103 research outputs found
Randomized approximation algorithms : facility location, phylogenetic networks, Nash equilibria
Despite a great effort, researchers are unable to find efficient algorithms for a number of natural computational problems. Typically, it is possible to emphasize the hardness of such problems by proving that they are at least as hard as a number of other problems. In the language of computational complexity it means proving that the problem is complete for a certain class of problems. For optimization problems, we may consider to relax the requirement of the outcome to be optimal and accept an approximate (i.e., close to optimal) solution. For many of the problems that are hard to solve optimally, it is actually possible to efficiently find close to optimal solutions. In this thesis, we study algorithms for computing such approximate solutions
Improved approximation algorithm for k-level UFL with penalties, a simplistic view on randomizing the scaling parameter
The state of the art in approximation algorithms for facility location
problems are complicated combinations of various techniques. In particular, the
currently best 1.488-approximation algorithm for the uncapacitated facility
location (UFL) problem by Shi Li is presented as a result of a non-trivial
randomization of a certain scaling parameter in the LP-rounding algorithm by
Chudak and Shmoys combined with a primal-dual algorithm of Jain et al. In this
paper we first give a simple interpretation of this randomization process in
terms of solving an aux- iliary (factor revealing) LP. Then, armed with this
simple view point, Abstract. we exercise the randomization on a more
complicated algorithm for the k-level version of the problem with penalties in
which the planner has the option to pay a penalty instead of connecting chosen
clients, which results in an improved approximation algorithm
New algorithms for approximate Nash equilibria in bimatrix games
We consider the problem of computing additively approximate Nash equilibria in non-cooperative two-player games. We provide a new polynomial time algorithm that achieves an approximation guarantee of 0.36392. Our work improves the previously best known (0.38197¿+¿e)-approximation algorithm of Daskalakis, Mehta and Papadimitriou [6]. First, we provide a simpler algorithm, which also achieves 0.38197. This algorithm is then tuned, improving the approximation error to 0.36392. Our method is relatively fast, as it requires solving only one linear program and it is based on using the solution of an auxiliary zero-sum game as a starting point. The first author was supported by NWO. The second and third author were supported by the EU Marie Curie Research Training Network, contract numbers MRTN-CT-2003-504438-ADONET and MRTN-CT-2004-504438-ADONET respectively
New Results on Optimizing Rooted Triplets Consistency
A set of phylogenetic trees with overlapping leaf sets is consistent if it can be merged without conflicts into a supertree. In this paper, we study the polynomial-time approximability of two related optimization problems called the maximum rooted triplets consistency problem (\textsc{MaxRTC}) and the minimum rooted triplets inconsistency problem (\textsc{MinRTI}) in which the input is a set of rooted triplets, and where the objectives are to find a largest cardinality subset of which is consistent and a smallest cardinality subset of whose removal from results in a consistent set, respectively. We first show that a simple modification to Wu’s Best-Pair-Merge-First heuristic [25] results in a bottom-up-based 3-approximation for \textsc{MaxRTC}. We then demonstrate how any approximation algorithm for \textsc{MinRTI} could be used to approximate \textsc{MaxRTC}, and thus obtain the first polynomial-time approximation algorithm for \textsc{MaxRTC} with approximation ratio smaller than 3. Next, we prove that f
Unbounded lower bound for k-server against weak adversaries
We study the resource augmented version of the -server problem, also known
as the -server problem against weak adversaries or the -server
problem. In this setting, an online algorithm using servers is compared to
an offline algorithm using servers, where . For uniform metrics, it
has been known since the seminal work of Sleator and Tarjan (1985) that for any
, the competitive ratio drops to a constant if . This result was later generalized to weighted stars (Young 1994) and
trees of bounded depth (Bansal et al. 2017). The main open problem for this
setting is whether a similar phenomenon occurs on general metrics.
We resolve this question negatively. With a simple recursive construction, we
show that the competitive ratio is at least , even as
. Our lower bound holds for both deterministic and randomized
algorithms. It also disproves the existence of a competitive algorithm for the
infinite server problem on general metrics.Comment: To appear in STOC 202
Facility Location in Evolving Metrics
Understanding the dynamics of evolving social or infrastructure networks is a
challenge in applied areas such as epidemiology, viral marketing, or urban
planning. During the past decade, data has been collected on such networks but
has yet to be fully analyzed. We propose to use information on the dynamics of
the data to find stable partitions of the network into groups. For that
purpose, we introduce a time-dependent, dynamic version of the facility
location problem, that includes a switching cost when a client's assignment
changes from one facility to another. This might provide a better
representation of an evolving network, emphasizing the abrupt change of
relationships between subjects rather than the continuous evolution of the
underlying network. We show that in realistic examples this model yields indeed
better fitting solutions than optimizing every snapshot independently. We
present an -approximation algorithm and a matching hardness result,
where is the number of clients and the number of time steps. We also
give an other algorithms with approximation ratio for the variant
where one pays at each time step (leasing) for each open facility
Parameterized Single-Exponential Time Polynomial Space Algorithm for Steiner Tree
"In the Steiner tree problem, we are given as input a connected n-vertex graph with edge weights in {1,2,...,W}, and a subset of k terminal vertices. Our task is to compute a minimum-weight tree that contains all the terminals. We give an algorithm for this problem with running time O(7.97^k n^4 log W) using O(n^3 log nW log k) space. This is the first single-exponential time, polynomial-space FPT algorithm for the weighted Steiner tree problem."
PLEASE NOTE:This is an author-created version that the author has self-archived to the "Aaltodoc" (aaltodoc.aalto.fi) faculty-level repository at Aalto University. The final publication is available at link.springer.com via the link http://dx.doi.org/10.1007/978-3-662-47672-7_40Peer reviewe
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