Problem
Previous work and our results
Quadratic algorithm for lower-intersecting
A 2-approximation for diagonal-intersecting
Bounds on the gap
Conclusions and extra results
Independent and Hitting Sets of Rectangles
Intersecting a Diagonal Line
José R. Correa1 , Laurent Feuilloley2 and José A. Soto3
1 Department
of Industrial Engineering, Universidad de Chile.
of Computer Science, ENS Cachan.
3 Department of Mathematical Engineering and CMM, Universidad de Chile.
2 Department
Monday, March 31, 2014
J.R. Correa, L. Feuilloley and J.A. Soto
Rectangles Intersecting a Diagonal Line
1 / 26
Problem
Previous work and our results
Quadratic algorithm for lower-intersecting
A 2-approximation for diagonal-intersecting
Bounds on the gap
Conclusions and extra results
Description of the problem
Rectangles in the plane
Axis-aligned
J.R. Correa, L. Feuilloley and J.A. Soto
Rectangles Intersecting a Diagonal Line
2 / 26
Problem
Previous work and our results
Quadratic algorithm for lower-intersecting
A 2-approximation for diagonal-intersecting
Bounds on the gap
Conclusions and extra results
Maximum independent set of rectangles (MIS)
A set of rectangles that do not intersect
Maximum cardinal
J.R. Correa, L. Feuilloley and J.A. Soto
Rectangles Intersecting a Diagonal Line
3 / 26
Problem
Previous work and our results
Quadratic algorithm for lower-intersecting
A 2-approximation for diagonal-intersecting
Bounds on the gap
Conclusions and extra results
Duality
The dual problem : minimum hitting set (MHS).
bc
bc
bc
The duality gap is max
MHS
MIS
J.R. Correa, L. Feuilloley and J.A. Soto
Rectangles Intersecting a Diagonal Line
4 / 26
Problem
Previous work and our results
Quadratic algorithm for lower-intersecting
A 2-approximation for diagonal-intersecting
Bounds on the gap
Conclusions and extra results
Rectangles intersecting a Diagonal Line : Several Cases.
diagonal-intersecting
diagonal-lower-intersecting
diagonal-corner-separated
diagonal-top splitting
diagonal-touching
Intersecting
a Diagonal Line
Figure: Examples of Rectangles
rectangle
families.
J.R. Correa, L. Feuilloley and J.A. Soto
5 / 26
Problem
Previous work and our results
Quadratic algorithm for lower-intersecting
A 2-approximation for diagonal-intersecting
Bounds on the gap
Conclusions and extra results
Previous work
MIS is NP-hard (Fowler, Paterson, Tanimoto, 1981)
No constant approximation algorithm is known, but ...
O(log log(n))-approximation in polynomial time
(Chalermsook, Chuzhoy, 2009)
(1 + ǫ)-approximation in time 2polylog (n)/ǫ
(Adamaszek, Wiese, 2013 )
Algorithms for some special cases (e.g. intervals)
Cubic algorithm for WMIS on diagonal-touching (Lubiw 1991)
6-approximation for MIS and MHS on diagonal-intersecting
(Chepoi, Felsner 2013)
J.R. Correa, L. Feuilloley and J.A. Soto
Rectangles Intersecting a Diagonal Line
6 / 26
Problem
Previous work and our results
Quadratic algorithm for lower-intersecting
A 2-approximation for diagonal-intersecting
Bounds on the gap
Conclusions and extra results
Our results
Quadratic algorithm for lower-intersecting
2-approximation for diagonal-intersecting
Bounds on the duality gap for diagonal-intersecting : between
2 and 4 (then a 4-approximation for MHS).
J.R. Correa, L. Feuilloley and J.A. Soto
Rectangles Intersecting a Diagonal Line
7 / 26
Problem
Previous work and our results
Quadratic algorithm for lower-intersecting
A 2-approximation for diagonal-intersecting
Bounds on the gap
Conclusions and extra results
Quadratic algorithm for lower-intersecting
Dynamic programming. Solve smaller instances : rectangles
that are in the same geometric shapes.
Good shapes : harpoons.
i
i
j
J.R. Correa, L. Feuilloley and J.A. Soto
j
Rectangles Intersecting a Diagonal Line
8 / 26
Problem
Previous work and our results
Quadratic algorithm for lower-intersecting
A 2-approximation for diagonal-intersecting
Bounds on the gap
Conclusions and extra results
Dynamic program for lower-intersecting
Dynamic Program to compute S(i , j) = WMIS on harpoon(i,j)
i
j
J.R. Correa, L. Feuilloley and J.A. Soto
Rectangles Intersecting a Diagonal Line
9 / 26
Problem
Previous work and our results
Quadratic algorithm for lower-intersecting
A 2-approximation for diagonal-intersecting
Bounds on the gap
Conclusions and extra results
Dynamic Program for lower-intersecting
The rectangle just above j define a strip.
i
j
J.R. Correa, L. Feuilloley and J.A. Soto
Rectangles Intersecting a Diagonal Line
10 / 26
Problem
Previous work and our results
Quadratic algorithm for lower-intersecting
A 2-approximation for diagonal-intersecting
Bounds on the gap
Conclusions and extra results
Dynamic Program for lower-intersecting
The rectangle just above j define a strip.
i
j
J.R. Correa, L. Feuilloley and J.A. Soto
Rectangles Intersecting a Diagonal Line
11 / 26
Problem
Previous work and our results
Quadratic algorithm for lower-intersecting
A 2-approximation for diagonal-intersecting
Bounds on the gap
Conclusions and extra results
Dynamic Program for lower-intersecting
Consider the rectangles in the harpoon (i , j), with a lower-left
corner in the strip.
i
k
j
J.R. Correa, L. Feuilloley and J.A. Soto
Rectangles Intersecting a Diagonal Line
12 / 26
Problem
Previous work and our results
Quadratic algorithm for lower-intersecting
A 2-approximation for diagonal-intersecting
Bounds on the gap
Conclusions and extra results
Dynamic Program for lower-intersecting
For k on the strip, the WMIS of the harpoon containing k is
mk = wk + S(i , k) + S(k, j).
i
k
j
J.R. Correa, L. Feuilloley and J.A. Soto
Rectangles Intersecting a Diagonal Line
13 / 26
Problem
Previous work and our results
Quadratic algorithm for lower-intersecting
A 2-approximation for diagonal-intersecting
Bounds on the gap
Conclusions and extra results
Dynamic Program for lower-intersecting
Actually : For k on the strip, the WMIS of the harpoon containing
k is mk = wk + max{S(i , k), S(k, i )} + S(k, j).
i
k
j
The WMIS for the harpoon is maxk∈strip mk .
J.R. Correa, L. Feuilloley and J.A. Soto
Rectangles Intersecting a Diagonal Line
14 / 26
Problem
Previous work and our results
Quadratic algorithm for lower-intersecting
A 2-approximation for diagonal-intersecting
Bounds on the gap
Conclusions and extra results
Complexity
Constant amortized cost by harpoon
A quadratic number of harpoons
Quadratic cost.
J.R. Correa, L. Feuilloley and J.A. Soto
Rectangles Intersecting a Diagonal Line
15 / 26
Problem
Previous work and our results
Quadratic algorithm for lower-intersecting
A 2-approximation for diagonal-intersecting
Bounds on the gap
Conclusions and extra results
A 2-approximation for diagonal-intersecting
Divide the rectangles into two sets.
J.R. Correa, L. Feuilloley and J.A. Soto
Rectangles Intersecting a Diagonal Line
16 / 26
Problem
Previous work and our results
Quadratic algorithm for lower-intersecting
A 2-approximation for diagonal-intersecting
Bounds on the gap
Conclusions and extra results
A 2-approximation for diagonal-intersecting
Divide the rectangles into two sets : the ones that touch the
diagonal with the upper side and the others.
J.R. Correa, L. Feuilloley and J.A. Soto
Rectangles Intersecting a Diagonal Line
17 / 26
Problem
Previous work and our results
Quadratic algorithm for lower-intersecting
A 2-approximation for diagonal-intersecting
Bounds on the gap
Conclusions and extra results
A 2-approximation for diagonal-intersecting
Compute the WMIS for the two sets, and output the max : a two
approximation in polynomial time.
J.R. Correa, L. Feuilloley and J.A. Soto
Rectangles Intersecting a Diagonal Line
18 / 26
Problem
Previous work and our results
Quadratic algorithm for lower-intersecting
A 2-approximation for diagonal-intersecting
Bounds on the gap
Conclusions and extra results
Duality gap
Projections on the axis : two sets of intervals.
J.R. Correa, L. Feuilloley and J.A. Soto
Rectangles Intersecting a Diagonal Line
19 / 26
Problem
Previous work and our results
Quadratic algorithm for lower-intersecting
A 2-approximation for diagonal-intersecting
Bounds on the gap
Conclusions and extra results
Upper bound on the gap
Projections on the axis.
J.R. Correa, L. Feuilloley and J.A. Soto
Rectangles Intersecting a Diagonal Line
20 / 26
Problem
Previous work and our results
Quadratic algorithm for lower-intersecting
A 2-approximation for diagonal-intersecting
Bounds on the gap
Conclusions and extra results
Upper bound on the gap
Compute the MISs and MHSs on the projections.
max(MISx , MISy ) gives an independent set for the rectangles.
J.R. Correa, L. Feuilloley and J.A. Soto
Rectangles Intersecting a Diagonal Line
21 / 26
Problem
Previous work and our results
Quadratic algorithm for lower-intersecting
A 2-approximation for diagonal-intersecting
Bounds on the gap
Conclusions and extra results
Upper bound on the gap
Compute the MISs and MHSs on the projections.
MHSx × MHSy gives a hitting set for the rectangles.
bc
bc
bc
bc
J.R. Correa, L. Feuilloley and J.A. Soto
Rectangles Intersecting a Diagonal Line
22 / 26
Problem
Previous work and our results
Quadratic algorithm for lower-intersecting
A 2-approximation for diagonal-intersecting
Bounds on the gap
Conclusions and extra results
Refining the hitting set
In the general case : a grid.
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
J.R. Correa, L. Feuilloley and J.A. Soto
Rectangles Intersecting a Diagonal Line
23 / 26
Problem
Previous work and our results
Quadratic algorithm for lower-intersecting
A 2-approximation for diagonal-intersecting
Bounds on the gap
Conclusions and extra results
Refining the hitting set
But we have a diagonal :
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
bc
Hitting set of size at most 2(MHSx + MHSy ).
For intervals MISx = MHSx and MISy = MHSy .
We get upper bound of 4 on duality gap.
J.R. Correa, L. Feuilloley and J.A. Soto
Rectangles Intersecting a Diagonal Line
24 / 26
Problem
Previous work and our results
Quadratic algorithm for lower-intersecting
A 2-approximation for diagonal-intersecting
Bounds on the gap
Conclusions and extra results
Extra results
Approximations for MIS on the other cases.
A 4-approximation for MHS on diagonal-intersecting.
A lower bound of 2 for duality gap.
After submission (with Pablo Pérez-Lantero) :
MIS on diagonal-intersecting is NP-hard.
J.R. Correa, L. Feuilloley and J.A. Soto
Rectangles Intersecting a Diagonal Line
25 / 26
Problem
Previous work and our results
Quadratic algorithm for lower-intersecting
A 2-approximation for diagonal-intersecting
Bounds on the gap
Conclusions and extra results
Thank you !
J.R. Correa, L. Feuilloley and J.A. Soto
Rectangles Intersecting a Diagonal Line
26 / 26