35 research outputs found
A Nearly Linear-Time PTAS for Explicit Fractional Packing and Covering Linear Programs
We give an approximation algorithm for packing and covering linear programs
(linear programs with non-negative coefficients). Given a constraint matrix
with n non-zeros, r rows, and c columns, the algorithm computes feasible primal
and dual solutions whose costs are within a factor of 1+eps of the optimal cost
in time O((r+c)log(n)/eps^2 + n).Comment: corrected version of FOCS 2007 paper: 10.1109/FOCS.2007.62. Accepted
to Algorithmica, 201
On the Approximability and Hardness of the Minimum Connected Dominating Set with Routing Cost Constraint
In the problem of minimum connected dominating set with routing cost
constraint, we are given a graph , and the goal is to find the
smallest connected dominating set of such that, for any two
non-adjacent vertices and in , the number of internal nodes on the
shortest path between and in the subgraph of induced by is at most times that in . For general graphs, the only
known previous approximability result is an -approximation algorithm
() for by Ding et al. For any constant , we
give an -approximation
algorithm. When , we give an -approximation
algorithm. Finally, we prove that, when , unless , for any constant , the problem admits no
polynomial-time -approximation algorithm, improving
upon the bound by Du et al. (albeit under a stronger hardness
assumption)
LP-based Covering Games with Low Price of Anarchy
We present a new class of vertex cover and set cover games. The price of
anarchy bounds match the best known constant factor approximation guarantees
for the centralized optimization problems for linear and also for submodular
costs -- in contrast to all previously studied covering games, where the price
of anarchy cannot be bounded by a constant (e.g. [6, 7, 11, 5, 2]). In
particular, we describe a vertex cover game with a price of anarchy of 2. The
rules of the games capture the structure of the linear programming relaxations
of the underlying optimization problems, and our bounds are established by
analyzing these relaxations. Furthermore, for linear costs we exhibit linear
time best response dynamics that converge to these almost optimal Nash
equilibria. These dynamics mimic the classical greedy approximation algorithm
of Bar-Yehuda and Even [3]
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Greedy δ-Approximation Algorithm for Covering with Arbitrary Constraints and Submodular Cost
This paper describes a simple greedy δ-approximation algorithm for any covering problem whose objective function is submodular and non-decreasing, and whose feasible region can be expressed as the intersection of arbitrary (closed upwards) covering constraints, each of which constrains at most δ variables of the problem. (A simple example is Vertex Cover, with δ=2.) The algorithm generalizes previous approximation algorithms for fundamental covering problems and online paging and caching problems. © 2012 Springer Science+Business Media, LLC
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Greedy δ-Approximation Algorithm for Covering with Arbitrary Constraints and Submodular Cost
This paper describes a simple greedy δ-approximation algorithm for any covering problem whose objective function is submodular and non-decreasing, and whose feasible region can be expressed as the intersection of arbitrary (closed upwards) covering constraints, each of which constrains at most δ variables of the problem. (A simple example is Vertex Cover, with δ=2.) The algorithm generalizes previous approximation algorithms for fundamental covering problems and online paging and caching problems. © 2012 Springer Science+Business Media, LLC
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Distributed algorithms for covering, packing and maximum weighted matching
This paper gives poly-logarithmic-round, distributed δ-approximation algorithms for covering problems with submodular cost and monotone covering constraints (Submodular-cost Covering). The approximation ratio δ is the maximum number of variables in any constraint. Special cases include Covering Mixed Integer Linear Programs (CMIP), and Weighted Vertex Cover (with δ = 2). Via duality, the paper also gives poly-logarithmic-round, distributed δ-approximation algorithms for Fractional Packing linear programs (where δ is the maximum number of constraints in which any variable occurs), and for Max Weighted c-Matching in hypergraphs (where δ is the maximum size of any of the hyperedges; for graphs δ = 2). The paper also gives parallel (RNC) 2-approximation algorithms for CMIP with two variables per constraint and Weighted Vertex Cover. The algorithms are randomized. All of the approximation ratios exactly match those of comparable centralized algorithms. © 2011 Springer-Verlag
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A Nearly Linear-Time PTAS for Explicit Fractional Packing and Covering Linear Programs
We give an approximation algorithm for fractional packing and covering linear programs (linear programs with non-negative coefficients). Given a constraint matrix with n non-zeros, r rows, and c columns, the algorithm (with high probability) computes feasible primal and dual solutions whose costs are within a factor of 1+ε of opt (the optimal cost) in time O((r+c)log(n)/ε2+n)
Distributed Algorithms for Covering, Packing and Maximum Weighted Matching
This paper gives poly-logarithmic-round, distributed δ-approximation algorithms for covering problems with submodular cost and monotone covering constraints (Submodular-cost Covering). The approximation ratio δ is the maximum number of variables in any constraint. Special cases include Covering Mixed Integer Linear Programs (CMIP), and Weighted Vertex Cover (with δ = 2). Via duality, the paper also gives poly-logarithmicround, distributed δ-approximation algorithms for Fractional Packing linear programs (where δ is the maximum number of constraints in which any variable occurs), and for Max Weighted c-Matching in hypergraphs (where δ is the maximum size of any of the hyperedges; for graphs δ = 2). The paper also gives parallel (RNC) 2-approximation algorithms for CMIP with two variables per constraint and Weighted Vertex Cover. The algorithms are randomized. All of the approximation ratios exactly match those of comparable centralized algorithms