119 research outputs found
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
The k-edge connected subgraph problem: Valid inequalities and Branch-and-Cut
International audienceIn this paper we consider the k-edge connected subgraph problem from a polyhedral point of view. We introduce further classes of valid inequalities for the associated polytope, and describe sufficient conditions for these inequalities to be facet defining. We also devise separation routines for these inequalities, and discuss some reduction operations that can be used in a preprocessing phase for the separation. Using these results, we develop a Branch-and-Cut algorithm and present some computational results
Complexity of the Steiner Network Problem with Respect to the Number of Terminals
In the Directed Steiner Network problem we are given an arc-weighted digraph
, a set of terminals , and an (unweighted) directed
request graph with . Our task is to output a subgraph of the minimum cost such that there is a directed path from to in
for all .
It is known that the problem can be solved in time
[Feldman&Ruhl, SIAM J. Comput. 2006] and cannot be solved in time
even if is planar, unless Exponential-Time Hypothesis
(ETH) fails [Chitnis et al., SODA 2014]. However, as this reduction (and other
reductions showing hardness of the problem) only shows that the problem cannot
be solved in time unless ETH fails, there is a significant
gap in the complexity with respect to in the exponent.
We show that Directed Steiner Network is solvable in time , where is a constant depending solely on the
genus of and is a computable function. We complement this result by
showing that there is no algorithm for
any function for the problem on general graphs, unless ETH fails
Approximation algorithms for network design and cut problems in bounded-treewidth
This thesis explores two optimization problems, the group Steiner tree and firefighter problems, which are known to be NP-hard even on trees. We study the approximability of these problems on trees and bounded-treewidth graphs. In the group Steiner tree, the input is a graph and sets of vertices called groups; the goal is to choose one representative from each group and connect all the representatives with minimum cost. We show an O(log^2 n)-approximation algorithm for bounded-treewidth graphs, matching the known lower bound for trees, and improving the best possible result using previous techniques. We also show improved approximation results for group Steiner forest, directed Steiner forest, and a fault-tolerant version of group Steiner tree. In the firefighter problem, we are given a graph and a vertex which is burning. At each time step, we can protect one vertex that is not burning; fire then spreads to all unprotected neighbors of burning vertices. The goal is to maximize the number of vertices that the fire does not reach. On trees, a classic (1-1/e)-approximation algorithm is known via LP rounding. We prove that the integrality gap of the LP matches this approximation, and show significant evidence that additional constraints may improve its integrality gap. On bounded-treewidth graphs, we show that it is NP-hard to find a subpolynomial approximation even on graphs of treewidth 5. We complement this result with an O(1)-approximation on outerplanar graphs.Diese Arbeit untersucht zwei Optimierungsprobleme, von welchen wir wissen, dass sie selbst in BĂ€umen NP-schwer sind. Wir analysieren Approximationen fĂŒr diese Probleme in BĂ€umen und Graphen mit begrenzter Baumweite. Im Gruppensteinerbaumproblem, sind ein Graph und Mengen von Knoten (Gruppen) gegeben; das Ziel ist es, einen Knoten von jeder Gruppe mit minimalen Kosten zu verbinden. Wir beschreiben einen O(log^2 n)-Approximationsalgorithmus fĂŒr Graphen mit beschrĂ€nkter Baumweite, dies entspricht der zuvor bekannten unteren Schranke fĂŒr BĂ€ume und ist zudem eine Verbesserung ĂŒber die bestmöglichen Resultate die auf anderen Techniken beruhen. DarĂŒber hinaus zeigen wir verbesserte Approximationsresultate fĂŒr andere Gruppensteinerprobleme. Im Feuerwehrproblem sind ein Graph zusammen mit einem brennenden Knoten gegeben. In jedem Zeitschritt können wir einen Knoten der noch nicht brennt auswĂ€hlen und diesen vor dem Feuer beschĂŒtzen. Das Feuer breitet sich anschlieĂend zu allen Nachbarn aus. Das Ziel ist es die Anzahl der Knoten die vom Feuer unberĂŒhrt bleiben zu maximieren. In BĂ€umen existiert ein lang bekannter (1-1/e)-Approximationsalgorithmus der auf LP Rundung basiert. Wir zeigen, dass die GanzzahligkeitslĂŒcke des LP tatsĂ€chlich dieser Approximation entspricht, und dass weitere EinschrĂ€nkungen die GanzzahligkeitslĂŒcke möglicherweise verbessern könnten. FĂŒr Graphen mit beschrĂ€nkter Baumweite zeigen wir, dass es NP-schwer ist, eine sub-polynomielle Approximation zu finden
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 terminal vertices or players needs to establish connectivity with the root. The social optimum is the minimum Steiner tree. We study 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 graph 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 goal of the designer is to choose a single, universal cost-sharing 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 graph metric help in the design? We first demonstrate that there exist classes of graphs where knowledge of the underlying graph metric can dramatically improve the performance of good network cost-sharing design. For outerplanar graph metrics, we provide a universal cost-sharing protocol with constant PoA, in contrast to protocols that, by ignoring the graph metric, cannot achieve PoA better than . Then, in our main technical result, we show that there exist graph metrics for which knowing the underlying graph 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 the players are activated according to some probability distribution that is known to the designer. We show that there exists a randomized ordered protocol that achieves constant PoA. If, further, each player is activated independently with some probability, by using standard derandomization techniques, we produce a deterministic ordered protocol that achieves constant PoA. We remark that the first result holds also for the black-box model, where the probabilities are not known to the designer, but she is allowed to draw independent (polynomially many) samples. Read More: https://epubs.siam.org/doi/10.1137/16M109669
Tight bounds for planar strongly connected Steiner subgraph with fixed number of terminals (and extensions)
(see paper for full abstract)
Given a vertex-weighted directed graph and a set of terminals, the objective of the SCSS problem is to find a
vertex set of minimum weight such that contains a
path for each . The problem is NP-hard, but
Feldman and Ruhl [FOCS '99; SICOMP '06] gave a novel algorithm for
the SCSS problem, where is the number of vertices in the graph and is
the number of terminals. We explore how much easier the problem becomes on
planar directed graphs:
- Our main algorithmic result is a algorithm
for planar SCSS, which is an improvement of a factor of in the
exponent over the algorithm of Feldman and Ruhl.
- Our main hardness result is a matching lower bound for our algorithm: we
show that planar SCSS does not have an algorithm
for any computable function , unless the Exponential Time Hypothesis (ETH)
fails.
The following additional results put our upper and lower bounds in context:
- In general graphs, we cannot hope for such a dramatic improvement over the
algorithm of Feldman and Ruhl: assuming ETH, SCSS in general graphs
does not have an algorithm for any computable
function .
- Feldman and Ruhl generalized their algorithm to the more general
Directed Steiner Network (DSN) problem; here the task is to find a subgraph of
minimum weight such that for every source there is a path to the
corresponding terminal . We show that, assuming ETH, there is no
time algorithm for DSN on acyclic planar graphs.Comment: To appear in SICOMP. Extended abstract in SODA 2014. This version has
a new author (Andreas Emil Feldmann), and the algorithm is faster and
considerably simplified as compared to conference versio
Bicriteria Network Design Problems
We study a general class of bicriteria network design problems. A generic
problem in this class is as follows: Given an undirected graph and two
minimization objectives (under different cost functions), with a budget
specified on the first, find a <subgraph \from a given subgraph-class that
minimizes the second objective subject to the budget on the first. We consider
three different criteria - the total edge cost, the diameter and the maximum
degree of the network. Here, we present the first polynomial-time approximation
algorithms for a large class of bicriteria network design problems for the
above mentioned criteria. The following general types of results are presented.
First, we develop a framework for bicriteria problems and their
approximations. Second, when the two criteria are the same %(note that the cost
functions continue to be different) we present a ``black box'' parametric
search technique. This black box takes in as input an (approximation) algorithm
for the unicriterion situation and generates an approximation algorithm for the
bicriteria case with only a constant factor loss in the performance guarantee.
Third, when the two criteria are the diameter and the total edge costs we use a
cluster-based approach to devise a approximation algorithms --- the solutions
output violate both the criteria by a logarithmic factor. Finally, for the
class of treewidth-bounded graphs, we provide pseudopolynomial-time algorithms
for a number of bicriteria problems using dynamic programming. We show how
these pseudopolynomial-time algorithms can be converted to fully
polynomial-time approximation schemes using a scaling technique.Comment: 24 pages 1 figur
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