Primal-Dual Approximation Algorithms for Feedback Problems in Planar Graphs

@article{Goemans1998PrimalDualAA,
  title={Primal-Dual Approximation Algorithms for Feedback Problems in Planar Graphs},
  author={Michel X. Goemans and David P. Williamson},
  journal={Combinatorica},
  year={1998},
  volume={18},
  pages={37-59}
}
A b s t r a c t . Given a subset of cycles of a graph, we consider the problem of finding a minimum-weight set of vertices that meets all cycles in the subset. This problem generalizes a number of problems, including the minlmum-weight feedback vertex set problem in both directed and undirected graphs, the subset feedback vertex set problem, and the graph bipartization problem~ in which one must remove a minimumweight set of vertices so that the remaining graph is bipartite. We give a… 
Planar Feedback Vertex Set and Face Cover: Combinatorial Bounds and Subexponential Algorithms
TLDR
This paper improves the algorithmic analysis of both problems by proving a series of combinatorial results relating the branchwidth of planar graphs with their face cover by combining this fact with duality properties of branchwidth.
Approximating Minimum Subset Feedback Sets in Undirected Graphs with Applications
TLDR
A polynomial time algorithm is provided for approximating the subset feedback edge set problem that achieves an approximation factor of two and a bootstrapping technique is employed to achieve the O(\log \tau^*) factor, which is the value of the optimal fractional solution.
Constant Factor Approximation for Subset Feedback Set Problems via a new LP relaxation
TLDR
New LP relaxations for S uBSet -FES and S ubset -fVS are introduced and their integrality gap is at most 13.5; the LP formulation and rounding are simple although the analysis is non-obvious.
Feedback Vertex Set on Graphs of Low Cliquewidth
TLDR
The algorithm applies a non-standard dynamic programming on a so-called k-module decomposition of a graph, as defined by Rao, which is easily derivable from a k-expression of the graph.
Feedback vertex set on graphs of low clique-width
Hitting Weighted Even Cycles in Planar Graphs
TLDR
The main result is a primal-dual algorithm that yields a 47/7 ≈ 6.71-approximation for ECT on node-weighted planar graphs, and an integrality gap of the same value for the standard LP relaxation on nodes- WeightedPlanar graphs.
On approximability and LP formulations for multicut and feedback set problems
  • V. Madan
  • Computer Science, Mathematics
  • 2018
TLDR
This thesis develops new LP relaxations with constant integrality gaps for subset feedback edge and vertex set problems and proves the optimality of the hitting-set LP for directed graphs.
A Linear Kernel for Planar Feedback Vertex Set
TLDR
This paper gives a polynomialtime algorithm, that given a planar graph G finds a equivalent planargraph G′ with at most 112k* vertices, where k* is the size of the minimumFeedback Vertex Set of G.
...
...

References

SHOWING 1-10 OF 37 REFERENCES
Approximation algorithms for the vertex feedback set problem with applications to constraint satisfaction and Bayesian inference
TLDR
It is shown how polynomial-time algorithms provided for approximating the problem of finding a vertex feedback set of G with a smallest weight can improve the search performance for constraint satisfaction problems and in the area of Bayesian inference of graphs with blackout vertices.
An 8-approximation algorithm for the subset feedback vertex set problem
  • G. Even, J. Naor, L. Zosin
  • Mathematics, Computer Science
    Proceedings of 37th Conference on Foundations of Computer Science
  • 1996
TLDR
An 8-approximation algorithm for the problem of finding a minimum weight subset feedback vertex set in undirected graphs and a subset of vertices S called special vertices is presented.
Approximating Minimum Subset Feedback Sets in Undirected Graphs with Applications
TLDR
A polynomial time algorithm is provided for approximating the subset feedback edge set problem that achieves an approximation factor of two and a bootstrapping technique is employed to achieve the O(\log \tau^*) factor, which is the value of the optimal fractional solution.
Approximating Minimum Feedback Sets and Multi-Cuts in Directed Graphs
TLDR
An approximation algorithm based on an approximation algorithm for the multi-cut problem in a special type of directed networks, and a combinatorial algorithm that computes a (1 + e) approximation to the fractional optimal feedback vertex set.
Approximating Minimum Feedback Sets and Multicuts in Directed Graphs
TLDR
A combinatorial algorithm that computes a (1+ɛ) approximation to the fractional optimal feedback vertex set, and a generalization of these problems, in which the feedback set has to intersect only a subset of the directed cycles in the graph.
When trees collide: an approximation algorithm for the generalized Steiner problem on networks
TLDR
This work gives the first approximation algorithm for the generalized network Steiner problem, a problem in network design, and proves a combinatorial min-max approximate equality relating minimum-cost networks to maximum packings of certain kinds of cuts.
An approximate max-flow min-cut relation for undirected multicommodity flow, with applications
TLDR
It is shown that for a feasible flow to exist in a multicommodity problem, it is sufficient that every cut's capacity exceeds its demand by a factor ofO(logClogD), whereC is the sum of all finite capacities andD is thesum of demands.
A general approximation technique for constrained forest problems
TLDR
The first approximation algorithms for many NP-complete problems, including the non-fixed point-to-point connection problem, the exact path partitioning problem and complex location-design problems are derived.
An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms
  • F. Leighton, Satish Rao
  • Computer Science
    [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science
  • 1988
TLDR
The main result is an algorithm for performing the task provided that the capacity of each cut exceeds the demand across the cut by a Theta (log n) factor.
...
...