SLIDE 1
Cheat Sheets for Hard Problems Some problems tend to be harder than - - PowerPoint PPT Presentation
Cheat Sheets for Hard Problems Some problems tend to be harder than - - PowerPoint PPT Presentation
Cheat Sheets for Hard Problems Some problems tend to be harder than others. NP NP P NP P NP-complete NP P X solveTSP{ blah blah blah blah blah } X TSP solveTSP{ blah blah blah blah blah } solveTSP{ blah blah X
SLIDE 2
SLIDE 3
Some problems tend to be harder than others.
SLIDE 4
NP
SLIDE 5
P NP
SLIDE 6
P NP NP-complete
SLIDE 7
P NP
SLIDE 8
X solveTSP{ blah blah blah blah blah }
SLIDE 9
TSP X solveTSP{ blah blah blah blah blah }
SLIDE 10
TSP X solveTSP{ blah blah blah blah blah }
SLIDE 11
SLIDE 12
Independent Set
SLIDE 13
Independent Set Clique
SLIDE 14
Independent Set Clique
Independent Set Clique
SLIDE 15
Independent Set Clique
SLIDE 16
Independent Set Clique
SLIDE 17
Independent Set Clique SolveIndSet { Return Clique( ); }
G
SLIDE 18
TSP X solveTSP{ blah blah blah blah blah }
SLIDE 19
Did you say NP-complete?
SLIDE 20
Did you say NP-complete?
SLIDE 21
Travelling Salesman Satisfiability Integer Linear Programming Minimum Vertex Cover
SLIDE 22
Minimum Multi-way cut ....
SLIDE 23
SLIDE 24
Heuristics
SLIDE 25
Heuristics
SLIDE 26
Formal analysis?
SLIDE 27
You have Polynomial Time.
SLIDE 28
You have Polynomial Time. WORK BACKWARDS!
SLIDE 29
Approximation & Randomized Algorithms
SLIDE 30
A no-compromise situation?
SLIDE 31
A no-compromise situation?
SLIDE 32
A no-compromise situation? Exploit additional structure in the input.
SLIDE 33
Parameterized & Exact Analysis
SLIDE 34
Parameterized & Exact Analysis
Chromatic ¡Number ¡is ¡easy ¡on ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡Graphs. Interval
SLIDE 35
Parameterized & Exact Analysis
Chromatic ¡Number ¡is ¡easy ¡on ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡Graphs. Planar*
SLIDE 36
Parameterized & Exact Analysis
Chromatic ¡Number ¡is ¡easy ¡on ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡Graphs. Bipartite
SLIDE 37
Good solutions tend to involve a combination of several techniques.
SLIDE 38
Vertex Cover
SLIDE 39
SLIDE 40
SLIDE 41
SLIDE 42
SLIDE 43
Vertex Cover
Every edge has at least one end point in a vertex cover.
SLIDE 44
Vertex Cover
Every edge has at least one end point in a vertex cover.
SLIDE 45
SLIDE 46
SLIDE 47
SLIDE 48
SLIDE 49
SLIDE 50
SLIDE 51
SLIDE 52
SLIDE 53
SLIDE 54
Is there a Vertex Cover with at most k vertices?
SLIDE 55
A vertex with more than k neighbors.
SLIDE 56
Throw away all vertices with degree (k+1) or more. (And decrease the budget appropriately.)
SLIDE 57
Throw away all vertices with degree (k+1) or more. After all the high-degree vertices are gone... (And decrease the budget appropriately.)
SLIDE 58
...any vertex can cover at most k edges.
SLIDE 59
...any vertex can cover at most k edges.
Suppose the current budget is (k-x).
SLIDE 60
...any vertex can cover at most k edges.
Suppose the current budget is (k-x).
If the number of edges in the graph exceeds k(k-x)...?
SLIDE 61
Lots of edges - no small vertex cover possible. Few edges - brute force becomes feasible.
SLIDE 62
Lots of edges - no small vertex cover possible. Few edges - brute force becomes feasible. win/ win situation
SLIDE 63
SLIDE 64
SLIDE 65
Common Sense
SLIDE 66
Common Sense Approximate
SLIDE 67
Common Sense Approximate Randomize
SLIDE 68
Common Sense Approximate Randomize Exploit Input Structure
SLIDE 69
Common Sense Approximate Randomize Exploit Input Structure
SLIDE 70