2/1/2016Version 09/093 Introduction: Minimum-Length Corridor Problem (MLC) MLC MLC-R
2/1/2016Version 09/091 Approximation Algorithms for
Minimum-Length Corridors Teofilo F. Gonzalez Version 10/09
Department of Computer Science University of California at Santa
Barbara (Co-author: Arturo Gonzlez-Gutirrez) Outline Introduction
Preliminaries and Related Problems Parameterized Approximation
Algorithm Alg(S) Selector Function S Constant Ratio Approximation
Additional Results and Conclusion 2/1/2016Version 09/092
2/1/2016Version 09/093 Introduction: Minimum-Length Corridor
Problem (MLC) MLC MLC-R 2/1/2016Version 09/094 Introduction: Formal
Definition of the MLC-R problem INPUT: A pair (F,R), where F is a
rectangular boundary partitioned into rectangles (or rooms) R={R
1,R 2,,R r }. OUTPUT: A corridor consisting of a set of connected
line segments each of which lies along the line segments that form
F and/or the boundary of the rooms, that includes at least one
point of F and at least one point from each of the rooms. OBJECTIVE
FUNCTION: Minimize the total edge length of the corridor.
2/1/2016Version 09/095 Introduction: Applications of the MLC
Problem Network Communication in Metropolitan Areas 2/1/2016Version
09/096 Introduction: Origin of the MLC Problem Demaine, E.D., and
ORourke, J. Open problems from CCCG In Proceedings of the 13 th.
Canadian Conference on Computational Geometry (2001), (by Naoki
Katoh). Eppstein, D. Some open problems in graph theory and
computational geometry.November
2001.http://www.ics.uci.edu/~eppstein/200-f01.pdf Jin, L. Y., and
Chong, O. W. The minimum touching tree problem.National University
of Singapore, School of Computing, No polynomial time algorithm
known Not even a constant ratio approximation algorithm Seems
likely to be NP-complete but no proof known 2/1/2016Version 09/097
Related Problems: Outline Tree Errand Cover (TEC) problem
Generalization of the Group Steiner Tree (GSTP) Problem
2/1/2016Version 09/098 Related Problems: Formal Definition of the
TEC problem INPUT: A connected undirected edge-weighted graph
G=(V,E,w), where w:E R + is an edge-weight function; a non-empty
set C, C V, of terminals; a non-empty set E ={e 1,e 2,...,e k } of
errands; a collection C={ C 1, C 2,, C k }, where C i C specifies
the vertices where errand e i can be performed. OUTPUT: A tree T(G,
C )=(V,E), where E E and V V, such that for each errand e i there
is at least one vertex v C i and v V, and the total length is
minimized GST problem: C= {C 1, C 2,, C k } is a partition of C
2/1/2016Version 09/099 Related Problems: MLC-R TEC E ={R i | 0 i 9}
C ={C i | 0 i 9} R1R1 R2R2 R3R3 R5R5 R6R6 R4R4 R8R8 R7R7 R9R9 v7v7
v5v5 v 10 v1v1 v2v2 v3v3 v4v4 v9v9 v6v6 v8v8 v 11 v 12 v 13 v 14 v
15 v 16 v 17 v 18 v 19 v 20 iCiCi a TEC solution 0{v 1,v 2, v 3,v
4,v 8,v 9,v 12, v 13,v 16,v 17,v 18,v 19,v 20 } v3v3 1{v 1,v 2,v
5,v 9,v 10 }v 5,v 10 2{v 2,v 3,v 5,v 6 }v 3,v 5,v 6 3{v 3,v 4,v 6,v
7,v 8 }v 3,v 6,v 7 4{v 9,v 10,v 13,v 14 }v 10,v 14 5{v 5,v 6,v 7,v
10,v 11,v 14,v 15,v 18,v 19 }v 5,v 6,v 7,v 10, v 11,v 14,v 15 6{v
7,v 8,v 11,v 12 }v 7,v 11 7{v 13,v 14,v 17,v 18 }V 14 8{v 11,v 12,v
15,v 16 }v 11,v 15 9{v 15,v 16,v 19,v 20 }v 15 2/1/2016Version
09/0910 Related Problems: TEC Problem Performance Ratio Slavik, P.
(1998) The TEC problem can be approximated to within a ratio of 2
in polynomial time, when each errand is assigned to at most
vertices. For the MLC problem there are errands that may be
assigned to an arbitrary number of vertices. GST problem (k-1) OPT
(Ihler, E., 1991) ( 1 + ln(k/2) ) k 0.5 OPT (Bateman, C. D. et al.,
1997) Polynomial time O(k )-approximation algorithms, for
arbitrarily small values of >0 (Helvig, C. S. et al.,2001) These
results do not generate a constant ratio approximation for the MLC
and MLC-R problems. 2/1/2016Version 09/0911 Parameterized
Approximation Algorithm: Alg(S) Hierarchy of the MLC Problem MLC
MLC-R p-MLC-R 2/1/2016Version 09/0912 Parameterized Approximation
Algorithm Alg(S): Outline MLC-R p-MLC-R Parameterized algorithm
Alg(S) for the p-MLC-R problem. 2/1/2016Version 09/0913
Parameterized Approximation Algorithm Alg(S): Selector Function S
and the p-MLC-R S problem I p-MLC-R p I p-MLC-R S p S(2OC+)
2/1/2016Version 09/0914 Parameterized Approximation Algorithm
Alg(S): Approximation Technique p-MLC-R problem p-MLC-R S problem
Feasible solutions of the p-MLC-R problem Feasible solutions after
relaxing (LP) the set of feasible solutions of the p-MLC-R S
problem Feasible solutions after rounding the solution found
2/1/2016Version 09/0915 t(I) = Parameterized Approximation
Algorithm Alg(S): For p-MLC-R problem t(I)2 k S r S opt(I)
I=(p,F,R) p-MLC-R J=(G=(V,E,w), C ) TEC Invoke Slaviks Algorithm:
t(I) 2 k opt(I); k = max{|V(R i )|} I S =(p,F,R,S) p-MLC-R S J S
=(G=(V,E,w), C S ) TEC S Invoke Slaviks Algorithm: t(I S ) 2 k S
opt(I S ); k S = |S| S opt(I S ) r S opt(I) t(I S ) 2 k S r S
opt(I) t(I S ) r S t(I) t(I S ) r S opt(I)opt(I S ) 2/1/2016Version
09/0916 Selector Function S Outline S selects four corners: S(4C)
Definition of Special Point S selects special points: S(+) S
selects two adjacent corners and one special point: S(2AC+) S
selects two opposite corners and one special point: S(2OC+)
2/1/2016Version 09/0917 Selector Function S: S selects four
corners: S(4C) k S(4C) =4 t(I S(4C) (j)) r S(4C) t(I(j))
2/1/2016Version 09/0918 How about if we select the middle vertex?
t(I S (j)) r S t(I(j)) 2/1/2016Version 09/0919 Selector Function S:
Definition of Special Point p SpP 2/1/2016Version 09/0920 R1R1 R2R2
R3R3 R5R5 R6R6 R4R4 R8R8 R7R7 R9R9 v7v7 v5v5 v 10 v1v1 v2v2 v3v3
v4v4 v9v9 v6v6 v8v8 v 11 v 12 v 13 v 14 v 15 v 16 v 17 v 18 v 19 v
20 Selector Function S: Definition of Special Point 3 2 CD(v 11,R)=
CD(R 5,R)= RjRj Candidates to be a special point u Min-connectivity
distance CD(u,R)=CD(R j,R) R1R1 {v 2,v 5 }2 R3R3 {v 3,v 6,v 7,v 8
}2 R5R5 {v 5,v 6,v 7,v 11,v 15,v 19 } 2/1/2016Version 09/0921
Selector Function S: S selects special point: S(+) k S(+) =1 t(I
S(+) (j)) r S(+) t(I(j)) 2/1/2016Version 09/0922 Selector Function
S: S selects special points: S(+) j k S(+) =1 t(I S(+) (j)) r S(+)
t(I(j)) 2/1/2016Version 09/0923 Selector Function S: S selects two
adjacent corners and one special point: S(2AC+) k S(2AC+) =3 t(I
S(2AC+) (j)) r S(2AC+) t(I(j)) 2/1/2016Version 09/0924 Selector
Function S: Gathering properties to construct S Special Point
Corners: Opposite ones 2/1/2016Version 09/0925 Constant Ratio
Approximation: Outline S selects two opposite corners and a special
point: S(2OC+) k S(2OC+) =3 and prove that r S(2OC+) 5 Ncpe/cpe
rectangles and tour Paths type-1 and type-2 for adjacent ncpe
rectangles CD(SpP i,R) is bounded by a portion of the corridor
Type-1 Type-2 2/1/2016Version 09/0926 Constant Ratio Approximation:
S selects two opposite corners and a special point: S(2OC+) I
p-MLC-R p T(I S ) k S(2OC+) =3 t(I S(2OC+) ) r S(2OC+) t(I) T(I)
TR, BL, SpP 2/1/2016Version 09/ Constant Ratio Approximation:
Ncpe/cpe rectangles and tour 2/1/2016Version 09/0928 SpP 3 Constant
Ratio Approximation: k S(2OC+) =3 t(I S(2OC+) ) r S(2OC+) t(I) r
S(2OC+) 5 2/1/2016Version 09/0929 Constant Ratio Approximation:
Paths type-1 and type-2 for adjacent ncpe rectangles n i-1 nini SpP
i n i-1 nini n i+1 SpP i 2/1/2016Version 09/0930 Constant Ratio
Approximation: CD(SpP i,R) is bounded by a portion of the corridor:
Type-1 2/1/2016Version 09/0931 Constant Ratio Approximation: CD(SpP
i,R) is bounded by a portion of the corridor : Type-1
2/1/2016Version 09/0932 Constant Ratio Approximation: CD(SpP i,R)
is bounded by a portion of the corridor: Type-1 2/1/2016Version
09/0933 Constant Ratio Approximation: CD(SpP i,R) is bounded by a
portion of the corridor : Type-1 2/1/2016Version 09/0934 n i-1 n
i+1 nini nini nini nini Constant Ratio Approximation: CD(SpP i,R)
is bounded by a portion of the corridor: Type-2 2/1/2016Version
09/0935 Constant Ratio Approximation: CD(SpP i,R)=CD(n i,R) l i-1
+h i +l i 2/1/2016Version 09/0936 Additional Results and
Conclusions: Outline MLC k problem, c-gons, ck Rectangular
group-TSP Edges may be visited more than once Other results
2/1/2016Version 09/0937 Additional Results and Conclusions: MLC k
problem Rectilinear c-gons for ck, and c6 2/1/2016Version 09/0938
Additional Results and Conclusions: New results Hans Bodlaender 1,
Corinne Feremans 2, Alexander Grigoriev 2, Eelko Penninkx 1, Ren
Sitters 3, Thomas Wolle 4. On the Minimum Corridor Connection
Problem and Other Generalized Geometric Problems. In 4 th. Workshop
on Approximation and Online Algorithms (WAOA) Zurich, Switzerland,
September NP-completeness of MLC problem Geographic Clustering
Problem: NP-complete? PTAS: (1+) OPT in time n(log n) O(1/)
-fatness rooms: NP-complete? (16/)-1 OPT; 0 1 If all the rooms are
squares then =1 and the solution is 15 times the optimal one. 1
Utrecht University, 2 Maastricht University, 3 Max-Planck-Institute
for Computer Science 4 National ICT Australia Ltd. 2/1/2016Version
09/0939 Additional Results and Conclusions: Rectangular group-TSP
Instead a tree we have a tour: 2*30=60 Slavik, P. (1998): The
Errand (Tour) Scheduling problem can be approximated to within a
factor of 3/2 in polynomial time, when each errand is assigned to
at most vertices. 2/1/2016Version 09/0940 Mark de Berg a, Joachim
Gudmundsson b, Mathew J. Katz c, Christos Levcopoulos d, Mark H.
Overmars e, A. Frank van der Stappenet e. TSP with neighborhoods of
varying sizes. J. of Algorithms 57 (2005) 1200 3 times the optimal,
1 93 times the optimal when the regions are squares Additional
Results and Conclusions: Rectangular group-TSP a TU Eindhoven, b
NICTA Sydney, c Ben-Gurion University, d Lund University, e Utrecht
University 2/1/2016Version 09/0941 Additional Results and
Conclusions: TRA-MLC,TRA-MLC-R,p-MLC-R,MLC-R problems are
NP-complete p-MLC-R S for S(2OC+),S(4C+) are NP-complete Solution
of the p-MLC-R and MLC-R problem is at most 2k s r s =2*3*5=30
times the optimal solution. Solution of the MLC k problem is at
most 15k-10 times the optimal solution The approximation ratio is a
constant! 2/1/2016Version 09/0942 Publications A.
Gonzalez-Gutierrez, T.F. Gonzalez, Complexity of the Minimum-Length
Corridor Problem, Computational Geometry: Theory and Applications,
37, 2 (2007), A. Gonzalez-Gutierrez, T.F. Gonzalez, Approximation
Algorithms for the Minimum- Length Corridor and Related Problems,
UCSB CS TR (to appear ACM TALG). 2/1/2016Version 09/0943 Thank you!
Questions Comments Suggestions 2/1/2016Version 09/0944
Introduction: Applications of the MLC Problem Circuit Board Layout
Design Wires for power supply Wires for clock signal Building
Wiring Design Optical Fiber for Data Communication Networks Wires
for Electrical Networks 2/1/2016Version 09/0945 FFFFF Example
2/1/2016Version 09/0946 TTTTT Example 2/1/2016Version 09/0947 FFTFF
Example 2/1/2016Version 09/0948 Parameterized Approximation
Algorithm Alg(S) 2/1/2016Version 09/0949 Parameterized
Approximation Algorithm Alg(S) 2/1/2016Version 09/0950 Related
Problems: Tree Vertex Cover INPUT: A connected undirected
edge-weighted graph G=(V,E,w), where w:E + is an edge- weight
function. OUTPUT: A tree T=(V,E), where E E, and V V is a vertex
cover (i.e. every edge in G includes at least one vertex in V) and
the total edge- weight is minimized. 2/1/2016Version 09/0951 INPUT:
A connected undirected edge-weighted graph G=(V,E,w), where w:E +
is an edge- weight function, a non-empty subset S, S V, of
terminals; and a partition {S 1,S 2,...,S k } of S. OUTPUT: A tree
T(S)=(V,E), where E E and V V, such that at least one terminal from
each set S i is in the tree T(S) and the total length is minimized.
Related Problems: Group Steiner Tree Problem Generalization of the
MLC Problem 2/1/2016Version 09/0952 Related Problems: Group Steiner
Tree Problem Reducing the TRA-MLC to the GST 2/1/2016Version
09/0953 Related Problems: Group Steiner Tree Problem Reducing the
TRA-MLC to the GST 2/1/2016Version 09/0954 Related Problems: Group
Steiner Tree Problem (k-1) OPTIhler, E. (1991) ( 1 + ln(k/2) ) k
0.5 OPTBateman, C. D. et. al. (1997) O(k ), for any >0 Helvig,
C. S. et. al. (2001) 2/1/2016Version 09/0955 MLC Problem: Hierarchy
MLC MLC-R TRA-MLC TRA-MLC-R p-MLC-R MLC n -fatness Rooms Problem
Geometric Clustering Problem 2/1/2016Version 09/0956 Introduction:
Hierarchy of the MLC Problem MLC MLC-R TRA-MLC TRA-MLC-R p-MLC-R
MLC n 2/1/2016Version 09/0957 Related Problems: Formal Definition
of the TFNS Problem INPUT: A connected undirected edge-weighted
graph G=(V,E,w), where w:E + is an edge- weight function. OUTPUT: A
tree T=(V,E), where E E, and V V is a feedback node set (i.e. every
cycle in G includes at least one vertex in V) and the total edge-
weight is minimized. 2/1/2016Version 09/0958 Related Problems:
TFNS-based algorithm for MLC (from TVC) A polynomial approximation
algorithm with performance 3.78 [Arkin, E.M. et. al. (1993)]. Uses
the approximation algorithm for the weighted vertex cover and then
uses an approximation algorithm for the Steiner tree problem.
Replace the approximation algorithm for the weighted vertex cover
by a constant ratio approximation algorithm for the weighted FNS
problem [Chudak, F. A. et. al. (1998)]. 2/1/2016Version 09/0959
Related Problems: Solving instances I j of MLC by TFNS-based
algorithm j p 2/1/2016Version 09/0960 S(Rk): selecting k points
randomly, k S(Rk) =k t(I S ) r S t(I) 2/1/2016Version 09/0961
S(k+): selecting k special points, k S(k+) =k j t(I S ) r S t(I)
2/1/2016Version 09/0962 Additional Results and Conclusions:
TRA-MLC,TRA-MLC-R,p-MLC-R,MLC-R problems are NP-complete p-MLC-R S
for S(2OC+),S(4C+) are NP-complete Solution of the p-MLC-R and
MLC-R problem is at most 2k s r s =2*3*5=30 times the optimal
solution. Solution of the MLC n problem is at most 15n-10 times the
optimal solution The approximation ratio is a constant! FFFFF
2/1/2016Version 09/0963 n i-1 njnj njnj njnj VMR HMD ND
Parameterized Approximation Algorithm Alg(S) 2/1/2016Version
09/0964 NP-Completeness: TRA-MLC-R MLC-R MLC-R TRA-MLC-R
2/1/2016Version 09/0965 FFFFF Example
