53 research outputs found
Lower bounds for several online variants of bin packing
We consider several previously studied online variants of bin packing and
prove new and improved lower bounds on the asymptotic competitive ratios for
them. For that, we use a method of fully adaptive constructions. In particular,
we improve the lower bound for the asymptotic competitive ratio of online
square packing significantly, raising it from roughly 1.68 to above 1.75.Comment: WAOA 201
Non-Preemptive Scheduling on Machines with Setup Times
Consider the problem in which n jobs that are classified into k types are to
be scheduled on m identical machines without preemption. A machine requires a
proper setup taking s time units before processing jobs of a given type. The
objective is to minimize the makespan of the resulting schedule. We design and
analyze an approximation algorithm that runs in time polynomial in n, m and k
and computes a solution with an approximation factor that can be made
arbitrarily close to 3/2.Comment: A conference version of this paper has been accepted for publication
in the proceedings of the 14th Algorithms and Data Structures Symposium
(WADS
Dynamic Windows Scheduling with Reallocation
We consider the Windows Scheduling problem. The problem is a restricted
version of Unit-Fractions Bin Packing, and it is also called Inventory
Replenishment in the context of Supply Chain. In brief, the problem is to
schedule the use of communication channels to clients. Each client ci is
characterized by an active cycle and a window wi. During the period of time
that any given client ci is active, there must be at least one transmission
from ci scheduled in any wi consecutive time slots, but at most one
transmission can be carried out in each channel per time slot. The goal is to
minimize the number of channels used. We extend previous online models, where
decisions are permanent, assuming that clients may be reallocated at some cost.
We assume that such cost is a constant amount paid per reallocation. That is,
we aim to minimize also the number of reallocations. We present three online
reallocation algorithms for Windows Scheduling. We evaluate experimentally
these protocols showing that, in practice, all three achieve constant amortized
reallocations with close to optimal channel usage. Our simulations also expose
interesting trade-offs between reallocations and channel usage. We introduce a
new objective function for WS with reallocations, that can be also applied to
models where reallocations are not possible. We analyze this metric for one of
the algorithms which, to the best of our knowledge, is the first online WS
protocol with theoretical guarantees that applies to scenarios where clients
may leave and the analysis is against current load rather than peak load. Using
previous results, we also observe bounds on channel usage for one of the
algorithms.Comment: 6 figure
Changing Bases: Multistage Optimization for Matroids and Matchings
This paper is motivated by the fact that many systems need to be maintained
continually while the underlying costs change over time. The challenge is to
continually maintain near-optimal solutions to the underlying optimization
problems, without creating too much churn in the solution itself. We model this
as a multistage combinatorial optimization problem where the input is a
sequence of cost functions (one for each time step); while we can change the
solution from step to step, we incur an additional cost for every such change.
We study the multistage matroid maintenance problem, where we need to maintain
a base of a matroid in each time step under the changing cost functions and
acquisition costs for adding new elements. The online version of this problem
generalizes online paging. E.g., given a graph, we need to maintain a spanning
tree at each step: we pay for the cost of the tree at time
, and also for the number of edges changed at
this step. Our main result is an -approximation, where is
the number of elements/edges and is the rank of the matroid. We also give
an approximation for the offline version of the problem. These
bounds hold when the acquisition costs are non-uniform, in which caseboth these
results are the best possible unless P=NP.
We also study the perfect matching version of the problem, where we must
maintain a perfect matching at each step under changing cost functions and
costs for adding new elements. Surprisingly, the hardness drastically
increases: for any constant , there is no
-approximation to the multistage matching maintenance
problem, even in the offline case
Parameterized Algorithms for Graph Partitioning Problems
We study a broad class of graph partitioning problems, where each problem is
specified by a graph , and parameters and . We seek a subset
of size , such that is at most
(or at least) , where are constants
defining the problem, and are the cardinalities of the edge sets
having both endpoints, and exactly one endpoint, in , respectively. This
class of fixed cardinality graph partitioning problems (FGPP) encompasses Max
-Cut, Min -Vertex Cover, -Densest Subgraph, and -Sparsest
Subgraph.
Our main result is an algorithm for any problem in
this class, where is the maximum degree in the input graph.
This resolves an open question posed by Bonnet et al. [IPEC 2013]. We obtain
faster algorithms for certain subclasses of FGPPs, parameterized by , or by
. In particular, we give an time algorithm for Max
-Cut, thus improving significantly the best known time
algorithm
A -Vertex Kernel for Maximum Internal Spanning Tree
We consider the parameterized version of the maximum internal spanning tree
problem, which, given an -vertex graph and a parameter , asks for a
spanning tree with at least internal vertices. Fomin et al. [J. Comput.
System Sci., 79:1-6] crafted a very ingenious reduction rule, and showed that a
simple application of this rule is sufficient to yield a -vertex kernel.
Here we propose a novel way to use the same reduction rule, resulting in an
improved -vertex kernel. Our algorithm applies first a greedy procedure
consisting of a sequence of local exchange operations, which ends with a
local-optimal spanning tree, and then uses this special tree to find a
reducible structure. As a corollary of our kernel, we obtain a deterministic
algorithm for the problem running in time
Vector Bin Packing with Multiple-Choice
We consider a variant of bin packing called multiple-choice vector bin
packing. In this problem we are given a set of items, where each item can be
selected in one of several -dimensional incarnations. We are also given
bin types, each with its own cost and -dimensional size. Our goal is to pack
the items in a set of bins of minimum overall cost. The problem is motivated by
scheduling in networks with guaranteed quality of service (QoS), but due to its
general formulation it has many other applications as well. We present an
approximation algorithm that is guaranteed to produce a solution whose cost is
about times the optimum. For the running time to be polynomial we
require and . This extends previous results for vector
bin packing, in which each item has a single incarnation and there is only one
bin type. To obtain our result we also present a PTAS for the multiple-choice
version of multidimensional knapsack, where we are given only one bin and the
goal is to pack a maximum weight set of (incarnations of) items in that bin
Spotting Trees with Few Leaves
We show two results related to the Hamiltonicity and -Path algorithms in
undirected graphs by Bj\"orklund [FOCS'10], and Bj\"orklund et al., [arXiv'10].
First, we demonstrate that the technique used can be generalized to finding
some -vertex tree with leaves in an -vertex undirected graph in
time. It can be applied as a subroutine to solve the
-Internal Spanning Tree (-IST) problem in
time using polynomial space, improving upon previous algorithms for this
problem. In particular, for the first time we break the natural barrier of
. Second, we show that the iterated random bipartition employed by
the algorithm can be improved whenever the host graph admits a vertex coloring
with few colors; it can be an ordinary proper vertex coloring, a fractional
vertex coloring, or a vector coloring. In effect, we show improved bounds for
-Path and Hamiltonicity in any graph of maximum degree
or with vector chromatic number at most 8
Mixing Color Coding-Related Techniques
Narrow sieves, representative sets and divide-and-color are three
breakthrough color coding-related techniques, which led to the design of
extremely fast parameterized algorithms. We present a novel family of
strategies for applying mixtures of them. This includes: (a) a mix of
representative sets and narrow sieves; (b) a faster computation of
representative sets under certain separateness conditions, mixed with
divide-and-color and a new technique, "balanced cutting"; (c) two mixtures of
representative sets, iterative compression and a new technique, "unbalanced
cutting". We demonstrate our strategies by obtaining, among other results,
significantly faster algorithms for -Internal Out-Branching and Weighted
3-Set -Packing, and a framework for speeding-up the previous best
deterministic algorithms for -Path, -Tree, -Dimensional -Matching,
Graph Motif and Partial Cover
- …