Matroids & Representative Sets Daniel Lokshtanov.
-
Upload
anis-gardner -
Category
Documents
-
view
223 -
download
0
Transcript of Matroids & Representative Sets Daniel Lokshtanov.
![Page 1: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/1.jpg)
Matroids & Representative Sets
Daniel Lokshtanov
![Page 2: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/2.jpg)
Alice vs Bob
F = {{a,b,c}, {a,c,d}, {b,c,e}}
{b, e}
{a,c,d}
{a, c}
![Page 3: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/3.jpg)
Rules of the game
Board: universe of size nAll Alice’s sets have size pBob a picks set B of size qAlice wins if she has a set disjoint from B
![Page 4: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/4.jpg)
Lazy Alice
Alice does not like remembering all those sets.Alice hates losing to Bob.
Can she forget a set A from F, and be sure this will not make the difference
between winning and losing?
![Page 5: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/5.jpg)
(Ir)relevant Sets
A F is irrelevant if: for every set B of size q such that A B = , there is a set A’ F such that A’ B = .
Alice may forget exactly the irrelevant sets
![Page 6: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/6.jpg)
Only relevant sets?
B1 B2 B3 Bm
F = { A2A1 A3 Am }…
![Page 7: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/7.jpg)
Bollobás’ Lemma [1966]
Let A1, A2, … , Am be sets of size p and
B1, B2, … , Bm be sets of size q s.t:
then .
No dependence on universe size n at all!
![Page 8: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/8.jpg)
Bollobás’ helps Alice
Bollobás’ lemma immediately implies that Alice only needs to remember at most sets.
yay!
![Page 9: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/9.jpg)
Proof of Bollobás’ Lemma
Consider a random permutation of the universe.
The events «all of Ai before all of Bi» and «All of Aj before all of Bj» are disjoint!
So and hence .
P[all of Ai is before all of Bi] = .Ai BiBj Aj
![Page 10: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/10.jpg)
Representative Sets
Let F be a family of p-sets. Then q-represents F if for every B of size q such that there exists an with there exists an with .
Corollary of Bollobás: For every F there is an of size at most that q-represents F.
![Page 11: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/11.jpg)
Computational Problem
Given a family F of p-sets and an integer q, compute a family of size at most that q-represents F.
![Page 12: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/12.jpg)
Computing Representative Sets
Will show: we can compute representative sets in time essentially where is the matrix multiplication constant < 2.38.
But first – an easy application
![Page 13: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/13.jpg)
d-Hitting Set
Input: Family F = {S1,…,Sm} of sets of size d over universe U = {v1, …, vn}, integer k.
Question: Does there exist a set X U of size at most k such that for every Si F, Si X ?
Easy branching in time dk
Next: kernel with O(kd) sets and elements
![Page 14: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/14.jpg)
d-Hitting Set as a Game
F = {{a,b,c}, {a,c,d}, {b,c,e}} Is {b, e} a hitting set?
No, since
{a,c,d}
![Page 15: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/15.jpg)
Kernel for d-Hitting Set
Compute a k-representative subfamily F’ F of size at most .
Remove all elements not in F’ (at most dkd)
Output the instance F’, k.
![Page 16: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/16.jpg)
Why is the kernel correct?
May not change a YES instance into a NO instance.
Can a NO instance change into a YES instance?NO instance = Alice always wins
YES instance = Bob can win
We did not forget any sets that made the difference between Alice winning and losing!
![Page 17: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/17.jpg)
Playing on a matroid
Suppose now that the universe is the edge set of a matroid.
A set A fits a set B if - A and B are disjoint and- A B is independent in the matroid.
![Page 18: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/18.jpg)
Alice vs Bob on a matroid
F = {{a,b,c}, {a,c,d}, {b,c,e}}
Do you have a set
that fits {b, e}?
&%¤&!!
Note: this game on a uniform matroid of rank p+q is exactly the old game.
![Page 19: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/19.jpg)
Representative Sets
Let F be a family of p-sets (in a matroid M). Then q-represents F if:
for every B of size q such thatthere exists an that fits B there exists an that also fits B.
Note: representation in a uniform matroid of rank p+q is exactly the old representation.
![Page 20: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/20.jpg)
Computing Representative Sets
Input: Family F of p-sets over a matroid, integer q, matrix M representing the matroid.
Task: Compute a q-representative subfamily of size at most .
![Page 21: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/21.jpg)
Playing on a matroidp=4, q=2
3049580290385029238402938502309058010120958303215203852923023023109580420935820230395283032023350203220225822202302350203209802104+4267429810983502239582820320502340958683040938323035802092309532029385308209821522998208357298739829872398253982359823987235239729019380205230958203958293958203958203958203958522938572938575292
M = p+q
F ?!
232401018920320110848338053002
Det
![Page 22: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/22.jpg)
Fit vs Determinant
If Alices set A and Bob’s set B overlap, then the same column is used twice determinant is 0!
Determinant is nonzero if and only if A fits B.
![Page 23: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/23.jpg)
Matrix game
ab
cd p+q
p
p+q
q
c
![Page 24: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/24.jpg)
Generalized Laplace Expansionalmost correct
MBMA p+q
p q
To compute Det
Compute the determinants of all
p p submatrices of MA
Compute the determinants of all
q q submatrices of MB
dimensional vector vA
dimensional vector vB
dot product!*
![Page 25: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/25.jpg)
Giant Vector game
a (𝑝+𝑞𝑝 )b c
d
c
(𝑝+𝑞𝑝 )
⋅ 0 ?
![Page 26: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/26.jpg)
Basis
If Alice keeps vectors v1,v2,v3 and v3 = v1 + v2
and v3 fits Bob’s vector vB
Then either v1 or v2 fits vB Alice only needs to keep linearly independent vectors!
At most of them, sincevectors are - dimensional
Finding the basis takes time.
![Page 27: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/27.jpg)
Wrap up
Alice has a family of p-sets, family of p (p+q) matricesfamily of - dimensonal vectors.
Keep linearly independent vectors, keep the corresponding sets!
![Page 28: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/28.jpg)
Computing Representative Sets
Theorem: we can compute representative sets of size in time essentially where is the matrix multiplication constant < 2.38.
![Page 29: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/29.jpg)
Application - Treewidth DP
Have seen several approaches for single exponential algorithms for connectivity problems parameterized by treewidth.
Representative sets gives yet another one
![Page 30: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/30.jpg)
Hamiltonian Path
![Page 31: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/31.jpg)
Representative Sets for Matroid Classes
Is it possible to compute representative sets for uniform matroids, graphic matroids or transversal matroids faster than for linear matroids in general?
For uniform matroids, the answer is yes(but proof is sort of complicated)
![Page 32: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/32.jpg)
Application – k-Path
Input: (directed) graph G, integer k.Question: Is there a simple directed path on k vertices?
Theorem: There is a deterministic time algorithm for k-Path.
![Page 33: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/33.jpg)
k-Path
Fix a source vertex u.
For vertex v and integer p, define P[v,p] to be the set of (vertex sets of) paths on exactly p vertices from u to v.
Goal: for every v and p k compute a set P’[v,p] that (k-p)-represents P[v,p].
![Page 34: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/34.jpg)
k-PathGoal: for every v, p k compute a set P’[v,p] that (k-p)-represents P[v,p].
X [v , p ]=¿wv∈ E(G)P′ [w , p−1 ] ⋅ \{v }P′ [u ,1 ]= {{u } }
P′ [v , p ]=reduce(X [v , p ] , k − p)
Need to prove: that (k-p)-represents P[v,p] assuming (k-p+1)-represents P[w,p-1]
Extend all paths that can beextended by v
![Page 35: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/35.jpg)
Need to prove: that (k-p)-represents P[v,p] assuming (k-p+1)-represents P[w,p-1]
BBu
wv
Size q+1
In
In
![Page 36: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/36.jpg)
k-Path
Size of family to reduce:
Time to reduce:
Total time:
Also works for weighted k-Path.
( 𝑘𝑝−1)2𝑜(𝑘) ⋅n
![Page 37: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/37.jpg)
Application – k-Cycle
Input: (directed) graph G, integer k.Question: Is there a simple directed cycle on at least k vertices?
Theorem: 8kpoly(n) algorithm.
![Page 38: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/38.jpg)
k-Cycle – main lemma
In a shortest cycle C on at least k vertices, we can replace any subpath on k vertices by any other path on k vertices, which is disjoint from the k vertices after it on C.
![Page 39: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/39.jpg)
k-Cycle – main lemma proof
Puv
Pvw
Pwv
v
u
w
![Page 40: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/40.jpg)
k-Cycle – algorithm
Guess a vertex u that a shortest cycle C of length at least k passes through.
For every vertex v and integer p, define P[u, p] to be the set of (vertex sets of) paths on exactly p vertices from u to v.
For every vertex v compute a set P’[v] that k-represents P[v,k] using the method from the k-path algorithm.
Size:
![Page 41: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/41.jpg)
k-Cycle – algorithm
Guess the k’th vertex v on the cycle C.
Main lemma there is a k-cycle containing a path from P’[v] as a subpath!
Check whether there is a path Q in P’[v] such that there is a path back from v to u in G\Q.
Time 8kpoly(n).
![Page 42: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/42.jpg)
Speeding up
Using similar methods, but trading off space for time one can speed up k-Path to 2.619k and k-Cycle to 6.75k.
![Page 43: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/43.jpg)
Exercises
Book: 12.9, 12.11, 12.13, 5.9
![Page 44: Matroids & Representative Sets Daniel Lokshtanov.](https://reader035.fdocuments.net/reader035/viewer/2022062421/56649d755503460f94a550f6/html5/thumbnails/44.jpg)
Thank You!