Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.
-
Upload
victor-elliott -
Category
Documents
-
view
216 -
download
1
Transcript of Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.
![Page 1: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/1.jpg)
Two Dimension Measures: A New Algorithimic Method for Solving
NP-Hard Problems
Yang Liu
![Page 2: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/2.jpg)
Outlines
Why Exact AlgorithmsWhy Fpt-algorithmsAlgorithmic techniques: branching search
tree and two dimension braching search tree
Previous worksFuture research
![Page 3: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/3.jpg)
Why NP-Hard Problems
Abundant aplications
Methods of tackling NP-hard problems:
1. Exact algorithms.
2. parameterized algorithms
3. Approximation algorithms
4. Heuristics
![Page 4: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/4.jpg)
Why Exact Algorithms
OptimalCorrect and meaningful for decision
problemsSubroutine for other methodsLimitation of the improvements of
processors
![Page 5: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/5.jpg)
Practical Examples (1)
The Rectlinear Steiner Tree problem Inputs of size up to 1000 Solved in 38 CPU hours.
![Page 6: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/6.jpg)
![Page 7: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/7.jpg)
Practical Examples (2)
The TSP problem (http://www.tsp.gatech.edu/sweden/)
German 15,112 towns
In 2001 22.6 cpu years
Sweden 24,978
Towns
In 2004 72,5000 km
Circuit 33,810 In 2005 66,048,945 units, 15.7 cpu years
Circuit 85,900 In 2006 136 cpu years
![Page 8: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/8.jpg)
Outlines
Why Exact AlgorithmsWhy Fpt-algorithmsAlgorithmic technique: branching search
tree and two dimensional braching search tree
Previous worksProposed future research
![Page 9: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/9.jpg)
Why FPT-algorithms
Still hard to design practical exact algorithms in general
An application may have a parameter k, small compared with input size n.
This parameter can give a better bound on time complexity, like O(f(k)nc).
![Page 10: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/10.jpg)
Practical Fpt-algorithms (1)
Type check: checking the compatibility of type declarations.
Parameter k: the maximum nesting depth of the type declarations.
Normally K<=6 Practical fpt-algorithm of time O(2kn)
![Page 11: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/11.jpg)
Practical Fpt-algorithms (2)
Individual halotyping problemParamters k1 and k2
K1<=n, but normally k1<=10
Usually k2<=19
Practical fpt-algorithm of complexity O(nk22k2+mlogm+mk1)
![Page 12: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/12.jpg)
Outlines
Why Exact AlgorithmsWhy Fpt-algorithmsAlgorithmic techniques: branching search
tree and two dimension braching search tree
Previous worksProposed future research
![Page 13: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/13.jpg)
Algorithmic Techniques
Kernelization, greedy localization, iterative compression, coloring coding, divide cand conquer, divide and coloring, branching search tree
Dynamic programming, prunning the search tree, preprocess the data, local search, measure and conquer
![Page 14: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/14.jpg)
Branching Search Tree
m
m2m1mp
m’m
mmmm
p
mf
xx
mfmfmf
p
)(
1
)()()()()(
1
1
m: measure
f(m): number of subproblems (poly-time solvable)
![Page 15: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/15.jpg)
Time Complexity of Branching Search Tree
Complexity: O(cmpoly(n))=O*(cm).
O*(cn) refers to O(cnpoly(n)).
![Page 16: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/16.jpg)
Example: Exact Algorithm
Finding Minimum Vertex Cover CPick an edge xy, let measure m=#vertices
m=n
m1=n-1 mp=n-2x in C
x not in C
y in C
mmfxx
mfmfmf
62.1)(1
)2()1()(21
![Page 17: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/17.jpg)
Example: fpt-Algorithm
Finding a vertex cover C of k vertices.Pick an edge xy, let measure m=#vertices needs to
be in C
m
m1=m-1 mp=m-1x in C
x not in C
y in C
mmfxx
mfmfmf
2)(1
)1()1()(11
![Page 18: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/18.jpg)
Weakness of Branching Search Tree
Only takes one measureFinding a feedback vertex set fvs of k
verticesChoose a cycle and one vertex must be in
fvs.
)()()1()1()( * kpOkfkfkpfkf
![Page 19: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/19.jpg)
Two Dimensional Braching Search Tree
Challenges:Finding two measures p,qhow to bound f(p,q)Boundary conditions: new combinatorial
properties
Acheivements:faster algorithms for some problems.
![Page 20: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/20.jpg)
Outlines
Why Exact AlgorithmsWhy Fpt-algorithmsAlgorithmic technique: branching search
tree and two dimensional braching search tree
Previous worksProposed future research
![Page 21: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/21.jpg)
3D Matching & 3-Set Packing
a1
a2
an
b1
b2
bn
c1
c2
cn
Symbols: ai, bi, ci
a3 b3c3
.
.
.
.
.
.
.
.
.
triples
Set U of symbols, triple t=<a1,a2,a3>
t1=<a1,a2,a3> and t2=<b1,b2,b3> conflicts if ai=bi for some i
A matching: set of mutually non-conflicting triples.
The maximum matching problem In the Karp’s list of NP-complete
problems Generalization of graph matching
![Page 22: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/22.jpg)
3D Matching & Packing
Finding a matching of k triples
Reference Random Determ
Downey et al. O*((3k)!(3k)9k+1)
Chen et al. O*((5.7k)k)
Koutis O*(10.883k) >O*(320003k)
Fellows et al. >O*(1263k)
Kneis O*(2.523k) O*(163k)
Chen et al. O*(2.523k) O*(12.83k)
Our result O*(2.323k) O*(2.773k)
![Page 23: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/23.jpg)
3D Matching & Packing
For the k-packing problem, most algorithms in the table above apply.
Our algorithm has complexity O*(4.613k)
Remark: Koutis gave a randomized algorithm of time O*(23k) for both k-matching and k-packing problems
![Page 24: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/24.jpg)
Multiway Cut
![Page 25: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/25.jpg)
Multiway Cutterminals
![Page 26: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/26.jpg)
Multiway Cutterminals separator
![Page 27: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/27.jpg)
Multiway Cut
Applications: distributed computing, VLSI, computer vision and more
Finding a separator of k vertices
An algorithm of time O*(4k)
the previous best algorithm of time )4(3* kO
![Page 28: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/28.jpg)
Feedback Vertex Set(undirected graphs)
![Page 29: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/29.jpg)
Feedback Vertex Set (undirected graphs)
Feedback vertex set Remaining graph is a forest
![Page 30: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/30.jpg)
Finding an FVS of k vertices (undirected graphs)
Reference Complexity
Bodlaender, Fellows O*(17(k4)!)
Dowey and Fellows O*((2k+1)k)
Raman et al. O*(max{12k,(4logk)k})
Kanj et al. O*((2logk+2loglogk+18)k)
Raman et al. O*((12logk/loglogk+6)k)
Guo et al. O*(37.7k)
Dehne et al. O*(10.6k)
Our results O*(5k)
![Page 31: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/31.jpg)
Feedback Vertex Set(directed graphs)
![Page 32: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/32.jpg)
Feedback Vertex Set(directed graphs)
Feedback vertex set Remaining graph is a DAG
![Page 33: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/33.jpg)
Finding an FVS of k vertices (directed graphs)Question: fpt or not? Had been an open problem for over 15
years
We developed an algorithm of time O*(4kk!) with our new approach
![Page 34: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/34.jpg)
Finding an FVS of k vertices (undirected graphs)
Reference Complexity
Bodlaender, Fellows O*(17(k4)!)
Dowey and Fellows O*((2k+1)k)
Raman et al. O*(max{12k,(4logk)k})
Kanj et al. O*((2logk+2loglogk+18)k)
Raman et al. O*((12logk/loglogk+6)k)
Guo et al. O*(37.7k)
Dehne et al. O*(10.6k)
Our results O*(5k)
![Page 35: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/35.jpg)
Finding an FVS of k vertices (directed graphs)
Question: fpt or not? Had been open for over 15 years
We developed an algorithm of time O*(4kk!)
Accepted by STOC 2008, JACM
![Page 36: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/36.jpg)
Max Leaf(undirected graphs)
Finding a Spanning tree With at least k leaves
• Equivalent to the minimum connected dominating set problem• Applications: design of communication networks, circuit layouts, and distributed systems.
![Page 37: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/37.jpg)
Finding a spanning tree with k leaves
Reference Complexity
Bodlaender O*((17k4)!)
Downey and Fellows O*((2k)4k)
Fellows et al. O*(14.23k)
Bonsma et al. O*(9.49k)
Bonsma and Zickfeld O*(6.75k)
Our result O*(4k)
![Page 38: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/38.jpg)
Out-branchingOut-branching: a rooted tree
such that there is a unique path
from the root to a leaf.
![Page 39: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/39.jpg)
Finding an out-branching with k leavesMore difficult
Graph minor theory on digraphs is not
mature enough to solve this problemProperties are harder to prove for
digraphs than for undirected graphs
Shown to be fpt only recently
![Page 40: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/40.jpg)
Finding an out-branching with k leaves for SCC and DAG for SCC and for DAG for digraphs for digraphs
Our results: O*(4k)
Remark: Kenis et al. developed similar algorithm.
)2( )log(* 2 kkOO
)2( )log(* 2 kkOO )2( )log(* kkOO
)2( )log(* 3 kkOO
)2( )log(* kkOO
![Page 41: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/41.jpg)
Max Leaf
Finding a spanning tree of maximum leaves
Equal to minimum connected dominating set
Applications: design of communication networks, circuit layouts, and distributed systems.
![Page 42: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/42.jpg)
Max Leaf on Digraphs
Out-branching: a rooted tree such that there is a unique path from the root to a leaf.
Max Leaf: finding an out-branching with maximum leaves
Approximable within )( nO
![Page 43: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/43.jpg)
Finding an out-branching with k leaves
More difficult
Graph minor theory on digraphs is not
mature enough to solve this problemSome property is harder to prove for
digraphs than for undirected graphs
Shown to be fpt 15 years later
![Page 44: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/44.jpg)
Finding an out-branching with k leaves
for SCC and DAG for SCC and for
DAG for digraphs for digraphs
Our results: O*(4k)
)2( )log(* 2 kkOO
)2( )log(* 2 kkOO )2( )log(* kkOO
)2( )log(* 3 kkOO
)2( )log(* kkOO
![Page 45: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/45.jpg)
Outlines
Why Exact AlgorithmsWhy Fpt-algorithmsAlgorithmic technique: branching search
tree and two dimensional braching search tree
Previous worksProposed future research
![Page 46: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/46.jpg)
Multiway Cut
Can we improve the algorithm of O*(4k)?Is there any faster or simpler randomized
algorithm?Is there any algebraic algorithms for this
problem?Is the multi-cut problem fpt?
![Page 47: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/47.jpg)
Feedback Vertex Set
Can we improve the current best algorithms?
Is there randomized algorithm better than O*(4k)?
Is there a deterministic algorithm of O*(4k) for the problem on undirected graphs?
Is there O*(ck) for the problem on digraphs?
Is there any algebraic algorithm?
![Page 48: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/48.jpg)
Max Leaf
Is there any better algorithm?Is there any better algorithm for the
problem on undirected graphs?Is there any faster or simple randomized
algorithm?Is there any algebraic algorithm?
![Page 49: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/49.jpg)
Two Dimension Measures: Further Study
Algorithms by this technique has complexity around O*(4k)
Can we do better? How?
Can we apply this or similar technique to design exact algorithms?
![Page 50: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/50.jpg)
Conclusion
Two dimension measures branching search tree is powerful in design fpt-algorithms.
Need improvement on this technique to design algorithms faster than O*(4k).
Generalize this technique to design exact algorithms.
![Page 51: Two Dimension Measures: A New Algorithimic Method for Solving NP-Hard Problems Yang Liu.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfca1a28abf838ca9683/html5/thumbnails/51.jpg)
Questions?
Thank You!