14 research outputs found
Online Network Design Algorithms via Hierarchical Decompositions
We develop a new approach for online network design and obtain improved
competitive ratios for several problems. Our approach gives natural
deterministic algorithms and simple analyses. At the heart of our work is a
novel application of embeddings into hierarchically well-separated trees (HSTs)
to the analysis of online network design algorithms --- we charge the cost of
the algorithm to the cost of the optimal solution on any HST embedding of the
terminals. This analysis technique is widely applicable to many problems and
gives a unified framework for online network design.
In a sense, our work brings together two of the main approaches to online
network design. The first uses greedy-like algorithms and analyzes them using
dual-fitting. The second uses tree embeddings and results in randomized -competitive algorithms, where is the total number of vertices in the
graph. Our approach uses deterministic greedy-like algorithms but analyzes them
via HST embeddings of the terminals. Our proofs are simpler as we do not need
to carefully construct dual solutions and we get competitive
ratios, where is the number of terminals.
In this paper, we apply our approach to obtain deterministic -competitive online algorithms for the following problems.
- Steiner network with edge duplication. Previously, only a randomized
-competitive algorithm was known.
- Rent-or-buy. Previously, only deterministic -competitive and
randomized -competitive algorithms by Awerbuch, Azar and Bartal
(2004) were known.
- Connected facility location. Previously, only a randomized -competitive algorithm by San Felice, Williamson and Lee (2014) was known.
- Prize-collecting Steiner forest. We match the competitive ratio first
achieved by Qian and Williamson (2011) and give a simpler analysis.Comment: Accepted to SODA 201
A Bicriteria Approximation for the Reordering Buffer Problem
In the reordering buffer problem (RBP), a server is asked to process a
sequence of requests lying in a metric space. To process a request the server
must move to the corresponding point in the metric. The requests can be
processed slightly out of order; in particular, the server has a buffer of
capacity k which can store up to k requests as it reads in the sequence. The
goal is to reorder the requests in such a manner that the buffer constraint is
satisfied and the total travel cost of the server is minimized. The RBP arises
in many applications that require scheduling with a limited buffer capacity,
such as scheduling a disk arm in storage systems, switching colors in paint
shops of a car manufacturing plant, and rendering 3D images in computer
graphics.
We study the offline version of RBP and develop bicriteria approximations.
When the underlying metric is a tree, we obtain a solution of cost no more than
9OPT using a buffer of capacity 4k + 1 where OPT is the cost of an optimal
solution with buffer capacity k. Constant factor approximations were known
previously only for the uniform metric (Avigdor-Elgrabli et al., 2012). Via
randomized tree embeddings, this implies an O(log n) approximation to cost and
O(1) approximation to buffer size for general metrics. Previously the best
known algorithm for arbitrary metrics by Englert et al. (2007) provided an
O(log^2 k log n) approximation without violating the buffer constraint.Comment: 13 page
Network Design with Coverage Costs
We study network design with a cost structure motivated by redundancy in data
traffic. We are given a graph, g groups of terminals, and a universe of data
packets. Each group of terminals desires a subset of the packets from its
respective source. The cost of routing traffic on any edge in the network is
proportional to the total size of the distinct packets that the edge carries.
Our goal is to find a minimum cost routing. We focus on two settings. In the
first, the collection of packet sets desired by source-sink pairs is laminar.
For this setting, we present a primal-dual based 2-approximation, improving
upon a logarithmic approximation due to Barman and Chawla (2012). In the second
setting, packet sets can have non-trivial intersection. We focus on the case
where each packet is desired by either a single terminal group or by all of the
groups, and the graph is unweighted. For this setting we present an O(log
g)-approximation.
Our approximation for the second setting is based on a novel spanner-type
construction in unweighted graphs that, given a collection of g vertex subsets,
finds a subgraph of cost only a constant factor more than the minimum spanning
tree of the graph, such that every subset in the collection has a Steiner tree
in the subgraph of cost at most O(log g) that of its minimum Steiner tree in
the original graph. We call such a subgraph a group spanner.Comment: Updated version with additional result
Online Matching with Set Delay
We initiate the study of online problems with set delay, where the delay cost
at any given time is an arbitrary function of the set of pending requests. In
particular, we study the online min-cost perfect matching with set delay
(MPMD-Set) problem, which generalises the online min-cost perfect matching with
delay (MPMD) problem introduced by Emek et al. (STOC 2016). In MPMD,
requests arrive over time in a metric space of points. When a request
arrives the algorithm must choose to either match or delay the request. The
goal is to create a perfect matching of all requests while minimising the sum
of distances between matched requests, and the total delay costs incurred by
each of the requests. In contrast to previous work we study MPMD-Set in the
non-clairvoyant setting, where the algorithm does not know the future delay
costs. We first show no algorithm is competitive in or . We then study
the natural special case of size-based delay where the delay is a
non-decreasing function of the number of unmatched requests. Our main result is
the first non-clairvoyant algorithms for online min-cost perfect matching with
size-based delay that are competitive in terms of . In fact, these are the
first non-clairvoyant algorithms for any variant of MPMD. Furthermore, we prove
a lower bound of for any deterministic algorithm and for any randomised algorithm. These lower bounds also hold for clairvoyant
algorithms
On the Extended TSP Problem
We initiate the theoretical study of Ext-TSP, a problem that originates in
the area of profile-guided binary optimization. Given a graph with
positive edge weights , and a non-increasing discount
function such that and for , for some
parameter that is part of the problem definition. The problem is to
sequence the vertices so as to maximize , where is the position of
vertex~ in the sequence.
We show that \prob{Ext-TSP} is APX-hard to approximate in general and we give
a -approximation algorithm for general graphs and a PTAS for some sparse
graph classes such as planar or treewidth-bounded graphs.
Interestingly, the problem remains challenging even on very simple graph
classes; indeed, there is no exact time algorithm for trees unless
the ETH fails. We complement this negative result with an exact time
algorithm for trees.Comment: 17 page
Online Matching with Set and Concave Delays
We initiate the study of online problems with set delay, where the delay cost at any given time is an arbitrary function of the set of pending requests. In particular, we study the online min-cost perfect matching with set delay (MPMD-Set) problem, which generalises the online min-cost perfect matching with delay (MPMD) problem introduced by Emek et al. (STOC 2016). In MPMD, m requests arrive over time in a metric space of n points. When a request arrives the algorithm must choose to either match or delay the request. The goal is to create a perfect matching of all requests while minimising the sum of distances between matched requests, and the total delay costs incurred by each of the requests. In contrast to previous work we study MPMD-Set in the non-clairvoyant setting, where the algorithm does not know the future delay costs. We first show no algorithm is competitive in n or m. We then study the natural special case of size-based delay where the delay is a non-decreasing function of the number of unmatched requests. Our main result is the first non-clairvoyant algorithms for online min-cost perfect matching with size-based delay that are competitive in terms of m. In fact, these are the first non-clairvoyant algorithms for any variant of MPMD. A key technical ingredient is an analog of the symmetric difference of matchings that may be useful for other special classes of set delay. Furthermore, we prove a lower bound of ?(n) for any deterministic algorithm and ?(log n) for any randomised algorithm. These lower bounds also hold for clairvoyant algorithms. Finally, we also give an m-competitive deterministic algorithm for uniform concave delays in the clairvoyant setting
The Online Broadcast Range-Assignment Problem
Let be a set of points in , modeling
devices in a wireless network. A range assignment assigns a range to
each point , thus inducing a directed communication graph in
which there is a directed edge iff , where denotes the distance between and
. The range-assignment problem is to assign the transmission ranges such
that has a certain desirable property, while minimizing the cost of the
assignment; here the cost is given by , for
some constant called the distance-power gradient.
We introduce the online version of the range-assignment problem, where the
points arrive one by one, and the range assignment has to be updated at
each arrival. Following the standard in online algorithms, resources given out
cannot be taken away -- in our case this means that the transmission ranges
will never decrease. The property we want to maintain is that has a
broadcast tree rooted at the first point . Our results include the
following.
- For , a 1-competitive algorithm does not exist. In particular, for
any online algorithm has competitive ratio at least 1.57.
- For and , we analyze two natural strategies: Upon the arrival of
a new point , Nearest-Neighbor increases the range of the nearest point to
cover and Cheapest Increase increases the range of the point for which
the resulting cost increase to be able to reach is minimal.
- We generalize the problem to arbitrary metric spaces, where we present an
-competitive algorithm.Comment: Preliminary version in ISAAC 202
Tight Bounds for Online Weighted Tree Augmentation
The Weighted Tree Augmentation problem (WTAP) is a fundamental problem in network design. In this paper, we consider this problem in the online setting. We are given an n-vertex spanning tree T and an additional set L of edges (called links) with costs. Then, terminal pairs arrive one-by-one and our task is to maintain a low-cost subset of links F such that every terminal pair that has arrived so far is 2-edge-connected in T cup F. This online problem was first studied by Gupta, Krishnaswamy and Ravi (SICOMP 2012) who used it as a subroutine for the online survivable network design problem. They gave a deterministic O(log^2 n)-competitive algorithm and showed an Omega(log n) lower bound on the competitive ratio of randomized algorithms. The case when T is a path is also interesting: it is exactly the online interval set cover problem, which also captures as a special case the parking permit problem studied by Meyerson (FOCS 2005). The contribution of this paper is to give tight results for online weighted tree and path augmentation problems. The main result of this work is a deterministic O(log n)-competitive algorithm for online WTAP, which is tight up to constant factors
Nested Active-Time Scheduling
The active-time scheduling problem considers the problem of scheduling preemptible jobs with windows (release times and deadlines) on a parallel machine that can schedule up to g jobs during each timestep. The goal in the active-time problem is to minimize the number of active steps, i.e., timesteps in which at least one job is scheduled. In this way, the active time models parallel scheduling when there is a fixed cost for turning the machine on at each discrete step.
This paper presents a 9/5-approximation algorithm for a special case of the active-time scheduling problem in which job windows are laminar (nested). This result improves on the previous best 2-approximation for the general case