MI Preprint SeriesMathematics for Industry

Kyushu University

Coalition Structure Generation

with Subadditivity Constraints

Naoyuki Kamiyama,

Akifumi Kira, Hirokazu Anai,

Hidenao Iwane & Kotaro Ohori

MI 2016-10

( Received August 8, 2016 )

Institute of Mathematics for IndustryGraduate School of Mathematics

Kyushu UniversityFukuoka, JAPAN

Coalition Structure Generation with Subadditivity Constraints

Naoyuki KAMIYAMAInstitute of Mathematics for Industry, Kyushu University

&JST PRESTO

Akifumi KIRAInstitute of Mathematics for Industry, Kyushu University

Hirokazu ANAI, Hidenao IWANE, and Kotaro OHORIKnowledge Information Processing Laboratory, Fujitsu Laboratories Ltd.

Abstract

Coalition structure generation is one of the most fundamental problems in the study of computationalaspects of cooperative games. The goal of this problem is to partition the ground set of a given cooperativegame in such a way that a certain property is satisfied. For example, one of typical goals is to find apartition that minimizes the sum of costs of all coalitions. In this paper, we introduce the problem offinding a minimum-size partition of the ground set such that a subgame on each coalition is subadditive.The contribution of this paper is twofold. In the first part of this paper, we consider this problem fromthe theoretical viewpoint. In this part, we prove that our problem in the minimum base game on matroidsintroduced by Nagamochi, Zeng, Kabutoya, and Ibaraki can be solved in polynomial time. In the secondpart, we study our problem from the practical viewpoint. In this part, we first propose heuristic algorithmsfor our problem in a general cooperative game. Our algorithms are based on heuristic algorithms for theset cover problem. Then, we apply our algorithms to the problem of sharing taxis that is our originalmotivation. More precisely, we consider the problem of partitioning customers for taxis in such a waythat customers in each group have incentive for sharing a taxi.

Keywords Coalition structure generation, matroid, column generation, taxi sharing.

1 Introduction

A cooperative game Γ = (N, v) is a pair of a finite set N of players and a characteristic function v : 2N → Rsuch that v(∅) = 0, where we denote by R the set of real numbers. One of the most fundamental problemsin computational aspects of cooperative games is coalition structure generation (see, e.g., [1, Chapter 6]for a survey of coalition structure generation). The goal of this problem is to partition the ground set ofa given cooperative game in such a way that a certain property is satisfied. For example, one of typicalgoals is to find a partition that minimizes the sum of costs of all coalitions. Several variations of coalition

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structure generation problems (e.g., a min-max type criterion) are considered in various fields (see, e.g.,[2, 3, 4, 5, 6, 7]).

In this paper, we introduce the problem of finding a minimum-size partition of the ground set such thata subgame on each coalition is subadditive. A cooperative game Γ = (N, v) is said to be subadditive, ifv(X ∪Y ) ≤ v(X)+v(Y ) for every pair of disjoint subsets X,Y of N . If a cooperative game is subadditive,then it is natural to assume that the agents form the grand coalition since the cost decreases by gettingtogether. Once the grand coalition is formed, we can use known solution concepts for sharing the cost. Thus,the subadditivity is one of fundamental properties that a cooperative game should satisfy.

1.1 Problem formulation

For each cooperative game Γ = (N, v), a partition N1, N2, . . . , Nk of N is called a feasible partition of Γ,if the cooperative game (Nt, v|Nt) is subadditive for every t = 1, 2, . . . , k, where v|Nt is the restriction of vto Nt. Then, the COALITION STRUCTURE GENERATION WITH SUBADDITIVITY CONSTRAINTS problem(CSGS for short) is formally defined as follows. In this problem, we are given a cooperative game Γ = (N, v).Then, the goal of this problem is to find a feasible partition P of Γ that minimizes |P| among all feasiblepartitions of Γ. Notice that for every cooperative game Γ = (N, v), since a family of {i} for all elementsi in N is a feasible partition of Γ, there always exists a solution of CSGS. Furthermore, it should be notedthat since the problem of checking the subadditivity of a cooperative game is computationally hard, ourproblem is also computationally hard. For example, Greco, Malizia, Palopoli, and Scarcello [8] prove thatthe special case of the variant of CSGS in which we decide whether its optimal objective value is equal toone is coNP-complete. Although CSGS is natural, to the best of our knowledge, this problem has not beeninvestigated.

In practical settings, it may be desirable that the size of each subset in a partition is bounded by someupper bound (e.g., a capacity of a taxi in our application). We will consider the following variant of CSGS.In this variant, we are given a cooperative game Γ = (N, v) and a positive integer ℓ. A feasible partitionP of Γ is called an ℓ-bounded feasible partition of Γ, if |X| ≤ ℓ for every member X in P . Then, ourgoal is an ℓ-bounded feasible partition P of Γ that minimizes |P| among all ℓ-bounded feasible partitionsof Γ. We call this problem the COALITION STRUCTURE GENERATION WITH SUBADDITIVITY AND SIZE

CONSTRAINTS problem (CSGSS for short).

1.2 Our contributions

The contribution of this paper is twofold. Our contributions are summarized as follows.

• In Section 2, we study CSGS from the theoretical viewpoint. Since the problem of checking the sub-additivity of a general cooperative game is hard, our problem is also computationally hard. Thus, itis a natural direction to find subclasses of cooperative games in which our problem can be efficientlysolved. The main contribution of this part is the polynomial-time solvability of CSGS and its variantsin the minimum base game on matroids introduced by Nagamochi, Zeng, Kabutoya, and Ibaraki [9].

• In Section 3, we study our problem from the practical viewpoint. In this part, we first propose heuristicalgorithms for CSGS in a general cooperative game. Our algorithms are based on heuristic algorithmsfor the set cover problem. Then, we apply our algorithms to the problem of sharing taxis that is ouroriginal motivation. More precisely, we consider the problem of partitioning customers for taxis insuch a way that customers in each group have incentive for sharing a taxi.

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2 Polynomial-Time Algorithms for the Minimum Base Game on Matroids

In this section, we prove that CSGS in the minimum base game on matroids can be solved in polynomialtime (Section 2.2). Furthermore, we prove that its variant with size constraints (Section 2.3) and a similarproblem related to a core (Section 2.4) can be solved in polynomial time.

2.1 Basics of Matroids

Here we give basics of matroids. See, e.g., [10] for more information about matroids. A pair M = (U, I) ofa finite set U and a family I of subsets of U is called a matroid, if it satisfies the following three conditions.

(I0) ∅ ∈ I.

(I1) For every pair of subsets X,Y of U such that X ⊆ Y , if Y ∈ I, then X ∈ I.

(I2) For every pair of members X,Y in I such that |X| < |Y |, there exists an element u in Y \X such thatX ∪ {u} ∈ I.

When we consider a problem in which some matroid M = (U, I) is given as an input, we assume that weare given a polynomial-time independence oracle. That is, we are not directly given the family I. Instead,for every subset X of U , we can check whether X ∈ I in polynomial time. That is, the polynomial-timesolvability of some problem related to a matroid M = (U, I) means that we can solve this problem in timebounded by a polynomial in |U | and the number of calls of an independence oracle for I. Notice that thecondition (I2) implies that for a given matroid M = (U, I), the size of every maximal member X in I (i.e,X ⊆ Y for any member Y in I) is the same, and we can find a maximum-size member in I in polynomialtime.

For each matroid M = (U, I), we define B(M) the set of maximal members in I. For each matroidM = (U, I) and each subset X of U , we define I|X as the family of subsets I of X such that I ∈ I, andM|X := (X, I|X). It is known [10] that M|X is a matroid. For each matroid M = (U, I) and each positiveinteger k, we define Ik as the family of subsets X of U such that there exists a partition X1, X2, . . . , Xk ofX such that Xt ∈ I for every t = 1, 2, . . . , k. Furthermore, for each matroid M = (U, I) and each positiveinteger k, we define Mk := (U, Ik). It is known [10] that Mk is a matroid. The following theorem plays animportant role in our algorithms.

Theorem 1 (See, e.g., [11, 12]). For every matroid M = (U, I) and every positive integer k, we can find amaximum-size member I∗ in Ik and a partition I1, I2, . . . , Ik of I∗ such that It ∈ I for every t = 1, 2, . . . , kin polynomial time.

Assume that we are given a matroid M = (U, I) and a positive integer ℓ. Then, we define the ℓ-truncation I ′ of I as the set of members I in I such that |I| ≤ ℓ. We define the ℓ-truncation M′ of M byM′ := (U, I ′). It is well known [10] that the ℓ-truncation of M is a matroid.

2.2 The Minimum Base Game on Matroids

In this subsection, we first formally define the minimum base game on matroids, and then we prove thatCSGS in the minimum base game on matroids can be solved in polynomial time.

For each matroid M = (U, I) and each function w : U → R, we define a function ρwM : 2U → R by

ρwM(X) := minY ∈B(M|X)

∑u∈Y

w(u).

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For each matroid M = (U, I) and each function w : U → R, the minimum base game [9] on the matroid Mwith the cost function w is defined as the cooperative game (U, ρwM).

Here we give an example of the minimum base game, called the minimum spanning tree game [9].Assume that we are given an undirected graph G = (V,E) and a cost function c : E → R. A subset F of Eis called a forest in G, if no cycle is contained in F . Let F be the family of forests in G. For each subset Xof E, we define F∗(X) as the set of forests F such that (i) F ⊆ X and (ii) there exists no forest F ′ such thatF ⊊ F ′ ⊆ X . That is, F∗(X) is the family of maximal forests contained in X . Then, we define a functionπ : 2E → R by

π(X) := minF∈F∗(X)

∑e∈F

c(e).

The minimum spanning tree game is defined as the cooperative game (E, π). This game can be representedby the minimum base game on matroids as follows. It is well known [10] that MG = (E,F) is a matroid.Furthermore, it is not difficult to see that for every subset X of E, we have B(MG|X) = F∗(X). Thus,the minimum spanning tree game is equivalent to the minimum base game on the matroid MG with the costfunction c.

For proposing a polynomial-time algorithm for CSGS in the minimum base game on matroids, we firstgive necessary facts.

Lemma 2. For every matroid M = (U, I) and every subset X of U , the cooperative game (X, ρwM|X) isequal to the minimum base game on M|X with the cost function w|X .

Proof. Assume that we are given a matroid M = (U, I) and a subset X of U . It is not difficult to see thatfor every subset Y of X , we have (M|X)|Y = M|Y . This implies that for every subset Y of X , we have

ρwM|X(Y ) = ρwM(Y ) = minZ∈B(M|Y )

∑u∈Z

w(u)

= minZ∈B((M|X)|Y )

∑u∈Z

w(u) = ρw|XM|X(Y ).

This completes the proof.

Theorem 3 (Nagamochi, Zeng, Kabutoya, and Ibaraki [9, Corollary 3.1]). Assume that we are given amatroid M = (U, I) and a function w : U → R. Then, the minimum base game on the matroid M with thecost function w is subadditive if and only if

{u ∈ U | w(u) < 0} ∈ I.

For each matroid M = (U, I), each function w : U → R, and each positive integer k, we define thedecision version of CSGS in the minimum base game on the matroid M with the cost function w as theproblem of deciding whether there exists a feasible partition P of the minimum base game on the matroidM with the cost function w such that |P| = k, and find such a partition if one exists. If we can solvethis decision version in polynomial time, then we can solve CSGS in the minimum base game on matroidsin polynomial time by solving the decision version O(log |U |) times. In what follows, we prove that thedecision version can be solved in polynomial time by using the following Algorithm 1.

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Algorithm 1

Step 1. Set ∆ := {u ∈ U | w(u) < 0}.

Step 2. Find a maximum-size member I∗ in (I|∆)k.

Step 3. If |I∗| < |∆|, then output null and halt (that is, there exists no feasible partition P of the minimumbase game on the matroid M with the cost function w such that |P| = k). Otherwise, go to Step 4.

Step 4. Find a partition I1, I2, . . . , Ik of I∗ such that It ∈ I for every t = 1, 2, . . . , k.

Step 5. Partition U \∆ into J1, J2, . . . , Jk.

Step 6. Set Ut := It ∪ Jt for each t = 1, 2, . . . , k.

Step 7. Output a family U1, U2, . . . , Uk and halt.

End of Algorithm

Lemma 4. For every matroid M = (U, I), every function w : U → R, and every positive integer k, thedecision version of CSGS in the minimum base game on the matroid M with the cost function w can besolved in polynomial time by using Algorithm 1.

Proof. Since it follows from Theorem 1 that Algorithm 1 can be implemented in polynomial time, it sufficesto prove its correctness. Assume that there exists a feasible partition N1, N2, . . . , Nk of the minimum basegame on the matroid M with the cost function w. That is, the cooperative game (Nt, ρ

wM|Nt) is subadditive

for every t = 1, 2, . . . , k. Then, Lemma 2 implies that the minimum base game on the matroid M|Nt

with the cost function w|Nt is subadditive. Theorem 3 implies that Nt ∩ ∆ is in I|Nt, i.e., I for everyt = 1, 2, . . . , k. Thus, ∆ (= ∪k

t=1(Nt ∩∆)) is in Ik, i.e., (I|∆)k. Furthermore, ∆ is clearly a maximum-size member in (I|∆)k. Thus, if Algorithm 1 outputs null, then there exists no feasible partition P of theminimum base game on the matroid M with the cost function w such that |P| = k.

Next we consider the case where Algorithm 1 outputs a family U1, U2, . . . , Uk. Since |I∗| = |∆| andI∗ ⊆ ∆, we have I∗ = ∆. Thus, the family U1, U2, . . . , Uk is a partition of U . Furthermore, for everyt = 1, 2, . . . , k, since Ut ∩ ∆ = It holds, Ut ∩ ∆ is in I. Thus, Theorem 3 implies that the minimumbase game on the matroid M|Ut with the cost function w|Ut is subadditive. This and Lemma 2 imply thatthe family U1, U2, . . . , Uk is a feasible partition of the minimum base game on the matroid M with the costfunction w. This completes the proof.

Lemma 4 and the argument above Algorithm 1 imply the following theorem.

Theorem 5. Assume that we are given a matroid M = (U, I) and a function w : U → R. Then, CSGS in theminimum base game on the matroid M with the cost function w can be solved in polynomial time.

2.3 Size Constraints

In this subsection, we consider CSGSS in the minimum base game on matroids. Assume that we are givena matroid M = (U, I), a function w : U → R, and a positive integer ℓ. As in the case of CSGSS in theminimum base game on matroids, it is suffice to prove that the decision version of CSGSS in the minimumbase game on matroids (i.e., the problem of deciding whether there exists an ℓ-bounded feasible partition Pof CSGSS in the minimum base game on the matroid M and the cost function w such that |P| = k). Thisproblem can be solved by using the following Algorithm 2.

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Algorithm 2

Step 1. If |U | > kℓ, then output null and halt (that is, there exists no ℓ-bounded feasible partition P ofCSGSS in the minimum base game on the matroid M and the cost function w such that |P| = k).Otherwise, go to Step 2.

Step 2. Set ∆ := {u ∈ U | w(u) < 0}.

Step 3. Let I ′ be the ℓ-truncation of I. Then, find a maximum-size member I∗ in (I ′|∆)k,

Step 4. If |I∗| < |∆|, then output null and halt. Otherwise, go to Step 5.

Step 5. Find a partition I1, I2, . . . , Ik of I∗ such that It ∈ I for every t = 1, 2, . . . , k.

Step 6. Partition U \∆ into J1, J2, . . . , Jk in such a way that |It|+ |Jt| ≤ ℓ for every t = 1, 2, . . . , k.

Step 7. Set Ut := It ∪ Jt for each t = 1, 2, . . . , k.

Step 8. Output a family U1, U2, . . . , Uk and halt.

End of Algorithm

It is not difficult to see that the following lemma can be proved in the same way as Lemma 4.

Lemma 6. For every matroid M = (U, I), every function w : U → R, and every positive integers k, ℓ, thedecision version of CSGSS in the minimum base game on the matroid M with the cost function w can besolved in polynomial time by using Algorithm 2.

Lemma 6 implies the following theorem.

Theorem 7. Assume that we are given a matroid M = (U, I), a function w : U → R, and a positive integerℓ. Then, CSGSS in the minimum base game on the matroid M with the cost function w can be solved inpolynomial time.

2.4 Non-empty cores

For each cooperative game Γ = (N, v), we define the core of Γ by

C(Γ) :={x ∈ RN

∣∣∣ ∑i∈N

x(i) = v(N), ∀S ⊆ N :∑i∈X

x(i) ≤ v(S)}.

The concept of a core is regarded as one of natural solution concepts in cooperative games. Thus, thenon-emptiness of a core is a desirable property of a cooperative game. In this subsection, we consider theCOALITION STRUCTURE GENERATION WITH NON-EMPTY CORE problem (CSGNC for short) defined asfollows. In this problem, we are given a cooperative game Γ = (N, v). Then, the goal of this problem is tofind a minimum-size partition N1, N2, . . . , Nk of N such that the core of the cooperative game (Nt, v|Nt)is not empty for any t = 1, 2, . . . , k, and find such a partition if one exists. We prove that CSGNC in theminimum base game on matroids can be solved in polynomial time. It should be noted that Deng andPapadimitriou [13] proved that the special case of CSGNC is NP-complete. The following theorem related tothe non-emptiness of a core of the minimum base game on matroids is known.

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Theorem 8 (Nagamochi, Zeng, Kabutoya, and Ibaraki [9, Theorem 4.1]). Assume that we are given a ma-troid M = (U, I) and a function w : U → R. Then, the core of the minimum base game on the matroid Mwith the cost function w is not empty if and only if

{u ∈ U | w(u) < 0} ∈ I.

Theorem 8 implies that for every matroid M = (U, I) and every function w : U → R, the core of theminimum base game on the matroid M with the cost function w is not empty if and only if this game issubadditive. Thus, Theorem 5 implies the following theorem.

Theorem 9. Assume that we are given a matroid M = (U, I) and a function w : U → R. Then, CSGNC inthe minimum base game on the matroid M with the cost function w can be solved in polynomial time.

3 Heuristic Algorithms for General Cooperative Games and Application forSharing Taxis

In this section, we propose heuristic algorithms for CSGS in a general cooperative game (Section 3.1), andthen we apply our algorithms to the problem of sharing taxis that is our original motivation (Section 3.2).

3.1 Heuristic algorithms based on a set cover

Our heuristic algorithms are based on the following observation. For every subadditive cooperative gameΓ = (N, v) and every subset X of N , the cooperative game Γ′ = (X, v|X) is subadditive. Thus, for eachcooperative game Γ = (N, v), CSGS in Γ can be formulated as the following integer program IP. We denoteby S(Γ) the family of subsets X of N such that (X, v|X) is subadditive.

Minimize∑

X∈S(Γ)

ξ(X)

subject to∑

X∈S(Γ) : i∈X

ξ(X) ≥ 1 (i ∈ N)

ξ ∈ {0, 1}S(Γ).

Once we can obtain the optimal solution of the problem IP, we can construct an optimal solution for CSGS

by resolving overlap.However, the size of S(Γ) is exponentially large compared to the input size, and enumerating all subsets

of N in S(Γ) is not realistic. Thus, our strategy is to use a subfamily of subsets in S(Γ). For each subfamilyL of S(Γ), we define the integer program IP(L) as follows.

Minimize∑X∈L

ξ(X)

subject to∑

X∈L : i∈Xξ(X) ≥ 1 (i ∈ N)

ξ ∈ {0, 1}L.

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Furthermore, for each subfamily L of S(Γ), we define the linear programming relaxation LP(L) of IP(L) asfollows.

Minimize∑X∈L

ξ(X)

subject to∑

X∈L : i∈Xξ(X) ≥ 1 (i ∈ N)

ξ ∈ RL+,

where R+ is the set of non-negative real numbers. In addition, the dual problem DP(L) of LP(L) can bedescribed as follows.

Maximize∑i∈N

λ(i)

subject to∑i∈X

λ(i) ≤ 1 (X ∈ L)

λ ∈ RN+ .

Our first heuristic algorithm is described as follows.

Algorithm 3

Step 1. Set L := {{i} | i ∈ N}.

Step 2. Set λ(i) := 1 for every element i in N .

Step 3. While∑

i∈Y λ(i) > 1 for some member Y in S(Γ), do the following steps.

(3-a) Update L := L ∪ {Y }.

(3-b) Find an optimal solution λ of DP(L).

Step 4. Find an optimal solution ξ of IP(L).

Step 5. Output a family {X ∈ L | ξ(X) = 1}, and halt.

End of Algorithm

Notice that Algorithm 3 is a straightforward application of the column generation [14, 15] for the linearprogramming relaxation of the set cover problem [16] (see also [17] for a survey of heuristic algorithmsfor the set cover problem). For the subfamily L that is finally output by the column generation scheme(Step 3), LP(L) yields an exact optimal solution of LP(S(Γ)). Therefore, IP(L) would also yields a goodapproximate solution of IP. Notice that since there may exists a gap between IP(L) and LP(L), Algorithm 3may fail to find an optimal solution. In Step 3 of Algorithm 3, since

∑i∈X λ(i) ≤ 1 for every member X in

L, the size of L is increasing. Thus, since S(Γ) is finite, Algorithm 3 halts. However, since the size of S(Γ)is very large, it is hard to decide whether

∑i∈X λ(i) > 1 for some member X in S(Γ) in Step 3. Thus, in

our second heuristic algorithm, we try to approximately solve this step by using the following Procedure 1.

Procedure 1

Step 1. Sort elements in N as a1, a2, . . . , a|N | in such a way that

λ(a1) ≥ λ(a2) ≥ · · · ≥ λ(a|N |).

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Step 2. Set X := ∅.

Step 3. For all t = 1, 2, . . . , |N |, do the following steps.

(3-a) Set X ′ := X ∪ {at}.

(3-b) If (X ′, v|X ′) is subadditive (we check this by brute force), then update X := X ′.

Step 4. Output X , and halt.

End of Procedure

Then, our second heuristic algorithm is described as Algorithm 4.

Algorithm 4

Step 1. Set L := {{i} | i ∈ N}.

Step 2. Set λ(i) := 1 for any element i in N .

Step 3. Set Y to be the output of Procedure 1 for λ.

Step 4. While∑

i∈Y λ(i) > 1, do the following steps.

(4-a) Update L := L ∪ {Y }.

(4-b) Find an optimal solution λ of DP(L).

(4-c) Set Y to be the output of Procedure 1 for λ.

Step 5. Find an optimal solution ξ of IP(L).

Step 6. Output a family {X ∈ L | ξ(X) = 1}, and halt.

End of Algorithm

Here we explain how to modify Algorithm 4 in such a way that it can solve CSGSS. For this purpose, itis sufficient to check whether |X ′| does not violate the upper bound in Procedure 1. More precisely, we usethe following Procedure 2 instead of Procedure 1.

Procedure 2

Step 1. Sort elements in N as a1, a2, . . . , a|N | in such a way that

λ(a1) ≥ λ(a2) ≥ · · · ≥ λ(a|N |).

Step 2. Set X := ∅.

Step 3. For all t = 1, 2, . . . , |N |, if |X| < ℓ, then do the following steps.

(3-a) Set X ′ := X ∪ {at}.

(3-b) If (X ′, v|X ′) is subadditive, then update X := X ′.

Step 4. Output X , and halt.

End of Procedure

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3.2 Application to the problem of sharing taxis

In this section, we apply our heuristic algorithms to CSGS arising from the problem of sharing taxis. Althoughseveral papers studied the problem of sharing vehicles from the theoretical (optimization) viewpoint, almostall papers consider the problem of finding an optimal routing. On the other hand, we focus on the problemof finding routing in which there exist incentive for participants. Furthermore, although several papers (e.g.,[18, 19]) consider routing problems from the viewpoint of the cooperative game theory, their focuses aremainly how to divide the cost of routing plans. To the best of knowledge, no one studied the problem ofsharing vehicles from the viewpoint of grouping.

Our model is defined as follows. We are given a set N of customers. For each customer i in N , we aregiven an origin oi and a destination di. In addition, we are given a special point s, called a depot. For eachsubset X of N , we define

P (X) := {s} ∪ {oi, di | i ∈ X}.

Furthermore, we are given a length function len : P (N)× P (N) → R+.Assume that we are given a subset X of N . A one-to-one mapping π from {1, 2, . . . , 2|X|+1} to P (X)

is called a feasible tour for X , if it satisfies the following conditions (R1) to (R3). For each element p inP (X), let π−1(p) be the integer t in {1, 2, . . . , 2|X|+ 1} such that π(t) = p.

(R1) π(2|X|+ 1) = s.

(R2) For every customer i in X , we have π−1(oi) < π−1(di).

(R3) There exist no integer t in {1, 2, . . . , 2|X| − 1} and no subset Y of X such that

{π(1), π(2), . . . , π(t)} = {oi, di | i ∈ Y }.

The condition (R1) means that the tour ends at s. The condition (R2) means that for every player i in X , thetaxi visits the origin of i before visiting its destination. The condition (R3) means the taxi does not becomeempty in the tour1. For each feasible tour π for X , the length of π is define by

len(s, π(1)) +

2|X|∑t=1

len(π(t), π(t+ 1)).

Here we define the cooperative game related to our motivation. For each subset X of N , we definevrs(X) as the minimum length of a feasible tour for X . In what follows, we consider the cooperative gameΓrs = (N, vrs).

In order to consider tCSGS in Γrs, we first consider the problem of computing vrs(X) for each subset Xof N . Assume that we are given a subset X of N . It is not difficult to see that the problem of computingvrs(X) is NP-complete. We compute vrs(X) by using the integer program RS(X) defined as follows. Fordefining RS(X), we first introduce a directed graph D(X). The vertex set of D(X) is P (X). The arc setA(X) of D(X) is defined by

A(X) := {(p, q) | p, q ∈ P (X) such that p = q},

where (p, q) is an arc from p to q. Notice that for every pair of vertices p, q in P (X), there exist two arcs(p, q) and (q, p) in A(X). For each vertex v in P (X), we denote by δ(v) (resp., ϱ(v)) the set of arcs (p, q)in A(X) such that p = v (resp., q = v). For each arc a = (p, q) in A(X), we define d(a) := len(p, q).

1In Japan, if a taxi becomes empty in a route, it is regarded as two routes and the base fare for each route is billed.

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We are now ready to give a concrete formulation of RS(X). The variables of RS(X) are x in {0, 1}A(X)

and y, z in RA(X)+ . The variable x represents which arcs the tour uses. That is, if x(a) = 1 for some arc a in

A(X), then the tour uses the arc a. In what follows, for each variable x in {0, 1}A(X), we define

S(x) := {a ∈ A(X) | x(a) = 1}.

Constraints for the variable x are the following.∑a∈δ(v)

x(a) = 1 and∑

a∈ϱ(v)

x(a) = 1 (v ∈ P (X)). (1)

The constraints in (1) ensure that S(x) is the collection of directed cycles in D(X). Thus, we need toensure that S(X) consists of a single directed cycle. For this, we use the variable y. Constraints for y arethe following (these constraints are the same as those used in the so-called “flow based formulation” of thetravelling salesman problem [20]).

y(a) ≤ (|P (X)| − 1) · x(a) (a ∈ A(X))∑a∈δ(s)

y(a) = |P (X)| − 1

∑a∈ϱ(v)

y(a)−∑

a∈δ(v)

y(a) = 1 (v ∈ P (X) \ {s}).

(2)

It is not difficult to see that the constraints in (2) ensure that the number of vertices in the directed cycle inS(x) containing s is |P (X)|, i.e., S(X) consists of a single directed cycle. In addition, we need that thedirected cycle in S(x) satisfies the condition (R2). This can be ensured by the following constraint.∑

a∈δ(oi)

y(a) ≥∑

a∈δ(di)

y(a) (i ∈ X). (3)

Lastly, we consider the condition (R3). For this, we use the variable z. Constraints for z are the following.

z(a) ≤ |X| · x(a) (a ∈ A(X))∑a∈δ(s)

z(a) = 0

∑a∈ϱ(oi)

z(a)−∑

a∈δ(oi)

z(a) = −1 (i ∈ X)

∑a∈ϱ(di)

z(a)−∑

a∈δ(di)

z(a) = 1 (i ∈ X)

∑a∈δ(di)

z(a) ≥ 1− x((di, s)) (i ∈ X).

(4)

It is not difficult to see that if z satisfies the constraints in (4), then the directed cycle in S(x) satisfies thecondition (R3). Furthermore, it is not difficult to see that we can construct variables x in {0, 1}A(X) and y, z

11

in RA(X)+ from a feasible tour for X . Thus, we can formulate RS(X) as follows.

Minimize∑

a∈A(X)

d(a) · x(a)

subject to (1), (2), (3), and (4)

x ∈ {0, 1}A(X)

y, z ∈ RA(X)+ .

In the following experiments, we solve this integer program by using an IP solver.

3.3 Computational experiments

To test our algorithms in a practical setting, we generate virtual taxi demand of customers on a real map.We make an instance of the problem of taxi sharing in the following manner: There are 196 bus stations inItoshima city, Fukuoka prefecture, Japan [21]. The authors depict them in Figure 1. By generating uniformrandom numbers on {1, 2, . . . , 196}, we randomly choose the locations of vertices in P (N) = {s}∪{oi, di |i ∈ N} from the bus stations, where the conditions s = oi, oi = di, and s = di are guaranteed for all i ∈ N .For each a = (p, q) in A(N) such that neither p nor q is not s, we set d(a) to the minimum travel time(min) from p to q along the real road network. The regulations of traffic such as one-way roads and restrictedturns are also taken into consideration. To do this, we solve the shortest path problem, in advance, by usinggeographic information system software ArcGIS for Desktop ver. 10.3 (Esri Inc., USA) and ArcGIS DataCollection Road Network 2015 (Esri Japan Corporation). Moreover, for each a = (p, q) such that either p orq is s, we set d(a) = h/2. Here, the value of h is interpreted as the base fare for a taxi converted in terms oftravel time (min).

(C) Esri Japan

Figure 1: Bus stations in Itoshima city

12

We implemented Algorithms 3 and 4 using Python Language and used Gurobi Optimizer ver. 6.5.1(Gurobi Optimization, Inc.) as the optimization engine to solve the problems DP(·), IP(·), and RS(·). Weexecuted it on a Linux server with Intel R⃝ Xeon R⃝-X5570 processor of 2.93GHz and 12.32GB memoryinstalled. Figure 2 shows our result of measuring the average computational time required to evaluate thecharacteristic function vrs : 2

N → R+. In this experiments, we first generated a virtual instance in the abovemanner, then we evaluated vrs : 2

N → R+ by solving RS(X), the traveling salesman problem (TSP) withtaxi sharing constraints, for all X ⊂ N . We repeated this process 100 times for each |N | = 5, 6, . . . , 10with changing the seed for random numbers. Here, we note again that solving the constrained TSP is alsoNP-hard, and the number of all subsets of N is 2|N |. The result clearly shows that the average computationaltime increases exponentially with the size of N .

Figure 2: Average computational time for evaluating vrs : 2N → R+

We repeated the process of generating a virtual instance and applying Algorithms 3 and 4 to it. Table1 sums up the average behavior of the experiments. More detailed results measuring the effect of the basefare for a taxi for problems with size form |N | = 13 up to 16 are shown in Table 2. We carried out theexperiments with changing the value of h. Figures 3 and 4 illustrate the outputs of the algorithms for aninstance with |N | = 8. As shown in these figures, each route may pass a road in a neighboring city ofItoshima city.

Estimating the optimization gap is quite hard in our challenging problem. However, since we can exactlysolve the TSP with taxi-sharing constraints, the gap owes the column generation scheme, and it is known tobe one of the most powerful heuristics for the set covering problem. Hence we can say by induction: Ourapproach is not a bad choice.

We note that it is not necessary to evaluate vrs(X) for all X ⊂ N in order to solve the problem ofsharing taxis. In our implementation, RS(X) is solved for the first time when X becomes a candidate whichis added to L and the value of vrs(X) is necessary to decide whether the cooperative game Γ′ = (X, vrs|X)is subadditive or not. A memoization technique is also used. That is, the evaluated value of vrs(X) is storedand reused when it is needed again to check the subadditivity of Γ′′ = (Y, vrs|Y ) for a set Y containing X .

It is found that our heuristic algorithms successfully work in a realistic time for problems with sizeup to |N | = 30. The computational time of our algorithms depends on the maximum cardinality of themembers in L rather than on the size of N . As expected, Table 2 indicates that a raise in the base fare fora taxi gives customers incentive to share taxis, and as a result, the computational time also increases. Inour experiments, the instances were randomly generated and the origin-destination points of customers were

13

Table 1: Average behavior of Algorithms 3 and 4Average

# of Problems Objective value CPU time (sec)|N | (h = 2, 4, 6, 8, 10) Alg. 3 Alg. 4 Alg. 3 Alg. 45 50 (10,10,10,10,10) 2.74 3.08 0.47 0.516 ↓ 2.94 3.36 1.19 1.347 ↓ 3.22 3.64 3.84 3.698 ↓ 3.44 4.48 6.59 7.789 ↓ 3.86 4.82 15.48 10.4710 ↓ 3.70 4.60 44.85 35.6811 ↓ 4.38 5.76 78.15 52.8712 ↓ 4.46 5.74 260.14 221.5513 ↓ 4.76 6.18 417.52 211.0114 ↓ 4.94 6.72 149.18 117.3615 ↓ 4.90 6.60 539.41 205.7316 ↓ 5.26 7.06 822.84 586.2517 ↓ 5.44 7.64 925.38 494.7618 20 (10,10) 7.30 10.20 84.87 66.1519 ↓ 7.10 11.20 72.40 38.4420 ↓ 7.55 11.25 98.90 48.8221 ↓ 7.85 11.50 122.38 88.3022 ↓ 7.85 11.90 422.22 58.1623 ↓ 8.25 14.20 189.83 56.8624 ↓ 8.25 12.05 487.11 247.0925 ↓ 8.75 13.85 617.57 309.7626 ↓ 8.20 13.20 887.69 400.4827 ↓ 8.86 13.38 740.19 302.3228 ↓ 8.86 15.19 920.14 284.1029 ↓ 9.33 14.95 739.99 390.0130 ↓ 9.33 13.81 1346.21 558.00

14

Table 2: Effect of the base fare for a taxi for problems with size |N | = 13, 14, 15, 16Average

# of Objective value CPU time (sec)|N | h Problems Alg. 3 Alg. 4 Alg. 3 Alg. 413 2 10 6.40 9.00 9.04 5.5713 4 10 5.70 6.50 11.86 12.8113 6 10 4.40 5.80 85.19 26.7513 8 10 4.00 5.10 164.61 91.6113 10 10 3.30 4.50 1816.90 918.2913 total 50 4.76 6.18 417.52 211.0114 2 10 7.00 9.00 7.53 5.3414 4 10 5.30 7.90 39.66 18.3614 6 10 4.60 6.20 53.27 48.9314 8 10 4.00 5.70 213.47 197.1014 10 10 3.80 4.80 431.97 317.0614 total 50 4.94 6.72 149.18 117.3615 2 10 6.10 8.80 16.97 14.4715 4 10 6.00 8.30 22.23 15.6415 6 10 4.90 6.10 163.79 173.8715 8 10 3.60 5.20 979.32 473.1915 10 10 3.90 4.60 1514.75 351.4715 total 50 4.90 6.60 539.41 205.7316 2 10 7.60 10.60 17.73 14.2616 4 10 5.30 7.40 57.76 39.4816 6 10 5.20 7.50 739.47 417.0316 8 10 4.40 5.40 780.08 249.8116 10 10 3.80 4.40 2519.18 2210.6816 total 50 5.26 7.06 822.84 586.25

15

(C) Esri Japan, Sumitomo Electric Industries, Ltd.

d1

d8

d4

o4

o1

o8depot

(a) Route 1

(C) Esri Japan, Sumitomo Electric Industries, Ltd.

d7

d2

o7

d3

o2d5

o5

d6

o3

o6

depot

(b) Route 2

Figure 3: Output of Algorithms 3 for an instance with |N | = 8.

16

(C) Esri Japan, Sumitomo Electric Industries, Ltd.

d7

d4d3

o7

o4

d1

o1o3

depot

(a) Route 1

(C) Esri Japan, Sumitomo Electric Industries, Ltd.

d6

d8

o6

d2o2

d5

o5o8

depot

(b) Route 2

Figure 4: Output of Algorithms 4 for the same instance as used in Figure 3.

17

distributed uniformly. Hence, the condition

maxX∈L

|X| ≤ 10

was satisfied for almost of the instances. On the other hand, it is still computationally hard to evaluatevrs : 2

X → R+ when some X such that |X| > 10 becomes a candidate being added to L. However, in a realworld situation, the size of each coalition sharing a taxi is bounded by some upper bound ℓ (ℓ ≤ 10) due tothe capacity of the taxi, customers’ time windows, and so on. Hence, we can use the modified algorithm forCSGSS.

4 Conclusion

In this paper, we introduce the problem of finding a minimum-size partition of the ground set such that asubgame on each coalition is subadditive. Then, we first prove that our problem in the minimum base gameon matroids can be solved in polynomial time. In the second part, we propose heuristic algorithms for ourproblem in a general cooperative game, and apply our algorithms to the problem of partitioning customersfor taxis in such a way that customers in each group have incentive for sharing a taxi.

Acknowledgements

Naoyuki Kamiyama was supported by JST, PRESTO. Akifumi Kira was supported in part by JSPS KAK-ENHI Grant Number 26730010.

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19

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MI2011-13 Vladimır CHALUPECKY & Adrian MUNTEANSemi-discrete finite difference multiscale scheme for a concrete corrosion model: ap-proximation estimates and convergence

MI2011-14 Jun-ichi INOGUCHI, Kenji KAJIWARA, Nozomu MATSUURA & Yasuhiro OHTAExplicit solutions to the semi-discrete modified KdV equation and motion of discreteplane curves

MI2011-15 Hiroshi INOUEA generalization of restricted isometry property and applications to compressed sens-ing

MI2011-16 Yu KAWAKAMIA ramification theorem for the ratio of canonical forms of flat surfaces in hyperbolicthree-space

MI2011-17 Naoyuki KAMIYAMAMatroid intersection with priority constraints

MI2012-1 Kazufumi KIMOTO & Masato WAKAYAMASpectrum of non-commutative harmonic oscillators and residual modular forms

MI2012-2 Hiroki MASUDAMighty convergence of the Gaussian quasi-likelihood random fields for ergodic Levydriven SDE observed at high frequency

MI2012-3 Hiroshi INOUEA Weak RIP of theory of compressed sensing and LASSO

MI2012-4 Yasuhide FUKUMOTO & Youich MIEHamiltonian bifurcation theory for a rotating flow subject to elliptic straining field

MI2012-5 Yu KAWAKAMIOn the maximal number of exceptional values of Gauss maps for various classes ofsurfaces

MI2012-6 Marcio GAMEIRO, Yasuaki HIRAOKA, Shunsuke IZUMI, Miroslav KRAMAR,Konstantin MISCHAIKOW & Vidit NANDATopological Measurement of Protein Compressibility via Persistence Diagrams

MI2012-7 Nobutaka NAKAZONO & Seiji NISHIOKA

Solutions to a q-analog of Painleve III equation of type D(1)7

MI2012-8 Naoyuki KAMIYAMAA new approach to the Pareto stable matching problem

MI2012-9 Jan BREZINA & Yoshiyuki KAGEISpectral properties of the linearized compressible Navier-Stokes equation aroundtime-periodic parallel flow

MI2012-10 Jan BREZINAAsymptotic behavior of solutions to the compressible Navier-Stokes equation arounda time-periodic parallel flow

MI2012-11 Daeju KIM, Shuichi KAWANO & Yoshiyuki NINOMIYAAdaptive basis expansion via the extended fused lasso

MI2012-12 Masato WAKAYAMAOn simplicity of the lowest eigenvalue of non-commutative harmonic oscillators

MI2012-13 Masatoshi OKITAOn the convergence rates for the compressibleNavier- Stokes equations with potential force

MI2013-1 Abuduwaili PAERHATI & Yasuhide FUKUMOTOA Counter-example to Thomson-Tait-Chetayev’s Theorem

MI2013-2 Yasuhide FUKUMOTO & Hirofumi SAKUMAA unified view of topological invariants of barotropic and baroclinic fluids and theirapplication to formal stability analysis of three-dimensional ideal gas flows

MI2013-3 Hiroki MASUDAAsymptotics for functionals of self-normalized residuals of discretely observed stochas-tic processes

MI2013-4 Naoyuki KAMIYAMAOn Counting Output Patterns of Logic Circuits

MI2013-5 Hiroshi INOUERIPless Theory for Compressed Sensing

MI2013-6 Hiroshi INOUEImproved bounds on Restricted isometry for compressed sensing

MI2013-7 Hidetoshi MATSUIVariable and boundary selection for functional data via multiclass logistic regressionmodeling

MI2013-8 Hidetoshi MATSUIVariable selection for varying coefficient models with the sparse regularization

MI2013-9 Naoyuki KAMIYAMAPacking Arborescences in Acyclic Temporal Networks

MI2013-10 Masato WAKAYAMAEquivalence between the eigenvalue problem of non-commutative harmonic oscilla-tors and existence of holomorphic solutions of Heun’s differential equations, eigen-states degeneration, and Rabi’s model

MI2013-11 Masatoshi　OKITAOptimal decay rate for strong solutions in critical spaces to the compressible Navier-Stokes equations

MI2013-12 Shuichi KAWANO, Ibuki HOSHINA, Kazuki MATSUDA & Sadanori KONISHIPredictive model selection criteria for Bayesian lasso

MI2013-13 Hayato CHIBAThe First Painleve Equation on the Weighted Projective Space

MI2013-14 Hidetoshi MATSUIVariable selection for functional linear models with functional predictors and a func-tional response

MI2013-15 Naoyuki KAMIYAMAThe Fault-Tolerant Facility Location Problem with Submodular Penalties

MI2013-16 Hidetoshi MATSUISelection of classification boundaries using the logistic regression

MI2014-1 Naoyuki KAMIYAMAPopular Matchings under Matroid Constraints

MI2014-2 Yasuhide FUKUMOTO & Youichi MIELagrangian approach to weakly nonlinear interaction of Kelvin waves and a symmetry-breaking bifurcation of a rotating flow

MI2014-3 Reika AOYAMADecay estimates on solutions of the linearized compressible Navier-Stokes equationaround a Parallel flow in a cylindrical domain

MI2014-4 Naoyuki KAMIYAMAThe Popular Condensation Problem under Matroid Constraints

MI2014-5 Yoshiyuki KAGEI & Kazuyuki TSUDAExistence and stability of time periodic solution to the compressible Navier-Stokesequation for time periodic external force with symmetry

MI2014-6 This paper was withdrawn by the authors.

MI2014-7 Masatoshi OKITAOn decay estimate of strong solutions in critical spaces for the compressible Navier-Stokes equations

MI2014-8 Rong ZOU & Yasuhide FUKUMOTOLocal stability analysis of azimuthal magnetorotational instability of ideal MHDflows

MI2014-9 Yoshiyuki KAGEI & Naoki MAKIOSpectral properties of the linearized semigroup of the compressible Navier-Stokesequation on a periodic layer

MI2014-10 Kazuyuki TSUDAOn the existence and stability of time periodic solution to the compressible Navier-Stokes equation on the whole space

MI2014-11 Yoshiyuki KAGEI & Takaaki NISHIDAInstability of plane Poiseuille flow in viscous compressible gas

MI2014-12 Chien-Chung HUANG, Naonori KAKIMURA & Naoyuki KAMIYAMAExact and approximation algorithms for weighted matroid intersection

MI2014-13 Yusuke SHIMIZUMoment convergence of regularized least-squares estimator for linear regression model

MI2015-1 Hidetoshi MATSUI & Yuta UMEZUSparse regularization for multivariate linear models for functional data

MI2015-2 Reika AOYAMA & Yoshiyuki KAGEISpectral properties of the semigroup for the linearized compressible Navier-Stokesequation around a parallel flow in a cylindrical domain

MI2015-3 Naoyuki KAMIYAMAStable Matchings with Ties, Master Preference Lists, and Matroid Constraints

MI2015-4 Reika AOYAMA & Yoshiyuki KAGEILarge time behavior of solutions to the compressible Navier-Stokes equations arounda parallel flow in a cylindrical domain

MI2015-5 Kazuyuki TSUDAExistence and stability of time periodic solution to the compressible Navier-Stokes-Korteweg system on R3

MI2015-6 Naoyuki KAMIYAMAPopular Matchings with Ties and Matroid Constraints

MI2015-7 Shoichi EGUCHI & Hiroki MASUDAQuasi-Bayesian model comparison for LAQ models

MI2015-8 Yoshiyuki KAGEI & Ryouta OOMACHIStability of time periodic solution of the Navier-Stokes equation on the half-spaceunder oscillatory moving boundary condition

MI2016-1 Momonari KUDOAnalysis of an algorithm to compute the cohomology groups of coherent sheaves andits applications

MI2016-2 Yoshiyuki KAGEI & Masatoshi OKITAAsymptotic profiles for the compressible Navier-Stokes equations on the whole space

MI2016-3 Shota ENOMOTO & Yoshiyuki KAGEIAsymptotic behavior of the linearized semigroup at space-periodic stationary solu-tion of the compressible Navier-Stokes equation

MI2016-4 Hiroki MASUDANon-Gaussian quasi-likelihood estimation of locally stable SDE

MI2016-5 Yoshiyuki KAGEI & Takaaki NISHIDAOn Chorin’s method for stationary solutions of the Oberbeck-Boussinesq equation

MI2016-6 Hayato WAKI & Florin NAEBoundary modeling in model-based calibration for automotive engines via the vertexrepresentation of the convex hulls

MI2016-7 Kazuyuki TSUDATime periodic problem for the compressible Navier-Stokes equation on R2 with an-tisymmetry

MI2016-8 Abulizi AIHAITI, Shota ENOMOTO & Yoshiyuki KAGEILarge time behavior of solutions to the compressible Navier-Stokes equations in aninfinite layer under slip boundary condition

MI2016-9 Fermın Franco MEDRANO, Yasuhide FUKUMOTO, Clara M. VELTE & AzurHODZICGas entrainment rate coefficient of an ideal momentum atomizing liquid jet

MI2016-10 Naoyuki KAMIYAMA, Akifumi KIRA, Hirokazu ANAI, Hidenao IWANE & KotaroOHORICoalition Structure Generation with Subadditivity Constraints