Traversing the
Machining Graph
Danny Chen, Notre Dame
Rudolf Fleischer, Li Jian,
Wang Haitao,Zhu Hong, Fudan
Sep,2006
2D-Milling
Example
[Arkin,Held,Smith’00]
Zigzag machining
Example
[Tang,Joneja’03]:
Example
[Tang,Joneja’03]:
The Model
The Model
Compulsory edge
The Model
(be traversed exactly once)
Non-compulsory edge
(be traversed at most once)
We are stuck
The Model
We are stuck: jump
The Model
Goal: minimize jumps
Greedy?
Greedy?
Greedy?
Greedy?
2 jumps
Greedy?
Greedy?
Greedy?
2 jumps
Greedy?
Greedy?
1 jump
Greedy?
1 jump
Greedy?
2 jumps
Greedy?
1 jump
Greedy?
1 jump
Greedy?
Greedy?
Greedy?
no jump
Greedy? May be exponential
What is Known
Simple polygon:
 NP-hard?
 Some heuristics
Polygon with h holes:
 NP-hard
 5OPT+6h jumps
 Opt+h+N jumps
[Held’91, Tang,Chou,Chen’98]
[Arkin,Held,Smith’00]
[AHS’00]
[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 time
 Opt 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)
Simple Pocket: The Dual Tree
Simple Pocket:
Dynamic Programming
start at the leaves
Simple Pocket:
Dynamic Programming
Dynamic Programming
Does path end here?  5 cases
constant time per node
Polygon with h Holes
time O(n)+O(1)O(h)
Polygon with h Holes
 Identify O(h) pivotal nodes.
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:
Boundary graph
Polygon with h Holes:
Minimum Restrict Path Cover
Original Pocket
Boundary graph
e_1
e_2
e_4
e_3
Forbidden pairs:
(e_1,e_4) and (e_2,e_3)
Polygon with h Holes:
Minimum Restrict Path Cover
 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)
1 communicaton
O(1) communicatons
Tree
Graph with bound treewidth
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:
Polygon with h Holes:
 k-outerplanar graph:
Peel off the outer
layer
Polygon with h Holes:
 k-outerplanar graph:
Peel off the outer
layer
Peel again
Polygon with h Holes:
 k-outerplanar graph:
Peel off the outer
layer
A 3-outplanar graph
Peel again
Peel again
--nothing left…
Theorem: if a graph is k-outerplanar, it has
treewidth 3k-1 .
[Bodlaender’88]
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.
Polygon with h Holes
 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
Approximation for general planar graphs
 Cut:
Original Pocket
After cut
Polygon with h Holes
Approximation for general planar graphs
 Cut an edge (in the dual):
Original dual
After cut
Polygon with h Holes
Approximation for general planar graphs
Decompose dual into a series of k-outerplanar graph
Baker’s technique
Polygon with h Holes
Approximation for general planar graphs
Decompose dual into a series of k-outerplanar graph
by cutting edges
Polygon with h Holes
Approximation for general planar graphs
 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 Holes
Approximation 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
Thank You!