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Primal-dual algorithms for node-weighted network design in planar graphs. Grigory Yaroslavtsev Penn State (joint work with Piotr Berman). Feedback Vertex Set Problems. Given: a collection of cycles in a graph Goal: break them, removing a small # of vertices.

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primal dual algorithms for node weighted network design in planar graphs

Primal-dual algorithms fornode-weighted network designin planar graphs

GrigoryYaroslavtsev

Penn State

(joint work with Piotr Berman)

feedback vertex set problems
Feedback Vertex Set Problems
  • Given: a collection of cycles in a graph
  • Goal: break them, removing a small # of vertices

Example:Collection = All cycles

X

X

Weighted vertices => remove set of min cost

fvs flavors and toppings
FVS: Flavors and toppings
  • All cycles = Feedback Vertex Set
  • All Directed cycles = DirectedFVS
  • All odd-length cycles = Bipartization
  • Cycles through a subset of vertices = Subset FVS

X

fvs in general graphs
FVS in general graphs
  • NP-hard (even in planar graph [Yannakakis])
  • MAX-SNP complete [Lewis, Yannakakis; Papadimitriou, Yannakakis]=>
    • 1.3606, if [Dinur, Safra]
    • under UGC [Khot, Regev]
fvs in planar graphs via primal dual
FVS in planar graphs (via primal-dual)
  • NP-hard (even in planar graph [Yannakakis])
bigger picture
Bigger picture

Graphs

Weights

General

Vertices

Planar

Edges

  • FeedbackEdgeSet in general graphs = Complement of MST
  • Planar edge-weightedBIP and D-FVS are also in P
  • Planaredge-weightedSteiner Forest has a PTAS[Bateni, Hajiaghayi, Marx, STOC’11]
  • PlanarunweightedFeedback Vertex Set has a PTAS [Baker; Demaine, Hajiaghayi, SODA’05]
class 1 uncrossing property
Class 1: Uncrossing property
  • Uncrossing:
  • Uncrossing property of a family of cycles :
  • Holdsfor all FVS problems, crucial for the algorithm of GW

  • For every two crossing cycles one of their two uncrossings has .
proper functions gw dhk
Proper functions [GW, DHK]
  • A function is proper if
    • Symmetry:
    • Disjointness: If and =>
  • A set is active, if
  • Boundary :
  • A boundary is active, if is active
class 2 hitting set ip dhk
Class 2: Hitting set IP [DHK]
  • The class of problems:

Minimize:

Subject to: , for all

,

where is a proper function

  • Theorem: is proper => the collection of all active boundaries forms an uncrossablefamily (requires triangulation)
  • Proof sketch: is proper => in one of the cases both interior sets are active => their boundaries are active

class 1 class 2
Class 1 = Class 2
  • Example: Node-Weighted Steiner Forest
    • Connect pairs via a subset of nodes of min cost
    • Proper function iff for some i.
primal dual method local ratio version
Primal-dual method (local-ratio version)
  • Given: G (graph), w (weights), (cycles)
    • set of all vertices of zero weight
    • While is not a hitting set for
      • collection of cycles returned by oracle Violation(G, C, )
      • # of cycles in M, which contain
      • = set of all vertices of zero weight
    • Return a minimal hitting set for
oracle 1 f ace minimal cycles gw
Oracle 1 = Face-minimal cycles [GW]
  • Example for Subset FVS with :
  • Oracle returns all gray cycles => all white nodes are selected
  • Cost = 3 * # blocks, OPT (1 + ) * # blocks
oracle 2 pocket removal gw
Oracle 2 = Pocket removal [GW]
  • Pocket defined by two cycles: region between their common points containing another cycle
  • New oracle: no pocket => all face-minimal cycles, otherwise run recursively inside any pocket.
  • Our analysis:
oracle 3 triple pocket removal
Oracle 3 = Triple pocket removal
  • Triple pocket = region defined by three cycles
  • Analysis:
open problems
Open problems

For our class of node-weighted problems:

  • Big question: APX-hardness or a PTAS?
  • Integrality gap = 2, how to approach it?
    • Pockets of higher multiplicities are harder to analyze
    • Pockets cannot go beyond 2 +
applications and ramifications
Applications and ramifications
  • Applications: from maintenance of power networks to computational sustainability
  • Example: VLSI design.
      • Primal-dual approximation algorithms of Goemans and Williamson are competitive with heuristics [Kahng, Vaya, Zelikovsky]
  • Connections with bounds on the size of FVS
      • Conjectures of Akiyama and Watanabe and Gallai and Younger [see GW for more details]
approximation factor
Approximation factor
  • Theorem [GW’96]: If for any minimal solution H the set M returned by the oracle satisfies:

then the primal-dual algorithm has approximation

  • Examples of oracles:
    • Single cycle: [Bar-Yehuda, Geiger, Naor, Roth]
    • Single cycle: 5[Goemans, Williamson]
    • Collection of all face-minimal cycles: 3[Goemans, Williamson]