Traversing the Machining Graph Danny Chen, Notre Dame Rudolf Fleischer, Li Jian, Wang Haitao,Zhu...
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Transcript of Traversing the Machining Graph Danny Chen, Notre Dame Rudolf Fleischer, Li Jian, Wang Haitao,Zhu...
Traversing the Machining Graph
Danny Chen, Notre Dame
Rudolf Fleischer, Li Jian,
Wang Haitao,Zhu Hong, Fudan
Sep,2006
The Model
We are stuck
Non-compulsory edge
(be traversed at most once)
Compulsory edge
(be traversed exactly once)
What is Known
Simple polygon:NP-hard?Some heuristics [Held’91,
Tang,Chou,Chen’98]
Polygon with h holes:NP-hard
[Arkin,Held,Smith’00]5OPT+6h jumps [AHS’00]Opt+h+N jumps [Tang,Joneja’03]
What we Show
Simple polygon:NP-hard? No, linear time (DP)Some heuristics [Held’91,
Tang,Chou,Chen’98]
Polygon with h holes:NP-hard [Arkin,Held,Smith’00]5OPT+6h jumps [AHS’00]Opt+h+N jumps [Tang,Joneja’03]OPT+εh jumps in polynomial timeOpt jumps in linear+O(1)O(h) time (DP)
lemma
Lemma [Arkin,Held,Smith’00]: There exists a optimal solution s.t.
(1) every path starts and ends with compulsory edges.
(2) No two non-compulsory edges are traversed consecutively. (alternating lemma)
Polygon with h Holes
Using arbitrary strategy to cut all the cycles gives a (O(1)^O(h))*O(n) algorithms.
Identify O(h) pivotal node whose removal s.t. 1.break all cycles.
2.each remaining (dual) tree is adjacent to O(1) pivotal nodes.
Then, we can do it in (O(1)^O(h))+O(n) time.
Polygon with h Holes:Minimum Restrict Path Cover
Boundary graphOriginal Pocket
Forbidden pairs:
(e_1,e_4) and (e_2,e_3)
e_1 e_2
e_4 e_3
A valid path: no forbidden pairs appear in one path.
MRPC: find min # valid paths cover all vertices.
Polygon with h Holes:Minimum Restrict Path Cover
Graph with Bounded Tree Width
(informal)
Polygon with h Holes:Minimum Restrict Path Cover
Tree
Graph with bound treewidth
O(1) communicatons
1 communicaton
Polygon with h Holes:Minimum Restrict Path
Cover(MRPC) It turns out MRPC can be solved in linear
time by dynamic programming if the boundary graph has bounded treewidth.
(assume its tree-decomposition is given)
Remark: If tree-decomposition is not given, find 3-approximation to treewidth in time O(n log n). [Reed’92]
Polygon with h Holes:
k-outerplanar graph:
Theorem: if a graph is k-outerplanar, it has treewidth 3k-1 . [Bodlaender’88]
Peel off the outer layer
Peel again
Peel again
--nothing left…
A 3-outplanar graph
Polygon with h Holes
Lemma:(1) If dual graph has a bounded treewidth and bounded degree, its corresponding boundary graph has bounded treewidth.
(2) If dual graph is a k-outplanar graph, its corresponding boundary graph is a 2k-outerplanar graph.
Thus, if the dual graph is (1) a graph with bounded treewidth and
bounded degree, or (2)a k-outerplanar graph,MRPC can be solved in polynomial time.
Polygon with h Holes
Cut an edge (in the dual):
Polygon with h HolesApproximation for general planar graphs
Original dual After cut
Polygon with h HolesApproximation for general planar graphs
Decompose dual into a series of k-outerplanar graph
k
Baker’s technique
Polygon with h HolesApproximation for general planar graphs
Decompose dual into a series of k-outerplanar graph by cutting edges
Intuitively, cutting one edge reduce the number of face by one.
use 2h/k cuts to decompose the dual (planar) graph into series of (k+1)-outerplanar graphs
Polygon with h HolesApproximation for general planar graphs
solve these (k+1)-outerplanar graphs optimally, then put solutions together for a solution with at most OPT+4h/k jumps
choose k=4/ε
OPT+εh jumps in polynomial time
Polygon with h HolesApproximation for general planar graphs