NCG Group
New Results and Open Problems
PoA for NSP
Simulator
NP-completeness of SC
Convergence speed of strong-PNSP
Open Problems
Table of Contents
1
PoA for NSP
2
Simulator
3
NP-completeness of SC
4
Convergence speed of strong-PNSP
5
Open Problems
2 / 21
PoA for NSP
Simulator
NP-completeness of SC
Convergence speed of strong-PNSP
Open Problems
PoA for NSP
3 / 21
PoA for NSP
Simulator
NP-completeness of SC
Convergence speed of strong-PNSP
Open Problems
PoA for NSP
Lemma
The PoA for the NSP is in Θ(diam(G )), even when only reachable
friendship situations are considered.
4 / 21
PoA for NSP
Simulator
NP-completeness of SC
Convergence speed of strong-PNSP
Open Problems
Average cost function
Upper bound:
X X
X
2|F | ≤
dG (u, v ) = 2
dG (u, v ) ≤ 2|F | diam(G )
v ∈V u∈F (v )
{u,v }∈F
⇒ worst-case PoA ≤ diam(G )
1
n−3
1
Lower bound:
reachable friendship example ⇒ PoA = Θ(diam(G ))
5 / 21
PoA for NSP
Simulator
NP-completeness of SC
Convergence speed of strong-PNSP
Open Problems
Average cost function
Upper bound:
X X
X
2|F | ≤
dG (u, v ) = 2
dG (u, v ) ≤ 2|F | diam(G )
v ∈V u∈F (v )
{u,v }∈F
⇒ worst-case PoA ≤ diam(G )
1
n−3
1
Lower bound:
reachable friendship example ⇒ PoA = Θ(diam(G ))
5 / 21
PoA for NSP
Simulator
NP-completeness of SC
Convergence speed of strong-PNSP
Open Problems
Maximum cost function
Upper bound:
|VF | ≤
X
v ∈V
max dG (u, v ) ≤ |VF | diam(G )
u∈F (v )
with VF := {v ∈ V | F (v ) 6= ∅}
⇒ worst-case PoA ≤ diam(G )
Lower bound:
n
n
n
non-reachable friendship example ⇒ PoA = Θ(diam(G ))
6 / 21
PoA for NSP
Simulator
NP-completeness of SC
Convergence speed of strong-PNSP
Open Problems
Maximum cost function
Upper bound:
|VF | ≤
X
v ∈V
max dG (u, v ) ≤ |VF | diam(G )
u∈F (v )
with VF := {v ∈ V | F (v ) 6= ∅}
⇒ worst-case PoA ≤ diam(G )
Lower bound:
n
n
n
non-reachable friendship example ⇒ PoA = Θ(diam(G ))
6 / 21
PoA for NSP
Simulator
NP-completeness of SC
Convergence speed of strong-PNSP
Open Problems
Maximum cost function
Reachable friendship example?
7 / 21
PoA for NSP
Simulator
NP-completeness of SC
Convergence speed of strong-PNSP
Open Problems
Maximum cost function
Graph without friendships:
(always case 1)
k−
1
1
k−
1
k−
1
1
k−
1
1
1
1
k−
1
8 / 21
PoA for NSP
Simulator
NP-completeness of SC
Convergence speed of strong-PNSP
Open Problems
Maximum cost function
Case 2:
9 / 21
PoA for NSP
Simulator
NP-completeness of SC
Convergence speed of strong-PNSP
Open Problems
Maximum cost function
Friendships:
13
12
8
7
3
2
4
1
9
14
6
5
11
10
15
10 / 21
PoA for NSP
Simulator
NP-completeness of SC
Convergence speed of strong-PNSP
Open Problems
Maximum cost function
k−
1
1
k−
1
k−
1
1
k−
1
1
1
1
k−
1
reachable friendship example with:
Social costs of NE: Θ(k 2 )
Social costs of optimum: Θ(k)
⇒ PoA = Θ(k) = Θ(diam(G ))
11 / 21
PoA for NSP
Simulator
NP-completeness of SC
Convergence speed of strong-PNSP
Open Problems
Simulator
12 / 21
PoA for NSP
Simulator
NP-completeness of SC
Convergence speed of strong-PNSP
Open Problems
NP-completeness of SC
13 / 21
PoA for NSP
Simulator
NP-completeness of SC
Convergence speed of strong-PNSP
Open Problems
Find best NE
We showed: 3-CNF-SAT p SCNE,dec
Ci−1
C1
Ci
2
Atj
Afj
Atm
Atl
Afm
C|ψ|C
Ci+1
2
Afl
5
Aj
A1
C10
0
Ci−1
2
Ci0
A|ψ|A
Al
Am
2
0
Ci+1
0
C|ψ|
C
14 / 21
PoA for NSP
Simulator
NP-completeness of SC
Convergence speed of strong-PNSP
Open Problems
Find best NE
Shortcut fixation:
c
c
c
c
c
v
c
c
15 / 21
PoA for NSP
Simulator
NP-completeness of SC
Convergence speed of strong-PNSP
Open Problems
Find social optimum
3-CNF-SAT reduction does not work
Graph problems: Vertex Cover, Dominating Set, ...
dynamic edges?
Knapsack Problem, Bin Packing, ...
code value v in a graph with size log(v )?
Set Cover: same as 3-CNF-SAT (with few nodes)...
Is it really NP-hard?
Seems related to Minimum Spanning Tree
solvable with greedy strategy?
No obvious formulation as matroid
Counterexample for greedy strategy!
16 / 21
PoA for NSP
Simulator
NP-completeness of SC
Convergence speed of strong-PNSP
Open Problems
Find social optimum
3-CNF-SAT reduction does not work
Graph problems: Vertex Cover, Dominating Set, ...
dynamic edges?
Knapsack Problem, Bin Packing, ...
code value v in a graph with size log(v )?
Set Cover: same as 3-CNF-SAT (with few nodes)...
Is it really NP-hard?
Seems related to Minimum Spanning Tree
solvable with greedy strategy?
No obvious formulation as matroid
Counterexample for greedy strategy!
16 / 21
PoA for NSP
Simulator
NP-completeness of SC
Convergence speed of strong-PNSP
Open Problems
Find social optimum
3-CNF-SAT reduction does not work
Graph problems: Vertex Cover, Dominating Set, ...
dynamic edges?
Knapsack Problem, Bin Packing, ...
code value v in a graph with size log(v )?
Set Cover: same as 3-CNF-SAT (with few nodes)...
Is it really NP-hard?
Seems related to Minimum Spanning Tree
solvable with greedy strategy?
No obvious formulation as matroid
Counterexample for greedy strategy!
16 / 21
PoA for NSP
Simulator
NP-completeness of SC
Convergence speed of strong-PNSP
Open Problems
Find social optimum
3-CNF-SAT reduction does not work
Graph problems: Vertex Cover, Dominating Set, ...
dynamic edges?
Knapsack Problem, Bin Packing, ...
code value v in a graph with size log(v )?
Set Cover: same as 3-CNF-SAT (with few nodes)...
Is it really NP-hard?
Seems related to Minimum Spanning Tree
solvable with greedy strategy?
No obvious formulation as matroid
Counterexample for greedy strategy!
16 / 21
PoA for NSP
Simulator
NP-completeness of SC
Convergence speed of strong-PNSP
Open Problems
Find social optimum
3-CNF-SAT reduction does not work
Graph problems: Vertex Cover, Dominating Set, ...
dynamic edges?
Knapsack Problem, Bin Packing, ...
code value v in a graph with size log(v )?
Set Cover: same as 3-CNF-SAT (with few nodes)...
Is it really NP-hard?
Seems related to Minimum Spanning Tree
solvable with greedy strategy?
No obvious formulation as matroid
Counterexample for greedy strategy!
16 / 21
PoA for NSP
Simulator
NP-completeness of SC
Convergence speed of strong-PNSP
Open Problems
Find social optimum
3-CNF-SAT reduction does not work
Graph problems: Vertex Cover, Dominating Set, ...
dynamic edges?
Knapsack Problem, Bin Packing, ...
code value v in a graph with size log(v )?
Set Cover: same as 3-CNF-SAT (with few nodes)...
Is it really NP-hard?
Seems related to Minimum Spanning Tree
solvable with greedy strategy?
No obvious formulation as matroid
Counterexample for greedy strategy!
16 / 21
PoA for NSP
Simulator
NP-completeness of SC
Convergence speed of strong-PNSP
Open Problems
Find social optimum
3-CNF-SAT reduction does not work
Graph problems: Vertex Cover, Dominating Set, ...
dynamic edges?
Knapsack Problem, Bin Packing, ...
code value v in a graph with size log(v )?
Set Cover: same as 3-CNF-SAT (with few nodes)...
Is it really NP-hard?
Seems related to Minimum Spanning Tree
solvable with greedy strategy?
No obvious formulation as matroid
Counterexample for greedy strategy!
16 / 21
PoA for NSP
Simulator
NP-completeness of SC
Convergence speed of strong-PNSP
Open Problems
Find social optimum
3-CNF-SAT reduction does not work
Graph problems: Vertex Cover, Dominating Set, ...
dynamic edges?
Knapsack Problem, Bin Packing, ...
code value v in a graph with size log(v )?
Set Cover: same as 3-CNF-SAT (with few nodes)...
Is it really NP-hard?
Seems related to Minimum Spanning Tree
solvable with greedy strategy?
No obvious formulation as matroid
Counterexample for greedy strategy!
16 / 21
PoA for NSP
Simulator
NP-completeness of SC
Convergence speed of strong-PNSP
Open Problems
Convergence speed of strong-PNSP
17 / 21
PoA for NSP
Simulator
NP-completeness of SC
Convergence speed of strong-PNSP
Open Problems
Known bounds
1
Reminder: weak-PNSP with average cost function:
At most |F |(diam(G ) − 1) improving moves until next NE
Reachable friendship example for tightness in O-Notation
Strong-PNSP (with maximum cost function):
Example with Θ(|VF |2 diam(G ))
2
Reminder: sorted
cost vector as potential
function
|VF | + diam(G ) − 1
At most
improving moves until
|VF |
next NE
⇒ Convergence speed somewhere in between!
18 / 21
PoA for NSP
Simulator
NP-completeness of SC
Convergence speed of strong-PNSP
Open Problems
Known bounds
1
Reminder: weak-PNSP with average cost function:
At most |F |(diam(G ) − 1) improving moves until next NE
Reachable friendship example for tightness in O-Notation
Strong-PNSP (with maximum cost function):
Example with Θ(|VF |2 diam(G ))
2
Reminder: sorted
cost vector as potential
function
|VF | + diam(G ) − 1
At most
improving moves until
|VF |
next NE
⇒ Convergence speed somewhere in between!
18 / 21
PoA for NSP
Simulator
NP-completeness of SC
Convergence speed of strong-PNSP
Open Problems
Known bounds
1
Reminder: weak-PNSP with average cost function:
At most |F |(diam(G ) − 1) improving moves until next NE
Reachable friendship example for tightness in O-Notation
Strong-PNSP (with maximum cost function):
Example with Θ(|VF |2 diam(G ))
2
Reminder: sorted
cost vector as potential
function
|VF | + diam(G ) − 1
At most
improving moves until
|VF |
next NE
⇒ Convergence speed somewhere in between!
18 / 21
PoA for NSP
Simulator
NP-completeness of SC
Convergence speed of strong-PNSP
Open Problems
Known bounds
1
Reminder: weak-PNSP with average cost function:
At most |F |(diam(G ) − 1) improving moves until next NE
Reachable friendship example for tightness in O-Notation
Strong-PNSP (with maximum cost function):
Example with Θ(|VF |2 diam(G ))
2
Reminder: sorted
cost vector as potential
function
|VF | + diam(G ) − 1
At most
improving moves until
|VF |
next NE
⇒ Convergence speed somewhere in between!
18 / 21
PoA for NSP
Simulator
NP-completeness of SC
Convergence speed of strong-PNSP
Open Problems
Approach
Start with easy graph (path) and easy friendships (path)
could give graph and friendship characterizations for
convergence speed
19 / 21
PoA for NSP
Simulator
NP-completeness of SC
Convergence speed of strong-PNSP
Open Problems
Open Problems
20 / 21
PoA for NSP
Simulator
NP-completeness of SC
Convergence speed of strong-PNSP
Open Problems
Open Problems
NP-completeness of SCOPT,dec
Convergence speed of strong-PNSP
PoS
Characterization of graphs/friendships for convergence/good
NE
Shortcut problem anyone?
21 / 21