1,843 research outputs found
Dial a Ride from k-forest
The k-forest problem is a common generalization of both the k-MST and the
dense--subgraph problems. Formally, given a metric space on vertices
, with demand pairs and a ``target'' ,
the goal is to find a minimum cost subgraph that connects at least demand
pairs. In this paper, we give an -approximation
algorithm for -forest, improving on the previous best ratio of
by Segev & Segev.
We then apply our algorithm for k-forest to obtain approximation algorithms
for several Dial-a-Ride problems. The basic Dial-a-Ride problem is the
following: given an point metric space with objects each with its own
source and destination, and a vehicle capable of carrying at most objects
at any time, find the minimum length tour that uses this vehicle to move each
object from its source to destination. We prove that an -approximation
algorithm for the -forest problem implies an
-approximation algorithm for Dial-a-Ride. Using our
results for -forest, we get an -
approximation algorithm for Dial-a-Ride. The only previous result known for
Dial-a-Ride was an -approximation by Charikar &
Raghavachari; our results give a different proof of a similar approximation
guarantee--in fact, when the vehicle capacity is large, we give a slight
improvement on their results.Comment: Preliminary version in Proc. European Symposium on Algorithms, 200
Designing Networks with Good Equilibria under Uncertainty
We consider the problem of designing network cost-sharing protocols with good
equilibria under uncertainty. The underlying game is a multicast game in a
rooted undirected graph with nonnegative edge costs. A set of k terminal
vertices or players need to establish connectivity with the root. The social
optimum is the Minimum Steiner Tree. We are interested in situations where the
designer has incomplete information about the input. We propose two different
models, the adversarial and the stochastic. In both models, the designer has
prior knowledge of the underlying metric but the requested subset of the
players is not known and is activated either in an adversarial manner
(adversarial model) or is drawn from a known probability distribution
(stochastic model).
In the adversarial model, the designer's goal is to choose a single,
universal protocol that has low Price of Anarchy (PoA) for all possible
requested subsets of players. The main question we address is: to what extent
can prior knowledge of the underlying metric help in the design? We first
demonstrate that there exist graphs (outerplanar) where knowledge of the
underlying metric can dramatically improve the performance of good network
design. Then, in our main technical result, we show that there exist graph
metrics, for which knowing the underlying metric does not help and any
universal protocol has PoA of , which is tight. We attack this
problem by developing new techniques that employ powerful tools from extremal
combinatorics, and more specifically Ramsey Theory in high dimensional
hypercubes.
Then we switch to the stochastic model, where each player is independently
activated. We show that there exists a randomized ordered protocol that
achieves constant PoA. By using standard derandomization techniques, we produce
a deterministic ordered protocol with constant PoA.Comment: This version has additional results about stochastic inpu
Online Steiner Tree with Deletions
In the online Steiner tree problem, the input is a set of vertices that
appear one-by-one, and we have to maintain a Steiner tree on the current set of
vertices. The cost of the tree is the total length of edges in the tree, and we
want this cost to be close to the cost of the optimal Steiner tree at all
points in time. If we are allowed to only add edges, a tight bound of
on the competitiveness is known. Recently it was shown that if
we can add one new edge and make one edge swap upon every vertex arrival, we
can maintain a constant-competitive tree online.
But what if the set of vertices sees both additions and deletions? Again, we
would like to obtain a low-cost Steiner tree with as few edge changes as
possible. The original paper of Imase and Waxman had also considered this
model, and it gave a greedy algorithm that maintained a constant-competitive
tree online, and made at most edge changes for the first
requests. In this paper give the following two results.
Our first result is an online algorithm that maintains a Steiner tree only
under deletions: we start off with a set of vertices, and at each time one of
the vertices is removed from this set: our Steiner tree no longer has to span
this vertex. We give an algorithm that changes only a constant number of edges
upon each request, and maintains a constant-competitive tree at all times. Our
algorithm uses the primal-dual framework and a global charging argument to
carefully make these constant number of changes.
We then study the natural greedy algorithm proposed by Imase and Waxman that
maintains a constant-competitive Steiner tree in the fully-dynamic model (where
each request either adds or deletes a vertex). Our second result shows that
this algorithm makes only a constant number of changes per request in an
amortized sense.Comment: An extended abstract appears in the SODA 2014 conferenc
Robust and MaxMin Optimization under Matroid and Knapsack Uncertainty Sets
Consider the following problem: given a set system (U,I) and an edge-weighted
graph G = (U, E) on the same universe U, find the set A in I such that the
Steiner tree cost with terminals A is as large as possible: "which set in I is
the most difficult to connect up?" This is an example of a max-min problem:
find the set A in I such that the value of some minimization (covering) problem
is as large as possible.
In this paper, we show that for certain covering problems which admit good
deterministic online algorithms, we can give good algorithms for max-min
optimization when the set system I is given by a p-system or q-knapsacks or
both. This result is similar to results for constrained maximization of
submodular functions. Although many natural covering problems are not even
approximately submodular, we show that one can use properties of the online
algorithm as a surrogate for submodularity.
Moreover, we give stronger connections between max-min optimization and
two-stage robust optimization, and hence give improved algorithms for robust
versions of various covering problems, for cases where the uncertainty sets are
given by p-systems and q-knapsacks.Comment: 17 pages. Preliminary version combining this paper and
http://arxiv.org/abs/0912.1045 appeared in ICALP 201
Greedy Algorithms for Steiner Forest
In the Steiner Forest problem, we are given terminal pairs ,
and need to find the cheapest subgraph which connects each of the terminal
pairs together. In 1991, Agrawal, Klein, and Ravi, and Goemans and Williamson
gave primal-dual constant-factor approximation algorithms for this problem;
until now, the only constant-factor approximations we know are via linear
programming relaxations.
We consider the following greedy algorithm: Given terminal pairs in a metric
space, call a terminal "active" if its distance to its partner is non-zero.
Pick the two closest active terminals (say ), set the distance
between them to zero, and buy a path connecting them. Recompute the metric, and
repeat. Our main result is that this algorithm is a constant-factor
approximation.
We also use this algorithm to give new, simpler constructions of cost-sharing
schemes for Steiner forest. In particular, the first "group-strict" cost-shares
for this problem implies a very simple combinatorial sampling-based algorithm
for stochastic Steiner forest
Parameterized Analysis of Online Steiner Tree Problems
Steiner tree problems occupy a central place in both areas of approximation and on-line algorithms. Many variants have been studied from the point of view of competitive analysis, and for several of these variants tight bounds are known. However, in several cases, worst-case analysis is overly pessimistic, which fails to explain the relative performance of algorithms. We show how adaptive analysis can help resolve this problem. As case studies, we consider the Steiner tree problem in directed graphs, and the Priority Steiner tree problem
Greedy Algorithms for Online Survivable Network Design
In an instance of the network design problem, we are given a graph G=(V,E), an edge-cost function c:E -> R^{>= 0}, and a connectivity criterion. The goal is to find a minimum-cost subgraph H of G that meets the connectivity requirements. An important family of this class is the survivable network design problem (SNDP): given non-negative integers r_{u v} for each pair u,v in V, the solution subgraph H should contain r_{u v} edge-disjoint paths for each pair u and v.
While this problem is known to admit good approximation algorithms in the offline case, the problem is much harder in the online setting. Gupta, Krishnaswamy, and Ravi [Gupta et al., 2012] (STOC\u2709) are the first to consider the online survivable network design problem. They demonstrate an algorithm with competitive ratio of O(k log^3 n), where k=max_{u,v} r_{u v}. Note that the competitive ratio of the algorithm by Gupta et al. grows linearly in k. Since then, an important open problem in the online community [Naor et al., 2011; Gupta et al., 2012] is whether the linear dependence on k can be reduced to a logarithmic dependency.
Consider an online greedy algorithm that connects every demand by adding a minimum cost set of edges to H. Surprisingly, we show that this greedy algorithm significantly improves the competitive ratio when a congestion of 2 is allowed on the edges or when the model is stochastic. While our algorithm is fairly simple, our analysis requires a deep understanding of k-connected graphs. In particular, we prove that the greedy algorithm is O(log^2 n log k)-competitive if one satisfies every demand between u and v by r_{uv}/2 edge-disjoint paths. The spirit of our result is similar to the work of Chuzhoy and Li [Chuzhoy and Li, 2012] (FOCS\u2712), in which the authors give a polylogarithmic approximation algorithm for edge-disjoint paths with congestion 2.
Moreover, we study the greedy algorithm in the online stochastic setting. We consider the i.i.d. model, where each online demand is drawn from a single probability distribution, the unknown i.i.d. model, where every demand is drawn from a single but unknown probability distribution, and the prophet model in which online demands are drawn from (possibly) different probability distributions. Through a different analysis, we prove that a similar greedy algorithm is constant competitive for the i.i.d. and the prophet models. Also, the greedy algorithm is O(log n)-competitive for the unknown i.i.d. model, which is almost tight due to the lower bound of [Garg et al., 2008] for single connectivity
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