18 research outputs found
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
A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms
Parameterization and approximation are two popular ways of coping with
NP-hard problems. More recently, the two have also been combined to derive many
interesting results. We survey developments in the area both from the
algorithmic and hardness perspectives, with emphasis on new techniques and
potential future research directions
Online Metric Matching with Delay
Traditionally, an online algorithm must service a request upon its arrival. In many practical situations,
one can delay the service of a request in the hope of servicing it more efficiently in the near future. As
a result, the study of online algorithms with delay has recently gained considerable traction. For most
online problems with delay, competitive algorithms have been developed that are independent of
properties of the delay functions associated with each request. Interestingly, this is not the case for
the online min-cost perfect matching with delays (MPMD) problem, introduced by Emek et al.(STOC
2016).
In this thesis we show that some techniques can be modified to extend to larger classes of delay
functions, without affecting the competitive ratio. In the interest of designing competitive solutions for
the problem in a more general setting, we introduce the study of online problems with set delay.
Here, the delay cost at any time is given by an arbitrary function of the set of pending requests, rather than the sum over individual delay functions associated with each request. In particular, we study the
online min-cost perfect matching with set delay (MPMD-Set) problem, which provides a
generalisation of MPMD. In contrast to previous work, the new model allows us to study the problem
in the non-clairvoyant setting, i.e. where the future delay costs are unknown to the algorithm.
We prove that for MPMD-Set in the most general non-clairvoyant setting, there exists no competitive
algorithm. Motivated by this impossibility, we introduce a new class of delay functions called sizebased
and prove that for this version of the problem, there exist both non-clairvoyant deterministic
and randomised algorithms that are competitive in the number of requests. Our results reveal that the
quality of an online matching depends both on the algorithm's access to information about future
delay costs, and the properties of the delay function
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum
LIPIcs, Volume 277, GIScience 2023, Complete Volume
LIPIcs, Volume 277, GIScience 2023, Complete Volum