Teoría Sociocultural: Lev Seminovitch Vygotsky

blocking, then this implies thatS is winning. This clarifies that self-dual games and decisive games are equivalent.

We give a brief outline of the chapter. In Section 10.2, computational aspects of computing the length of simple games are considered. Section 10.3 examines the complexity of questions related to duality of games. In Section 10.4, com- plexity of bribery in simple games is explored in the light of previous sections. In Section 10.5, a summary of results and future directions of study are given. Throughout, we assume that the voting games have integer weights.

10.2 Computing length of games

Now, game theoretic version of definitions from Ramamurthy’s book [184] are provided:

Definitions 10.3.The set Wk(v) is the set of winning coalitions of size k of the simple game v. Moreover, Bk(v)is the set of blocking coalitions of v of size k.

These definitions can be used to define the length and width of simple games:

Definitions 10.4.Thelengthof a simple game is the smallest integer k such that Wk(v), ∅. Thewidthof a simple game is the smallest integer k such that Bk(v), ∅

The length is an important indicator of a game which signifies in a sense the ease with which the status quo can be changed. We examine the complexity of computing the length of a simple game.

Name: LENGTH

Instance: Simple gamev

Output: Length ofv

Name: WIDTH

Instance: Simple gamev

138 10 Complexity of length, duality and bribery

10.2.1 Complexity of computing length

It is evident that LENGTH(v) is equivalent to WIDTH(vd). Moreover, for some special types of simple games, it is easy to observe their length and the width:

Observation 10.5 For a simple game v,

1. If v is a unanimity game, LENGT H(v)= n and W IDT H(v)=1. 2. If v is a singleton game, LENGT H(v)= 1and W IDT H(v)=n.

3. If v is a majority game, LENGT H(v)= dn/2eand W IDT H(v)=d(n+1)/2e

Proof. (Follows from the definitions). ut

Now the complexity of computing LENGTH for a simple game represented by (N,W), (N,Wm_{), WVG or MWVG is analysed:}

Observation 10.6 The problem LENGTH for a simple game represented by

(N,W),(N,Wm)or WVG is in P.

Proof. For a simple gamevrepresented by (N,W) or (N,Wm_{), LENGTH(v) can}

be computed in linear time by scanning the winning coalitions and identifying the smallestksuch that coalitionS is inWorWm_and|S|= _k.

For the case of WVG, the weights of the players are already sorted. So start off withw1and keep adding more players with decreasing weights until

Pk

i=1wi ≥q.

It is then claimed that LENGTH(v) is k. It is easy to see this since any other approach, apart from the greedy approach to pick up weights, will require at least

k weights for the sum of the weights to be more than q. The greedy method outlined for LENGTH(v) for WVGs also computes the coalition which has the smallest feasible length. ut

Proposition 10.7.The problem LENGTH for a simple game represented by a

MWVG is NP-hard.

Proof. We provide a reduction from a special case of the minimization version of multidimensional 0-1 knapsack problem (MKP) [92].

10.2 Computing length of games 139

Name: MKP

Instance: A collection ofnitems andmknapsacks where the capacity of theith knapsack isbi, the jth item requiresai j units of resource consumption in theith

knapsack and has corresponding profitcj

Output: MaximizePn_j=1cjxj such thatPnj=1ai jxj ≤ bi, i∈ M = {1,2, . . .m}, and

xj ∈ {0,1}, j∈N = {1,2, . . .n}.

The goal in MKP is to find a subset of items that yields maximum profit with- out exceeding the resource capacities. MKP is equivalent to the minimization version of the problem (MIN-MKP) since maximizing the profit of a set of items is equivalent to minimizing the profit of items not in the set. The transformations needed areyj = 1−xj for j ∈ N anddi = (

Pn

j=1ai j)−bi for j ∈ N andi ∈ M.

Therefore the following problem is as hard as MKP:

Name: MIN-MKP

Instance: A collection ofnitems andmknapsacks where each knapsackishould

have at least di capacity filled and the jth item has corresponding profitcj and

requiresai j units of resource consumption.

Output: MinimizePnj=1cjyj such that Pn

j=1ai jyj ≥ di, i ∈ M = {1,2, . . .m}and

yj ∈ {0,1}, j∈N.

Gens and Levner [96] point out that Dinic and Karzanov [61] proved that even the special case of MIN-MKP where m = 2 and cj = 1 for all j = 1 to n is

NP-hard. It is easy to see that by renaming some variables (ai j towi_jandditoqi),

the NP-hard special case of MIN-MKP is equivalent to computing the length of a MWVG of dimension 2. ut

10.2.2 Approximating the length of a MWVG

Although this may seem a paradox, all exact science is dominated by the idea of approximation.

140 10 Complexity of length, duality and bribery

Although all NP-complete problems share the same worstcase complexity, they have little else in common. When seen from almost any other per- spective, they resume their healthy, confusing diversity. Approximability is a case in point.

- Christos Papadimitriou (1993)

Although the length of a MWVG cannot be computed efficiently, it is observed that it can be approximated efficiently:

Proposition 10.8.For a MWVG, v with dimension m, there exists a polynomial

time approximation algorithm which computes LENGTH(v) with an absolute er- ror of m−1.

Proof. This result uses the same approach as in [34] where the authors use LP- relaxation to provide an m − 1 absolute approximation algorithm for the Safe Deposit Boxes (SDB) problem:

Name: Safe Deposit Boxes (SDB) problem Instance:aji≥ 0 fori=1, . . .nand j= 1, . . .m.

Output: Minimize Pn_i=1xi such that Pajixi ≥ Aj, j = 1, . . .m; x ∈ {0,1},

i=1, . . . ,n.

A complete proof is given as follows. Let vbe a MWVG with nplayers and

m constituent WVGs [qt_;wt

1, . . . ,w

t

n] for 1 ≤ t ≤ m. We assume that m < n.

The problem of computing LENGTH(v) is an integer program. An LP-relaxation changes it into a problem where we want to minimizePni=1xiwhere

Pn

i=1xiwti ≥ q t

for all 1 ≤ t ≤ m and 0 ≤ xi ≤ 1 for all i ∈ N. The inequalities of the linear

program can be changed into equalities by introducingn+mslack variable where one slack variable is used for each inequality.

10.2 Computing length of games 141 minPni=1xi s.t. Pni=1xiwt_i+si = qtfor i=1,. . . , m, xi+ sn+i = 1 for i=1,. . . , n, xi ≥ 0 for i=1,. . . , n si ≥ 0 for i=1,. . . , m+n (10.1)

The resultant LP has a total of n+ m constraints and 2n+ m variables (all of which are non-negative) where n+m is the number of slack variables. Any extreme point in the feasible region of formulation requiresnbinding constraints. It follows that any basic feasible solution contains at leastn zero values. Out of thesenzero values, a maximum ofmvalues can be attributed to slack variables related to the quota constraints. Out of the remainingn−mzero values, either one of the original variables is zero, or a slack variable related to the inequalityxi ≤1

is zero. In either of the cases, the original variable is non-fractional. Therefore, there are at most m original variables(xis) which may have fractional values in

the LP solution. The LP solution is of course solvable in polynomial time [44]. If none of the xis are fractional, then the LP solution is also the length of the

MWVG. If not, then letl be the number of xis equal to one in the LP solution.

Then, the length of the MWVG is at least l+1. If we round up every fractional values of the LP solution, then all the constraints are still satisfied. Moreover, the maximum value of the sum of the ceilings of values of the LP solution isl+m. Therefore the maximum error between the length of the MWVG and the sum of the ceilings of values of the LP solution ism−1. ut

Although computing the length of an MWVG has a PTAS, there is no FP- TAS (fully polynomial time approximation scheme) [96]. The argument is that if there is an-approximation algorithm polynomial innand 1/ to approximate the length of a MWVG, then this implies that there is a polynomial algorithm to compute the length of a MWVG.

142 10 Complexity of length, duality and bribery