Report No. 94.166
On the Complexity of Testing Membership in the Core of Min-cost Spanning Tree Games by U. Faigle, S. Fekete, W. Hochstattler, W. Kern 1994
Ulrich Faigle Walter Kern Department of Applied Mathematics University of Twente P.O.Box 217 NL-7500 AE Enschede The Netherlands Sandor P. Fekete Winfried Hochstattler ZPR, Zentrum fur paralleles Rechnen Universitat zu Koln Albertus-Magnus-Platz D-50923 Koln Germany Telefon (0221) 470{6024 (6023) e{Mail
[email protected] (WH@ZPR...)
ON THE COMPLEXITY OF TESTING MEMBERSHIP IN THE CORE OF MIN{COST SPANNING TREE GAMES Ulrich Faigle Walter Kern
Department of Applied Mathematics University of Twente P.O.Box 217 NL-7500 AE Enschede The Netherlands S andor P. Fekete Winfried Hochst attler
ZPR, Zentrum fur paralleles Rechnen Universitat zu Koln Albertus-Magnus-Platz D-50923 Koln Germany
Abstract = f1 g be a nite set of players and N the complete graph on the node set [ f0g. Assume that the edges of N have nonnegative weights and associate with each coalition of players as cost ( ) the weight of a minimal spanning tree on the node set [ f0g.
Let
N
; : : :; n
K
N
K
S
N
c S
S
Using reduction to EXACT COVER BY 3-SETS, we exhibit the following problem to be NP-complete. Given the vector 2 c S
Keywords: Cooperative game, core, spanning tree, NP-complete, X3C 1991 Mathematics Subject Classi cation: 90C27, 90D12
1 Introduction In this note we investigate the computational complexity of nding optimal core allocations of a much studied cooperative N-person game. The minimum cost spanning tree game, MCST-game for short, is a game on the set = f1 g of players, the grand coalition, that is to be connected to the supply node 0. Establishing a direct link between any pair ( ), where 2 [ f0g, is assumed to cost the nonnegative amount ( ) = ( ). The objective is to create a connected graph on the node set [ f0g and to divide the resulting total cost among the players. The N
;::: ;n
i; j
w i; j
i; j
w j; i
N
n
1
N
latter is represented by an allocation vector 2
x S
> c S
S
S
G N; E
g
E
g
S
S
( ) = ( + 1) j \ f1 ( ) = ( + 2)j \ f1
c S
q
q S
x S
q
S
3 gj + ( + 1)j \ f3 + 1 3 + gj + 2 ? 1 3 gj + j \ f3 + 1 3 + gj
;::: ;
;::: ;
q
q
q
q S
4
S
q
q
;:::
;:::
q
k
q
:
k
q
;
(1) (2)
Hence
( ) ? ( ) = j \ f1 3 gj ? j \ f3 + 1 3 + gj ? 2 + 1 implying j \ f3 + 1 3 + gj . Every set-player covers at most 3 element players, thus we have 0
< x S
S
c S
q
;:::
S
q
;::: ;
k
S
q
;:::
q
k
q
(3)
q
j \ f1 S
q
;::: ;
3 gj 3j \ f3 + 1 q
S
q
;:::
3 + gj q
k
(4)
:
Together with (3) this yields 2j \ f3 + 1 S
implying j \ f3 + 1 get S
q
;:::
q
;:::
3 + gj 2 ? 1 q
k
>
3 + gj and hence j \ f3 + 1 q
k
q
S
0 j \ f1
c S
x
2
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[7] M. R. Garey, D. S. Johnson [1979]: Computers and Intractability. A Guide to the Theory of NP-Completeness. W. H. Freeman, New York [8] M. Grotschel, L. Lovasz, and A. Schrijver [1988]: Geometric Algorithms and Combinatorial Optimization. Springer-Verlag, Berlin [9] D. Granot and F. Granot [1993]: Computational complexity of a cost allocation approach to a xed cost spanning forest problem. Mathematics of Operations Research 17, 765{780 [10] D. Granot and G. Huberman [1981]: Minimum cost spanning tree games. Mathematical Programming 21, 1{18 [11] D. Granot and G. Huberman [1982]: The relationship between convex games and minimum cost spanning tree games: A case for permutationally convex games. SIAM Journ. Algebraic and Discrete Methods 3, 288{292 [12] J. Kuipers [1994]: Combinatorial Methods in Cooperative Game Theory. Ph.D. Thesis, University of Limburg, Maastricht [13] N. Meggido [1978]: Computational complexity of the game theory approach to cost allocation for a tree. Mathematics of Operations Research 3, 189{196
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