1. Hardness of an Asymmetric Stackelberg Network Pricing Game
Davide Bilò, Luciano Gualà, Guido Proietti

2. Pricing problems: a wide class of optimization problems
pricing problems for Unit-Demand Consumers:
Min-buying model
Max-buying model
Rank-buying model
With/without price ladder constraint …
pricing problems for Single-Minded Consumers:
general
on a network
Network Pricing games
Bilevel optimization problems
with a game-theoretic
flavour

3. Network pricing game 2-player game we have a (di)graph G=(V,E=RB)
each eR has a fixed cost c(e)
leader sets a price p(e) for each eB
the follower computes (buys) a (feasible) subgraph H(p) of G which minimizes (H,c,p)
leader’s revenue: rev(H(p))
goal: find prices in order to maximize the revenue

4. Example: StackMST The follower computes an MST T(p) of G1 6 6
(T,c,p) =  c(e)+ p(e) 7
eT
eT 4 3 8 2 2 1 1 1
rev(T(p)) = p(e) 4
eT(p) 1 1

5. Example: StackMST The follower computes an MST T(p) of G6 6 1
(T,c,p) =  c(e)+ p(e) 7
eT
eT 4 3 8 2 2 1 1 1
rev(T(p)) = p(e) 4
eT(p) 1 1
The revenue is 13

6. Related work: StackSP [Roch et al., Networks’05]
strongly NP-hard
O(log |B|)-apx
[Joret, tech. report’08]
APX-hard
[Briest,Khanna, tech. report’09]
Not approximable within 2-

7. Related work: StackMST
[Cardinal et al., WADS’07]
APX-hard
O(log |B|)-apx (single-price algorithm)
[Cardinal et al., WINE’09]
NP-hard for planar graphs
Polynomial-time solvable for bounded treewidth graphs

8. Related work [Briest et al., STACS’08]:
O(log |B|+log f)-apx for binary demand Stackelberg games with f followers
binary demand:
Single-price algorithm
(H,c,p) =  c(e)+ p(e)
eH
eH
rev(H(p)) = p(e)
eH(p)

9. Related work [Bilò,Gualà,Proietti,Widmayer, WINE’08]
(symmetric) StackSPT
NP-hard
Efficient poly-time algorithm when |B| is constant
(T,c,p) =  dT(r,v)
=  loadT(e)c(e)+ loadT(e)p(e) v
eT
eT
rev(T(p)) = loadT(e)p(e)
eT(p)

10. Related work [Briest,Hoefer,Gualà,Ventre, WINE’09]:
What when follower’s problem is NP-hard?
Follower uses a 2-apx algorithm for Min-Knapsack
Strong NP-hard
(2+)-apx
Follower uses a primal-dual algorithm for Set Cover
APX-hard when element frequency f>2
exact poly-time algorithm when f=2

11. Asymmetric Stackelberg Shortest Path Tree (ASSPT) game
The follower computes
an SPT T(p) rooted at r r
(T,c,p) =  dT(r,v) 1 4 v 4 ?
=  loadT(e)c(e)+ loadT(e)p(e) 8
eT
eT ? 6 ?
rev(T(p)) = p(e) 1
eT(p)

12. Our results Not approximable within n1/2- (n-1)-apx algorithm
unweighted star
Equivalence with the Rooted Maximum Leaf Outbranching (RMLO) problem
APX-hard in general, NP-hard for DAGs [Alon et al., Discrete Math., 2009]
92-apx algorithm [Daligault, Thomassé,IWPEC’09]
APX-hard for DAGs
unweighted chain
Exact poly-time algorithm when G=(V, RB) is a DAG
NP-hard otherwise

13. Theorem
For every >0, the ASSPT game is not approximable within a factor n1/2-, even on DAGs and when c(e){0,1}, for each eR
reduction from Maximum Independent Set (MIS) problem
MIS not approximable within n1-, for every >0, unless P=NP

14. G G’
of n nodes
of N=(n2) nodes v v
n copies of v 1 u u r
n copies of u 1 1 v’ v’
n copies of v’
Claim 1:
There exists an optimal pricing p* with the form p*:B  {0,1}
Claim 2:
Any pricing p:B  {0,1} defines an independent set Ip and the
revenue of p is at most |Ip|n+n-|Ip|
Claim 3:
rev(p*) is at least |I*|n + n-|I*|, where I* is a MIS

15. G G’ ≥ ≥ > > rev(p*) n|I*| + n - |I*| n|I*| n1-’
of n nodes
of N=(n2) nodes v v
n copies of v 1 u u r
n copies of u 1 1 v’ v’
n copies of v’
rev(p*)
n|I*| + n - |I*|
n|I*| ≥ ≥
>
n1-’
n|Ip| + n - |Ip|
rev(p)
n|Ip| + n
|I*|
>
n1-
|Ip|

16. Theorem The exists an (n-1)-approximation algorithm for the ASSPT game
price every edge e=(u,v) as p(e)=max{0,d(v)-d(u)}
d(x): red distance, i.e. distance from r to x in G=(V,R)
Notice:
every blue edge e with p(e)>0 is selected by the follower

17. T(p*) r : the path in T(p*) with maximum revenue rev()≥ 1/(n-1) r*
we’ll show that
rev(T(p)) ≥ rev()

18. since every ei with p(ei)>0 is selected
k-1  
d(vi)+wi+1 ≥ d(ui+1)
i=1,…,k-1 r
i=1
w1 = d(u1) w1
d(uk) = d(uk) u1 e1 k k v1  
(d(vi)-d(ui)) ≥ d(vk) wi w2
i=1
i=1 u2 e2 v2 k  w3
rev()  d(vk)- wi
i=1 .
since every ei with p(ei)>0 is selected uk k  ek
rev(T(p)) ≥ (d(vi)-d(ui)) vk
i=1

19. Open problems Close the gap between lower and upper bounds
Is the single-price algorithm the best algorithm one can design?

20. Positioning MST problem
leader places non-parallel blue edges and price them
Then, the follower computes an MST
Leader collects the revenue
Constraints:
Leader has a budget B
a placement cost (u,v), for each feasible position
must place edges with total placement cost at most B T 4 4 8 1 1 3 6 7 2

21. Positioning MST problem
3 variants:
without placement costs
Uniform placement costs
Unrestricted placement costs
a modification of the
single-price algorithm
always guarantees
a O(log n)-apx
Subcase of StackMST
???
can we say
more?
generalizes StackMST

22. Thank you