Our approximation ratio is essentially best possible, since it is NP-hard to approximate the UCVRP on trees to better than a 1.5 factor
A Tight -Approximation for Unsplittable Capacitated Vehicle Routing on Trees
In the unsplittable capacitated vehicle routing problem (UCVRP) on trees, we
are given a rooted tree with edge weights and a subset of vertices of the tree
called terminals. Each terminal is associated with a positive demand between 0
and 1. The goal is to find a minimum length collection of tours starting and
ending at the root of the tree such that the demand of each terminal is covered
by a single tour (i.e., the demand cannot be split), and the total demand of
the terminals in each tour does not exceed the capacity of 1.
For the special case when all terminals have equal demands, a long line of
research culminated in a quasi-polynomial time approximation scheme
[Jayaprakash and Salavatipour, SODA 2022] and a polynomial time approximation
scheme [Mathieu and Zhou, ICALP 2022].
In this work, we study the general case when the terminals have arbitrary
demands. Our main contribution is a polynomial time
-approximation algorithm for the UCVRP on trees. This is the
first improvement upon the 2-approximation algorithm more than 30 years ago
[Labb\'e, Laporte, and Mercure, Operations Research, 1991]. Our approximation
ratio is essentially best possible, since it is NP-hard to approximate the
UCVRP on trees to better than a 1.5 factor