Group-Strategyproof Mechanisms for Network Design Games via Primal-Dual Algorithms Jochen K¨onemann∗
Stefano Leonardi†
Abstract
Guido Sch¨afer‡
David Wheatley§
S ⊆ V . We will furthermore assume that f satisfies maximality, i.e.,
About 15 years ago, Goemans and Williamson formally introduced the primal-dual framework for approximation algorithms and applied it to a class of network design optimization problems. Since then literally hundreds of results appeared that extended, modified and applied the technique to a wide range of optimization problems. In this paper we define a class of cost-sharing games arising from Goemans and Williamson’s original network design problems. We then show how to derive a groupstrategyproof (i.e., collusion resistant) mechanism for such a game, using an existing primal-dual algorithm for the underlying optimization problem as a black box. The budgetbalance factor of this mechanism is proportional to the performance ratio of the primal-dual algorithm if the optimization problem satisfies an additional technical condition. Most existing collusion-resistant cost-sharing mechanisms are obtained through skillful adaptation of existing primal-dual algorithms for the associated optimization problems. This paper shows that, at least for a large class of games arising from network design problems, no such adaptation is necessary.
f (A) = f (B) = 0
=⇒
f (A ∪ B) = 0
for any pair of disjoint vertex sets A, B ⊆ V . The f connectivity problem is NP-hard in general as it captures many well-known NP-hard optimization problems such as Steiner tree, Steiner forest, etc. In [5], Goemans and Williamson presented a suite of primal-dual approximation algorithms whose performance guarantee is 2 if the cut-requirement function f has certain additional structural properties. In this paper, we first propose a class of network design games arising from network design optimization problems. We then show how existing primal-dual algorithms for the optimization problem can be used to derive mechanisms for the associated game. Network design games. At the centre of our attention are network design games where we are given a set N of players each of which has a cut-requirement function that she wishes to satisfy. Let f i : 2V → {0, 1} be the function of player i ∈ N. Player i receives service in a subgraph H of G, if degH (S) ≥ f i (S) for all S ⊆ V , where degH (S) denotes the number of edges in H that have exactly one endpoint in S. We assume that each player i has a private valuation vi that she derives from receiving service.
1 Introduction In a typical instance of a network design problem one is asked to find a minimum-cost subgraph in a given input graph with certain connectivity properties. Specifically, in this paper we consider the following class of f -connectivity problems: given an undirected graph G = (V, E), nonnegative costs ce for all edges e ∈ E, and a cut-requirement function f : 2V → {0, 1}, find a minimum-cost subgraph H of G such that H has at least f (S) edges crossing each set
Example 1 (Steiner forest game). Consider the Steiner forest game where each player i has a terminal-pair (si ,ti ) that she likes to connect. The cut-requirement function encoding this player’s goal is f i (S) = 1 for all sets S ⊆ V that contain exactly one of si and ti , and f (S) = 0 otherwise. A max-flow/min-cut argument suffices to show that i receives service in subgraph H iff degH (S) ≥ f i (S) for all S ⊆ V .
∗ Department of Combinatorics and Optimization, University of Waterloo, Waterloo, ON N2L 3G1, Canada. Email:
[email protected]. Research support by NSERC grant OGP0288340. † Dipartimento di Informatica e Sistemistica, Universit` a di Roma “La Sapienza”, Via Ariosto 25, 00185 Roma, Italy. Email:
[email protected]. ‡ Technische Universit¨ at Berlin, Institut f¨ur Mathematik, Straße des 17. Juni 136, 10623 Berlin, Germany. Email:
[email protected]. Supported by the DFG Research Center Matheon “Mathematics for key technologies”. § Department of Combinatorics and Optimization, University of Waterloo, Waterloo, ON N2L 3G1, Canada. Email:
[email protected].
The main goal in this paper is to design direct revelation cost-sharing mechanisms for network design games. Such a mechanism solicits bids bi from each of the players i ∈ N and uses these to select a set Q ⊆ N of players to service. Our mechanism will then compute a subgraph H(Q) of G that services the players in Q; it also determines a service fee or price pi to charge to each i ∈ Q. Our mechanisms should adhere to the following three economic constraints: (i) no player is payed for receiving 1
service, i.e., pi ≥ 0 (no positive transfer), (ii) a player is charged only if she receives service and the charged price does not exceed her bid (individual rationality), (iii) a player is guaranteed to receive service if only her bid is high enough (consumer sovereignty). Apart from this, we want our mechanisms to be approximately budget-balanced, i.e., 1 · optQ ≤ ∑ pi ≤ optQ β i∈Q
cost-sharing methods for a given game is to modify an existing primal-dual approximation algorithm. The duals produced by such an algorithm often give rise to cost shares in a natural way. These are usually not necessarily crossmonotone and modifying the algorithm such that they are is an art. In particular, a general framework for the design of approximately budget-balanced and group-strategyproof mechanisms for cost-sharing games from existing primaldual algorithms is elusive. In this paper we make a step in this direction and show how an existing primal-dual algorithm for a network design problem from the class defined above can be turned into a group-strategyproof mechanism. The budget-balance factor of the resulting mechanism is proportional to the performance guarantee of the primal-dual algorithm if the problem satisfies an additional technical property. We demonstrate the power of our framework by applying it to network design games arising from downwards monotone and proper cut-requirement functions.
(1)
for some small β ≥ 1, where optQ is the minimum cost of any subgraph servicing Q. We assume that players act strategically, i.e., each player’s goal is to maximize his own (quasi-linear) utility function. The utility of player i is defined as ui = vi − pi if i ∈ Q, and ui = 0 otherwise. In this paper, we are interested in devising mechanisms that are group-strategyproof, i.e., players should be motivated to bid truthfully, even if collusion is allowed. That is, no coordinated bidding of a coalition S ⊆ U can ever strictly increase the utility of some player in S without strictly decreasing the utility of another player in S.
Previous work. The framework of Moulin and Shenker has been used extensively to obtain cost sharing mechanisms that are group-strategyproof and approximately budget balanced for several combinatorial optimization problems [2, 3, 6, 7, 10, 11, 12, 14, 17]. Most of these crossmonotonic cost sharing methods were derived by adapting existing primal-dual approximation algorithms. Immorlica et al. [9] showed that all group-strategyproof mechanism that satisfy a certain technical fairness condition can be obtained using Moulin and Shenker’s framework. The authors also gave strong lower bounds on the budget balance approximation factor β of cross-monotonic cost sharing methods for several problems. Subsequently, lower bounds on the budget balance factor were also obtained for other problems [2, 3, 13]. Very recently, Mehta et al. [15] introduced a more general framework for designing truthful cost sharing mechanisms termed acyclic mechanisms. Acyclic mechanisms implement a slightly weaker notion of group-strategyproofness, but therefore leave more flexibility to improve upon the approximation guarantees of desirable objectives such as budget balance.
Moulin mechanisms. Moulin and Shenker [16] presented a powerful framework that reduces the task of designing a group-strategyproof cost sharing mechanism for a game to that of giving a cross-monotonic cost sharing method. A cost sharing method ξ is a polynomial-time algorithm that, given any subset Q ⊆ N of players, computes a solution to service Q and for each j ∈ Q determines a non-negative cost share ξQ ( j). Analogously to the definition in (1), we say that ξ is β -budget balanced if 1 · c(Q) ≤ β
∑ ξQ ( j) ≤ optQ .
j∈Q
A cost sharing method ξ is cross-monotone if, for any two sets Q and S such that Q ⊆ S, and any player j ∈ Q we have ξS ( j) ≤ ξQ ( j). In other words, the cost share of any player under the given cost sharing method does not increase if the size of the player set increases. Given a budget balanced and cross-monotonic cost sharing method ξ for a game, the following Moulin mechanism Mξ from [16] satisfies budget balance and groupstrategyproofness: Initially, let Q = N. If, for every player j ∈ Q, the cost share ξQ ( j) is less than or equal to her bid b j , stop. Otherwise, remove from Q all players whose cost shares in Q are larger than their bids, and repeat. Eventually, let p j = ξQ ( j) be the prices that are charged to players in the final set Q. Jain and Vazirani [10] later proved that the result of Moulin and Shenker also holds true if one considers cross-monotonic cost sharing methods that are approximately budget balanced.
2 Primal-dual algorithms for network design problems
We first review Goemans and Williamson’s algorithm (subsequently referred to by GW) for the class of network design problems introduced in the previous section. The algorithm is primal-dual and hence constructs a feasible primal solution for a given integer linear program. It also computes a feasible solution for the dual of its linear programming reFrom primal-dual algorithms to mechanisms. A com- laxation. In the performance analysis we compare the cost mon approach for designing approximately budget-balanced of the primal solution to the objective value of the computed 2
Theorem 1 ([8], p. 162). The solution computed by GW has dual solution. We start by giving an integer programming formulation cost at most γ · ∑S⊆V yS ≤ γ ·opt f , if for any infeasible A ⊆ E for the f -connectivity problem. The IP has a binary indica- and for all minimal augmentations B of A tor variable xe for every edge e ∈ E where we intend xe = 1 (3) ∑ degB (S) ≤ γ · |V (A)|. to mean that edge e is part of the selected subgraph. In the S∈V (A) following, we let δ (S) denote the set of edges in E with exWilliamson et al. showed that (3) holds with γ = 2 whenactly one endpoint in S ⊆ V . ever f is uncrossable [18]. This class of cut-requirement functions captures the Steiner tree and forest problems as (IP) well as many other classical network design problems. opt f := min ∑ ce · xe e∈E
∑
s.t.
xe ≥ f (S) ∀S ⊆ V
3 From primal-dual algorithms to mechanisms
e∈δ (S)
xe ∈ {0, 1} ∀e ∈ E We relax the integrality requirement and replace it by xe ≥ 0 for all e ∈ E. Let the resulting linear program be denoted by (LP). The linear programming dual of (LP) has a variable yS for each set S ⊆ V and a constraint for each edge e ∈ E. max
∑
f (S) · yS
Consider an instance of a network design game (see Section 1) defined by an undirected graph G = (V, E), nonnegative edge costs ce for all e ∈ E, player set N and a cutrequirement function f i that satisfies maximality for each player i ∈ N. For a subset Q ⊆ N, we define function f Q : 2V → {0, 1} as f Q (S) = maxi∈Q f i (S). It is easy to see that f Q satisfies maximality if each of the f i does. Assume that GW is γ -approximate for f Q for all Q ⊆ N, i.e., GW computes a feasible solution A to (IP) with f = f Q of cost at most γ ·opt f Q for all Q ⊆ N. In particular, we assume that the condition in (3) stated in Theorem 1 is satisfied for all Q ⊆ N. We describe a naive approach to turn GW’s dual solution into cost shares. For this let Ri = {v ∈ V : f i ({v}) = 1} be the set of terminals of player i; maximality implies that Ri 6= 0/ unless f i is the all-zero function. Without loss of generality, we also assume that Ri ∩ R j = 0/ for any two distinct players i and j. This can alwaysSbe achieved by copying vertices suitably. Finally let RQ = i∈Q Ri for any Q ⊆ N. Now suppose that we want to compute cost shares ξQ ( j) for some Q ⊆ N and j ∈ Q. For this, we run algorithm GW on input f Q . The algorithm maintains a cost share ξQ (v) for each v ∈ RQ and sets their values to 0 initially. For each time τ and for each set S ∈ V τ , the algorithm GW distributes the growth of dual variable yS evenly over the active terminals in S. Terminal v ∈ Ri ∩ S is active at time τ if f i (S) = 1 and inactive otherwise. At termination we then let for every player i ∈ Q ξQ (i) = ∑ ξQ (v).
(D)
S⊆V
s.t.
∑
yS ≤ ce
∀e ∈ E
(2)
S⊆V :e∈δ (S)
yS ≥ 0 ∀S ⊆ V For an edge set A ⊆ E let degA (S) be the number of edges in A that cross S, i.e., degA (S) = |δ (S)∩A|. The execution of algorithm GW is best explained as a continuous process over time. For each time τ ≥ 0, GW maintains a partial solution Aτ ⊆ E and a feasible dual yτ . Initially, at time τ = 0, we let A0 = 0/ and y0S = 0 for all S ⊆ V . For τ ≥ 0 we say that a set S ⊆ V is violated if f (S) = 1 and degAτ (S) = 0. The algorithm now computes the set of all inclusion-wise minimal violated sets. This is in general not an easy problem but can be done efficiently in our special case. Lemma 1 (see [8], p. 163). Let f : 2V → {0, 1} be a cutrequirement function satisfying the maximality property. The minimal violated sets of an infeasible edge set A ⊆ E are the violated connected components of the induced subgraph G[A]. Let V τ be the set of minimal violated sets at time τ . The algorithm increases the dual variables for all S ∈ V τ simultaneously and at the same rate until a constraint of type (2) for some edge e becomes tight (ties are broken arbitrarily). At this point we include e into A and continue. The algo∗ rithm stops at the first time τ ∗ where Aτ is feasible. In a final reverse delete step, we clean up the feasible solution. ∗ We consider the edges of Aτ in reverse order of addition ∗ ∗ and remove edge e from Aτ if Aτ \ {e} remains feasible. In the following main technical result of [8] we let V (A) be the set of violated connected components of G[A] for any infeasible A ⊆ E. A set B ⊆ E is called a minimal augmentation of A if A ∪ B is feasible and A ∪ B \ {e} is infeasible for all e ∈ B.
v∈Ri
By construction, we have 1 · optQ ≤ ∑ ξQ (i) = ∑ yS ≤ optQ γ S⊆V i∈Q as GW is a γ -approximation algorithm. The resulting costsharing method is not cross-monotone as the following example shows. Example 2. Figure 1 shows an instance of the Steiner forest game defined in Example 1. In the figure, the edges are labeled by their costs. The instance has 3 players N = {1, 2, 3} 3
2 s1
1 s2
1 s3
1 t3
The cost share of player i ∈ Q is the sum of its terminals’ cost shares: 1 ξQ+ (i) = · ∑ ξQ+ (v). (4) α v∈Ri
2 t2
t1
Figure 1: Distributing the dual growth evenly among active terminals does not lead to a cross-monotonic cost sharing Here α is a scaling factor that we will need to determine method. in a problem-specific way; scaling is necessary in order to ensure budget-balance. and the terminal pair of player i ∈ N is (si ,ti ). For player set Q = N, distributing the dual growth uniformly as proposed yields a cost share of ξQ (1) = 3 for player 1. On the other hand, running GW on player set Q′ = {1, 2} and distributing duals accordingly yields ξQ′ (1) = 5/2, violating cross-monotonicity.
4 Analysis
Recall that we used Aτi for the edge set maintained by GW at time τ when run on f i . Similarly, we used AτQ for the corresponding edge set in the run of GW+ on Q ⊆ N. We also use Ciτ and CQτ for the set of connected components of Aτi and AτQ , respectively. Let Aτi (v) ∈ Ciτ and AτQ (v) ∈ CQτ Death-time concept. The inherent problem with the pre- be the connected components containing v in GW and GW+ , vious approach is that the time during the algorithm at which respectively. The following lemma is essential in the subsequent proof terminals of player 1 become inactive depends on the presence of player 3. In the following we revise GW slightly in of feasibility of AQ . order to obtain independent activity times. τ For a player i ∈ N we run GW on function f i . Let Aτi be Lemma 2 (Refinement I). Consider a set S ∈ C′ i forτ some the set of edges maintained by the algorithm at time τ . We i ∈ Q and ′ for some τ ≥ 0. There exists a set S ∈ CQ such then use Aτi (v) to denote the connected component of Aτi that S ⊆ S . containing v. Let the death-time d(v) of terminal v ∈ Ri be Proof. We will prove this lemma for a fixed i ∈ Q by inducthe first time τ for which f i (Aτi (v)) = 0. tion over time τ . The claim clearly holds for τ = 0 as all We now modify algorithm GW in order to determine cross- connected components are singleton vertices. monotone cost shares for the players in a given set Q ⊆ N. Now consider some τ ≥ 0 and assume that the lemma The new algorithm constructs a solution iteratively much holds at this time: for all sets S ∈ C τ there exists a set i like GW and we let AτQ be the partial edge-set constructed S′ ∈ C τ that contains it. The only sets S ∈ C τ that may poi Q until time τ . Recall from the previous section that algorithm tentially violate the lemma are those that are violated (i.e., GW raises the dual variables of all violated connected compo- f i (S) = 1). Consider such a set S. It is straightforward to nents of AτQ simultaneously and at the same rate. The new al- see from the description of GW that maximality implies that gorithm instead raises the dual variables of connected com- there is at least one terminal v ∈ S that has been in a violated ponents of AτQ that are active. We say that terminal v ∈ RQ is connected component of Aτ ′ for all 0 ≤ τ ′ ≤ τ . Therefore i active at time τ if τ < d(v), and a set S ⊆ V is active at that we must have d(v) > τ . By assumption v ∈ S′ and hence S′ time if it contains an active terminal. is active at time τ in GW+ as well. Our modified algorithm GW+ raises the dual of all active connected components of the current forest at all times and The lemma has the following useful consequence: is otherwise equivalent to GW. The algorithm terminates at τ the first time τ ∗ where no active terminal remains. We will Corollary 1. A set S ∈ CQ is active whenever it is violated, ∗ Q later show that the edge set AτQ produced by the algorithm is i.e., whenever f (S) = 1. feasible. Finally, prune the solution in a reverse delete step Proof. Consider a set S ∈ CQτ for some τ ≥ 0 and assume that is identical to that in GW: consider the edges added by i GW+ in reverse order of addition. Remove edge e from the that f (S) = 1 for some i ∈ Q. The above Lemmaτ 2 implies output set of edges if feasibility (with respect to f Q ) is not that S is the disjoint union of sets S1 , . . . , S p ∈ Ci . Assume now, for the sake of contradiction, that S is inactive. This affected. Let the final edge set be AQ . implies that all terminals v ∈ Ri have d(v) ≤ τ . But this Computing cost shares is almost like in the naive api 1 ≤ j ≤ p. It follows from proach: for each terminal v ∈ RQ and for each time τ we implies that f (S j ) =i 0 for all i (S) = 0 as well, and this is a the maximality of f that f τ τ let AQ (v) be the connected component in AQ that contains v. Then let aτQ (v) be the number of active terminals in contradiction. RQ ∩ AτQ (v). The cost share of terminal node v ∈ RQ is then The corollary implies the feasibility of AQ . given by Z d(v) Corollary 2. The edge set AQ produced by algorithm GW+ 1 + dτ . ξQ (v) = for player set Q ⊆ N is feasible for f Q . τ τ =0 a (v) 4
Proof. Corollary 1 shows that violated connected compo- Lemma 4. The edge set AQ computed by GW+ (Q) has cost nents are always active. This implies that GW+ always in- at most αγ · ∑i∈Q ξQ+ (i) if creases the duals of violated sets and therefore eventually (5) adds edges to satisfy them. ∑ degAQ (S) ≤ γ · |AQτ | S∈AQτ
4.1 Cross-monotonicity
for all times 0 ≤ τ ≤ τ ∗ .
prove the cross-monotonicity of GW+
In order to we consider Proof. Let y be the dual produced by GW+ (Q). We first show an arbitrary player i ∈ Q ⊆ N and let Q′ = Q \ {i}. In this that section we study the effect of the removal of i on the cost (6) c(AQ ) ≤ γ · ∑ yS . ′ ∗ ∗ shares of all other players j ∈ Q . Let τQ and τQ′ be the S⊆V termination times of GW+ (Q) and GW+ (Q′ ). The following A standard primal-dual argument gives the following: lemma whose proof is similar to that of Lemma 2 shows Z τ∗ Z τ∗ that CQτ′ refines CQτ for all τ . Q Q c(A ) = c = (S) d deg τ ≤ γ · |AQτ | d τ e Q ∑ ∑ AQ ∗ Lemma 3 (Refinement II). For all times τ ≤ τQ and for all 0 0 τ e∈AQ S∈AQ S′ ∈ CQτ′ there must be a vertex set S ∈ CQτ such that S′ ⊆ S. Proof. The proof is by induction on the time τ . It is clear where the second equality uses the fact that edges are inthat the claim is true for τ = 0 since CQ0′ = CQ0 = V . Consider cluded in AQ only if they are tight and the final inequality a point in time 0 ≤ τ < τQ∗ ′ and assume that the lemma holds. uses (5). The integral on the right-hand side equals ∑S⊆V yS and this proves (6). Algorithm GW+ (Q′ ) grows active sets in CQτ ′ and these are By construction, we have the only sets that can potentially violate the lemma at any time τ + ε for ε > 0. Let S′ ∈ CQτ ′ be an active set at time ξ + (v) = y .
∑
′
∑
Q
S
S⊆V τ in GW+ (Q′ ), i.e., there exists a terminal v ∈ S′ ∩ RQ with v∈RQ d(v) > τ . From the induction hypothesis we know that there + is a connected component S of AτQ that contains S′ . Then v ∈ The definition of ξQ (i) in (4) finishes the proof. S and hence S must be active in GW+ (Q) at time τ . Therefore We finish proving approximate budget balance by showGW+ (Q) grows S at time τ and the lemma follows. ing that the cost shares computed by GW+ are competitive if The previous lemma immediately implies cross- α is chosen sufficiently large. We use d(T ) to denote the maximum death time among all terminals in a tree T . monotonicity.
Lemma 5. The cost shares computed by GW+ are competitive, i.e., ∑ ξQ+(i) ≤ optQ
Corollary 3. The cost shares computed by algorithm GW+ are cross-monotonic. Proof. It suffices to prove that ξQ+′ (v) ≥ ξQ+ (v) for all v ∈
i∈Q
RQ . Consider a terminal v ∈ RQ and let AτQ (v) ∈ CQτ and if there is a forest A¯ whose cost is at most ω · opt and Q Q AτQ′ (v) ∈ CQτ′ be the connected components containing it at α ¯ Q. − 1) · c(T) for every tree T of A d(T ) ≤ ( ω time τ in GW+ (Q) and GW+ (Q′ ), respectively. Also let aτQ (v) and aτQ′ (v) be the number of active terminals in AτQ (v) ∩ RQ Proof. Pick an arbitrary complete order ≺ on the terminals in RQ such that u ≺ v whenever d(u) < d(v). Consider a con′ and AτQ′ (v) ∩ RQ . nected component S of AτQ for some time τ ≥ 0 during the Lemma 3 implies that AτQ′ (v) ⊆ AτQ (v) and hence aτQ′ (v) ≤ execution of GW+ (Q). We say the terminal v ∈ S is responsiaτQ (v) for all τ ≤ τQ∗ . We obtain ble for an active set S if it has highest rank with respect to our Z d(v) Z d(v) order among all terminals in S, i.e., if u ≺ v for all u ∈ RQ ∩ S 1 1 + + ξQ (v) = d τ = ξQ′ (v) dτ ≤ with u 6= v. It is not hard to see that each terminal v ∈ RQ τ τ τ =0 aQ (v) τ =0 aQ′ (v) is responsible in some consecutive interval [0, r(v)] where ′ we call r(v) the responsibility time of v (see also [12]). By Q for all v ∈ R and the corollary follows. construction it is clear that ′
′
∑
4.2 Budget-balance
v∈RQ
r(v) =
∑
v∈RQ
ξQ+ (v).
We first show that the cost shares computed by GW+ recover a substantial fraction of the cost of the computed forest. ConWe now bound the total responsibility time of all termisider a time τ ≥ 0 in the execution of algorithm GW+ on nals. In our analysis we consider each tree T in the given player set Q. Let AQτ be the set of active connected com- forest A¯ Q in isolation. Let RQ (T ) = {v1 , . . . , v p } be the terponents of AτQ . minals in T such that v1 ≺ v2 ≺ . . . ≺ v p . 5
Consider a time τ < r(vi ) for some 1 ≤ i < p. At this time, each of the terminals in vi , . . . , v p is responsible for a distinct active set. Since the terminals in RQ (T ) are connected by T , we know that each set AτQ (v j ), i ≤ j ≤ p, has a non-empty intersection with T . We therefore immediately obtain
It is an easy exercise to show that the Steiner forest cut requirement function f Q introduced in Example 1 is symmetric and hence proper for all Q ⊆ N. Consider a player i ∈ N in this game and let si and ti be the two terminals of the player. It can then be checked that the death time of both terminals equals half the shortest path distance between si p−1 and ti in G. r(v ) ≤ c(T ) (7) ∑ i Now consider a tree T in any optimum Steiner forest FQ∗ i=1 for a given player set Q ⊆ N. Assume that si is the respon(see also [12]). The responsibility time of v p can be at most sible terminal for T . Feasibility now implies that ti is in T α d(v p ) = d(T ) ≤ ( ω − 1) · c(T ) by assumption. Summing as well and hence d(T ) ≤ c(T )/2 for all trees T in FQ∗ . In ¯ over all trees in AQ together with (7) gives Lemma 5 we may now choose A¯ Q to be an optimum Steiner α forest for the terminals of players in Q; we can then set ∑ ξQ+(v) ≤ ω · c(A¯ Q ) ≤ α · optQ α = 3/2 and ω = 1. Together with Lemma 6 we obtain v∈RQ the following theorem: and the definition of cost shares in (4) completes the proof. Theorem 2. The cost-sharing method ξ + is cross-monotone and 3-budget-balanced for the Steiner forest game.
5 Applications
K¨onemann et al. [12] showed that the analysis can be strengthened in order to prove that ξ + is 2-budget-balanced and this is best possible [13]. Unfortunately, the argument leading to Theorem 2 is not sufficient to show that ξ + is approximately budget-balanced for an arbitrary proper game as the following example shows.
We provide a few concrete example classes of cutrequirement functions f for which the above mechanism approximates budget-balance well.
5.1 Proper functions
Example 3 (Parity game). In the parity game, every player i ∈ N has an even number of terminals Ri . The player’s goal is to pair up her terminals and to connect the terminals of each pair. This is possible in a given forest if each connected component of the forest contains an even number of terminals. The cut-requirement function of player i is therefore given by f i (S) = 1 if |S ∩ Ri | is odd and f i (S) = 0 otherwise. It can be verified easily that this function is proper. Figure 2 shows an instance with two players. Player 1 has terminals v1 , . . . , v6 and player 2 has terminals u1 , . . . , u4 . Each edge is labeled with its cost, where 0 < ε ≪ M are some constants. Running GW on the terminals of player 1 yields a death-time of roughly M/2 for terminals v1 and v6 . The unique optimum solution that serves both players is given by the solid edges in the figure and shows that v1 and v6 are in trees of small cost. Choosing M large enough shows that the conditions of Lemma 5 do not hold for any fixed α if A¯ Q is chosen to be an optimum solution. Notice however that choosing A¯ Q as the set of all edges yields a tree whose cost is only slightly higher than that of an optimum forest. The resulting tree T has cost about M + 10 > 2 · d(T ). Choosing ω = 2 and α = 4 in Lemma 5 would certainly work in this case.
A cut-requirement function f is called proper if it satisfies maximality and symmetry, i.e., f (S) = f (V \ S) for all S ⊆ V . In the following we say that a game is proper if f i is proper for all i ∈ N. This implies that f Q is proper for all Q ⊆ N. First we show that Lemma 4 holds with γ = 2 for such games. Lemma 6. In a proper game, the edge set AQ computed by GW+ (Q) has cost at most 2α · ∑i∈Q ξQ+ (i) for all Q ⊆ N. Proof. Assume that GW+ (Q) produces forest AQ . Now consider a time τ ≥ 0 during the execution. We construct a graph H from (V, AQ ) by shrinking all connected components of AτQ . Let W be the set of those nodes in H that correspond to active connected components of AτQ . Proving (5) amounts to showing
∑ degH (v) ≤ 2 · |W |.
v∈W
Goemans and Williamson [8, proof of Theorem 4.7 on p. 170] showed that degH (v) = 1 only if the corresponding set Sv has f Q (Sv ) = 1 and hence, using Corollary 1, only if v ∈ W . We then have !
∑ degH (v) ≤ 2|V (H)| − 2 − ∑ degH (v)
We note that the cut requirement function in the above parity game is not, what Chekuri and Shepherd [4] call fastidious. A proper function f is fastidious if there do not ≤ 2|V (H)| − 2 − 2|V(H) \ W | = 2|W | − 2 exist sets S, S1 , . . . , Sl ⊆ V such that f (S) = 1, f (Si ) = 0 where the first inequality uses the fact that H is a forest, and for all 1 ≤ i ≤ l and S = S1 ∪ . . . ∪ Sl . The authors showed the second inequality follows as every node that is not in W that fastidious cut-requirement functions encode generalized has degree at least 2 in H. Steiner network problems. As such the results in Theorem v∈W
v6∈W
6
v1 1 u1 1 v2
2−ε
v3
M
1
v4
2−ε
v5 1 u3 1 v6 1
u2
u4
Figure 2: A two-player proper game that defies a Steiner forest-style analysis. 2 and [12] apply and show that GW+ and its cost-sharing method is cross-monotonic and 2-budget-balanced. We conclude this section with the observation that GW+ yields a cross-monotonic and approximately budgetbalanced cost-sharing method for the parity game in certain cases. Consider the run of GW( f i ) on the terminal set of player i and fix a terminal u ∈ Ri . At time d(u) the connected component of u is united with another connected component that contains a terminal u¯ ∈ Ri with d(u) ¯ = d(u). It is easy to verify that u¯ is unique in the parity game. We call u¯ the mate of u and define d(u, u) ¯ = d(u). The proof of the following theorem is given in Section 6.
Lemma 7. In a downwards monotone game, the edge set AQ computed by GW+ (Q) has cost at most 2α · ∑i∈Q ξQ+ (i) for all Q ⊆ N.
Proof. Assume that GW+ (Q) produces forest AQ . Now consider a time τ ≥ 0 during the execution. We construct a graph H from (V, AQ ) by shrinking all connected components of AτQ . Let W be the set of those nodes in H that correspond to active connected components of AτQ . Goemans and Williamson (see Thm 4.5 in Chapter 4 of [8]) showed that each connected component of H has at most one vertex v whose corresponding set Sv has f Q (Sv ) = 0. Corollary 1 implies that v ∈ W if f Q (Sv ) = 1 and this together with Goemans and Williamson’s lemma implies that Theorem 3. For any β ≥ 1, ξ + is a cross-monotonic and each connected component of H contains at most one vertex (2β + 4)-budget-balanced cost-sharing method for the par- that is not in W . We therefore have ity game if cu,u¯ ≤ β · d(u, u) ¯ for all u ∈ Ri and for all i ∈ N. ∑ degH (v) ≤ 2(|H| − c) ≤ 2|W | v∈W A concrete application of the above theorem is provided by instances of the parity game where each player i has a where c is the number of connected components of H. constant number of terminals. Consider a mate pair u, u¯ for some player i ∈ N and let Pu,u¯ be the unique tight path in We next show that each a terminal v ∈ RQ must lie in a the solution computed by GW( f i ). As Ri has constant car- heavy connected component in any feasible solution AQ for dinality, there can only be a constant number of active con- Q. nected components at any time τ ≥ 0 that load the edges of this path. As Pu,u¯ becomes tight at time d(u, u), ¯ its cost is Lemma 8. Let AQ be an arbitrary feasible solution for cutQ O(d(u, u)) ¯ as well. Therefore, the above theorem implies requirement function f and let AQ (v) be the connected + that ξ is a constant-budget-balanced cost-sharing method component of AQ that contains terminal v. We must have Q for the parity game when each player has a constant number c(AQ (v)) ≥ d(v) for all v ∈ R . of players. Proof. Assume that v is a terminal of player i ∈ Q. From We conjecture that ξ + is constant budget-balanced for the definition of death-time it follows that v’s connected proper games in general. component Aτi (v) in the run GW( f i ) is violated for all that Conjecture 1. ξ + is a cross-monotonic and constant 0 ≤ τ ≤ d(v). Assume for the sake of contradiction τ (v) for c(A (v)) < d(v). This implies that A (v) ⊆ A Q Q i budget-balanced cost-sharing method for proper games. some τ ∈ (c(AQ (v)), d(v)). As Aτi (v) is violated we have As outlined above, this is true for fastidious proper games, f i (Aτi (v)) = 1, and this together with downwards monoand Theorem 3 indicates that potential counterexamples tonicity implies that f i (AQ (v)) = 1. This contradicts the fact need to be relatively complex. that AQ is feasible for f Q . Lemma 8 implies that we may choose α = 2 and ω = 1 in Lemma 5 in order to achieve competitiveness. Together A cut-requirement function f is called downwards mono- with Lemma 7 we obtain the following theorem: tone if it satisfies f (B) ≤ f (A) for all A ⊆ B. Downwards Theorem 4. The cost-sharing method ξ + is cross-monotone monotone functions clearly satisfy maximality. Call a game and 4-budget-balanced for downwards monotone games. downwards monotone if f i is downwards monotone for all i ∈ N. This implies that f Q is downwards monotone for all We need to choose α ≥ 3/2 for downwards monotone Q ⊆ N. In this section we analyze the budget-balance of games in order to attain competitiveness as the following example shows. algorithm GW+ for downwards monotone games.
5.2 Downwards monotone functions
7
restore the v, v-connectivity ¯ in F ∗ + F. In the first step, we identify a candidate v, v-path. ¯
Example 4 (Edge cover game). In the edge cover game, we are given a weighted undirected graph G = (V, E). Each player i ∈ N owns a vertex vi ∈ V and wants vi to be covered by an edge, i.e., f i ({vi }) = 1 and f i (S) = 0 for all other sets S ⊆ V . Consider the instance of this game where G consists of a single edge (v1 , v2 ) of cost 1 and where the player set is N = {1, 2}. The death times for each of the terminals is 1 and running GW+ on N will result in a cost share of 3/4 for each of the players, leading to a cost share sum of 3/2 > 1.
6.1 Finding a v, v-path ¯ We construct an auxiliary graph G that has one vertex vT ′ for each tree T ′ of F ∗ + F. Let T (u) be the tree in F ∗ + F that contains terminal u ∈ Ri , and add an edge (vT (u) , vT (u) ¯ ) to G if (u, u) ¯ is separated.
The method presented in Section 3 can be adapted in order to obtain the following improvement of the last theorem; the Lemma 9. G is an Eulerian graph that has a cycle through vT (v) and vT (v) ¯ . proof is given in Appendix A. Theorem 5. There is a group-strategyproof and 2-budget- Proof. For a tree T ′ of F ∗ + F, let Ri (T ′ ) be the set of mate balanced mechanism for downwards monotone games. pairs of player i that are separated by T ′ , i.e., those pairs (u, u) ¯ ∈ Ri with |{u, u} ¯ ∩ V (T ′ )| = 1. Parity implies that This result is tight as the edge-cover game from Example i ′ R (T ) has an even number of pairs for all trees T ′ , and hence 4 is a downwards monotone game and Immorlica et al. [9] every node of G has even degree. showed that no cross-monotonic cost-sharing method with Now consider an arbitrary cut set S ⊆ V (G ) \ {vT (v) ¯ }, and budget-balance factor smaller than 2 exists. let S be the set of vertices of G that are spanned by the trees corresponding to the nodes in S. Assume that there is a single edge in G that crosses S, and hence δG (S) = 6 A mechanism for the parity game {(vT (v) , vT (v) ¯ )}. By definition of G this means that v is the only terminal in S whose mate is in V \ S . Therefore, the In this section, we consider the parity game from Example + number of terminals spanned by the trees in S is odd. This 3. The cost shares induced by GW are cross-monotonic, contradicts the feasibility of F ∗ + F. but showing budget-balance is not straight-forward as we pointed out in Example 3. In particular, an optimum soluLemma 9 implies that G \ {(vT (v) , vT (v) tion FQ∗ for some Q ⊆ N may have trees T with d(T ) > c(T ) ¯ )} has a as the example shows. In this section we show that, un- vT (v) , vT (v) ¯ -path P = hvT1 , . . . , vTp i, where T1 = T (v) and ¯ Each edge of the path corresponds to a mate pair; der certain conditions, the forest FQ∗ may be patched such Tp = T (v). that the resulting, slightly more expensive forest satisfies the let (ui , u¯i ) be the mate pair for edge (vTi , vTi+1 ), 1 ≤ i ≤ p − 1. conditions in Lemma 5. This together with Lemma 6 will then give us the sought-after approximate budget-balance. The plan of attack is as follows: Fix a player-set Q ⊆ N 6.2 Charging edges and let F ∗ be a minimum-cost solution for cut function f Q . Our plan is to connect some of the separated mate pairs We will now add edges to F ∗ iteratively such that, in the in {(u , u¯ ), . . . , (u , u¯ )} in order to place v in a non1 1 p−1 p−1 end, no tree in the resulting forest is violated; a tree T is violated tree in F ∗ + F. We modify graph G to obtain a simviolated if its death-time is larger than its cost. The cost of plified auxiliary graph H to facilitate charging. Let yi be the added edges is charged to the edges of F ∗ ; we mark a tree the feasible dual computed by GW when executed on f i . AsT of F ∗ if its edges have been used in the charging process. sume, w.l.o.g., that G is a complete graph and that c equals uv Consequently all trees of F ∗ are unmarked initially. In the the minimum-cost of any u, v-path in G for all u, v ∈ V . following, let F be the forest of edges that we add to F ∗ and Initialize H = G, and let the cost c¯e of each edge e ∈ let F ∗ + F be the resulting patched forest. We maintain the E(H ) be ce initially. Also let y¯iS be a new dual, and let following two invariants: y¯i = yi initially. Each of the following steps modifies H , c, ¯ ∗ or y¯i . We maintain that (a) y¯ is feasible for (D) with respect (I1) No marked tree of F + F is violated. to H , and (b) the metric induced by c¯ on H dominates that (I2) The maximum death-time of all violated trees is at most induced by c on G, i.e., distances in H with respect to c¯ are ∗ the cost of any marked tree of F + F. bigger than or equal to distances in G with respect to c. As a first step, we will shorten edge (ui , u¯i ) and let c¯ui ,u¯i = In each iteration, pick a violated tree T of F ∗ + F of highest death-time d(T ). Assume that d(T ) = d(v) for some ter- 2d(ui , u¯i ) for all 1 ≤ i ≤ p − 1. This does not affect the ¯ and, furthermore, c¯ continues to be a metminal v ∈ V (T ) ∩ Ri . We first observe that 2 · d(v) is a lower- feasibility of y, bound on the minimum-cost of any path between v and its ric. We want the vertex set of our auxiliary graph to be ¯ u1 , u¯1 , . . . , u p−1 , u¯ p−1 }. As long as H has mate v¯ in G. This and the fact that T is violated immedi- V (H ) = {v, v, ately implies that the mate pair (v, v) ¯ is separated in F ∗ + F. a vertex w 6∈ V (H ), remove w and all incident edges from One way of ensuring that v’s component is not violated is to H . Also project down the dual onto the new vertex set; let 8
T1 v
Tp−1
T2 u1
u2
u¯1
u¯ p−2
u¯2
Tp
u p−1
u¯ p−1
v¯
Figure 3: Auxiliary graph H (red edges are drawn in bold). the dual value of set S ⊆ V (H ) be
∑
The corollary lets us charge the dual growth of moats Aτi (w) for w ∈ V \ {v, v} ¯ to the black edges in E.
y¯iS′ .
Corollary 5. Let w ∈ V \ {v, v} ¯ be a terminal with d(w) ≤ d(v, v). ¯ Then, there is at least one black edge incident to τ Clearly, this step does maintain properties (a) and (b) defined A¯ i (w) for at all times 0 ≤ τ < d(w). above. ¯ and Aτi (w) are pairwise disjoint for Proof. As Aτi (v), Aτi (v), Finally, we want the edge-set of H to be E(H ) = Eb ∪ any 0 ≤ τ < d(w) by Lemma 10, it follows that |δ (Aτ (w)) ∩ i Er , where E| ≥ 2 for all τ , and hence, by Corollary 4, there must be at least one black edge incident to w’s moat at all times 0 ≤ Eb = {(v, u1 ), (u¯1 , u2 ), . . . , (u¯ p−2 , u p−1 ), (u¯ p−1 , v)}, ¯ and τ < d(w). Er = {(u1 , u¯1 ), . . . , (u p−1 , u¯ p−1 )} Armed with the above lemmas and corollaries, we will now prove that there is an index r ∈ {2, . . . , p} such that are the sets of black and red edges, respectively. Remove all S′ ⊆V (H )∪{w},S=S′ \{w}
edges in E(H ) that are neither black nor red. Once again, this preserves properties (a) and (b). The final graph H which is a v, v-path ¯ is depicted in Figure 3. We now simulate the run GW( f i ) on graph H by executing GW+ on H with the player i death times. That is, terminal w ∈ V is active at time τ iff τ < d(w). Recall that Aτi (w) is the connected component of tight edges containing vertex w at time τ in GW( f i ). Similarly, we use A¯ τi (w) to be w’s component in the GW+ run. The following lemma shows that A¯ τi (w) and A¯ τi (w′ ) are disjoint in H whenever Aτi (w) and Aτi (w′ ) are disjoint in G. Its proof is similar to that of Lemma 2 and we omit it here.
r−1
2 ∑ d(ui , u¯i ) ≤ i=1
∑
c(Ti ),
(8)
1≤i≤r Ti unmarked
and adding edges (u1 , u¯1 ), . . . , (ur−1 , u¯r−1 ) creates a new marked tree that is not violated. We branch into two cases. Case 1: None of the trees T1 , . . . , Tp is marked and d(ui , u¯i ) ≤ d(v, v) ¯ for all 1 ≤ i ≤ p − 1. Corollary 5 immediately implies that p−1
p
p
i=1
i=1
i=1
2 ∑ d(ui , u¯i ) ≤ ∑ cu¯i−1 ,ui ≤ ∑ c(Ti )
Lemma 10. For all w ∈ V and for all τ ≥ 0, we have where we let u¯0 = v, and u p = v. ¯ This implies (8) with r = p. A¯ τi (w) ⊆ Aτi (w). Furthermore, edge (ui , u¯i ) becomes tight The choice of v implies that d(Ti ) ≤ c(Ti ) or d(Ti ) ≤ d(v, v) ¯ at time d(ui , u¯i ) for all 1 ≤ i ≤ p − 1. for all 1 ≤ i ≤ p. Hence, the component created by adding (u1 , u¯1 ), . . . , (u p−1 , u¯ p−1) is not violated. It is easy to check The lemma has the following important consequence. that invariants (I1) and (I2) continue to hold. For the remaining two cases, let Corollary 4. For all 1 ≤ i ≤ p − 1, w ∈ {ui , u¯i }, and for all τ ¯ (w) is crossed by (u , u ¯ ), and has 0 ≤ τ < d(w), the cut Ai i i l = min{2 ≤ i ≤ p : Ti marked or d(ui−1 , u¯i−1 ) > d(v, v)}. ¯ no other incident red edges. Case 2: d(ul−1 , u¯l−1 ) ≤ d(v, v). ¯ In this case Tl must be marked. Lemma 10 and Corollary 5 imply that, at all times 0 ≤ τ < d(v, v), ¯ there can be at most one active moat that contains u¯l−1 . This moat may not intersect any of the components T1 , . . . , Tl−1 , and its dual growth can therefore not be charged. Lemma 10 and Corollary 5 therefore imply
Proof. The edge (ui , u¯i ) has cost 2d(ui , u¯i ) and it is therefore clear that A¯ τi (w) is incident to this edge. Furthermore, the edge (ui , u¯i ) is tight at time d(ui , u¯i ). Now consider two terminals s,t ∈ V and a time τ < min{d(s), d(t)}. From the definition of the algorithm GW, it is easy to see that s and t are the only active terminals in their respective moats, and hence Aτi (s) and Aτi (t) must be disjoint. Lemma 10 implies that A¯ τi (s) and A¯ τi (t) are disjoint as well. It therefore follows that, for any 1 ≤ j ≤ p − 1, moats A¯ τi (u¯ j ) and A¯ τi (u j ) are disjoint from any other growing moat at all times 0 ≤ τ ≤ d(u j , u¯ j ). Thus no moat other than those around u j and u¯ j can be incident to edge (u j , u¯ j ). The lemma follows.
l−1
l−1
l−1
i=1
i=1
i=1
d(v) + 2 ∑ d(ui , u¯i ) − d(v) ≤
∑ cu¯i−1,ui ≤ ∑ c(Ti ).
This implies (8) for r = l. The choice of v and our assumptions imply that d(Ti ) ≤ c(Ti ) or d(Ti ) ≤ d(v, v) ¯ for all 1 ≤ i ≤ r − 1. Hence, the component created by adding (u1 , u¯1 ), . . . , (ur−1 , u¯r−1 ) is not violated. It is easy to check that invariants (I1) and (I2) continue to hold. 9
Case 3: d(ul−1 , u¯l−1 ) > d(v, v). ¯ Tree Tl−1 is not marked by assumption and hence d(ul−1 ) ≤ c(Tl−1 ). Furthermore, by Lemma 10, A¯ τi (v), A¯ τi (ul−1 ), and A¯ τi (w) are pairwise disjoint for all 0 ≤ τ < d(w), and for all w ∈ {u1 , u¯1 , . . . , ul−2 , u¯l−2 }. Hence, l−2
l−1
l−1
i=1
i=1
i=1
2 ∑ d(ui , u¯i ) ≤
∑ cu¯i−1,ui ≤ ∑ c(Ti ).
[6] A. Gupta, J. K¨onemann, S. Leonardi, R. Ravi, and G. Sch¨afer. An efficient cost-sharing mechanism for the prize-collecting Steiner forest problem. In Proceedings, ACM-SIAM Symposium on Discrete Algorithms, pages 1153–1162, 2007. ´ Tardos. Cost-sharing mech[7] A. Gupta, A. Srinivasan, and E. anisms for network design. In Proceedings, International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, 2004. [8] D. Hochbaum, editor. Approximation Algorithms for NP-hard
Problems. PWS Publishing Company, 1997. This implies (8) for r = l − 1. Adding (u1 , u¯1 ), . . . , (ul−1 , u¯l−1 ) creates a new marked compo- [9] N. Immorlica, M. Mahdian, and V. S. Mirrokni. Limitations of cross-monotonic cost sharing schemes. In Proceedings, nent that is easily checked to be non-violated. Invariants ACM-SIAM Symposium on Discrete Algorithms, pages 602– (I1) and (I2) continue to hold as before. 611, 2005. Repeating the procedure until no violated components re∗ main yields a forest F + F. We are ready to prove Theorem [10] K. Jain and V. V. Vazirani. Applications of approximation algorithms to cooperative games. In Proceedings, ACM Sym3. posium on Theory of Computing, pages 364–372, 2001.
Proof of Theorem 3. Let A be the set of all mate pairs that [11] K. Kent and D. Skorin-Kapov. Population monotonic cost allocation on mst’s. In Operational Research Proceedings were added in the patching process. By construction we KOI, pages 43–48, 1996. have ∗ [12] J. K¨onemann, S. Leonardi, and G. Sch¨afer. A group2 ∑ d(u, u) ¯ ≤ c(F ),
strategyproof mechanism for Steiner forests. In Proceedings, ACM-SIAM Symposium on Discrete Algorithms, pages 612 – 619, 2005.
(u,u)∈A ¯
and hence
∑
cu,u¯ ≤ β ·
(u,u)∈A ¯
∑
(u,u)∈A ¯
d(u, u) ¯ ≤
β c(F ∗ ). 2
[13] J. K¨onemann, S. Leonardi, G. Sch¨afer, and S. van Zwam. From primal-dual to cost shares and back: A stronger LP relaxation for the Steiner forest problem. In Proceedings, International Colloquium on Automata, Languages and Processing, pages 930–942, 2005.
Thus, the cost of F ∗ +F is at most (β +2)/2·c(F ∗ ). In other words, we may choose ω = (β + 2)/2 in Lemma 5. None [14] S. Leonardi and G. Sch¨afer. Cross-monotonic cost sharof the trees T of F ∗ + F is violated and hence d(T ) ≤ c(T ). ing methods for connected facility location games. Theoret. Therefore, we may choose α = 2ω = β + 2 in Lemma 5. Comput. Sci., 326(1-3):431–442, 2004. Using Lemma 6 we obtain that the cost-sharing method is [15] A. Mehta, T. Roughgarden, and M. Sundararajan. Beyond (2β + 4)-budget-balanced. Moulin mechanisms. In Proceedings, ACM Conference on Electronic Commerce, 2007.
References [1] A. Agrawal, P. Klein, and R. Ravi. When trees collide: An approximation algorithm for the generalized Steiner problem in networks. SIAM J. Comput., 24:440–456, 1995. [2] Y. Bleischwitz and B. Monien. Fair cost-sharing methods for scheduling jobs on parallel machines. In Proceedings, International Conference on Algorithms and Complexity, volume 3998 of Lecture Notes in Computer Science, pages 175–186. Springer, 2006. [3] J. Brenner and G. Sch¨afer. Cost sharing methods for makespan and completion time scheduling. In Proceedings, Symposium on Theoretical Aspects of Computer Science, pages 670–681, 2007. [4] C. Chekuri and F. B. Shepherd. Approximate integer decompositions for undirected network design problems. Manuscript, 2004. [5] M. X. Goemans and D. P. Williamson. A general approximation technique for constrained forest problems. SIAM J. Comput., 24:296–317, 1995.
10
[16] H. Moulin and S. Shenker. Strategyproof sharing of submodular costs: budget balance versus efficiency. Econom. Theory, 18(3):511–533, 2001. ´ Tardos. Group strategyproof mechanisms [17] Martin P´al and Eva via primal-dual algorithms. In Proceedings, IEEE Symposium on Foundations of Computer Science, pages 584–593, 2003. [18] D. P. Williamson, M. X. Goemans, M. Mihail, and V. V. Vazirani. A primal-dual approximation algorithm for generalized steiner network problems. Combinatorica, 15(3):435–454, 1995.
for all S ⊆ V . Assume that the forest F τ at time τ is infeasible. We use F¯ τ to denote the subgraph of G that is induced by the tight edges for dual yτ . In the following, we will also the term moat to refer to a connected component S of In the following discussion we assume that we are given an use ¯τ. F instance of a downwards monotone game with player set As before, a terminal v ∈ RQ is active at time τ if d(v) > τ N and downwards monotone cut-requirement function f i of S of F¯ τ is active if it contains player i. As before we use Ri to denote the set of terminals and a connected component τ Let A be the set of all connected comof player i and let R be the set of all terminals in the game. an active terminal. ¯ τ that are active at time τ . KLS raises the dual ponents of F i For all players i ∈ N and for all v ∈ R we let variables for all sets in A τ uniformly at all times τ ≥ 0. We i i Sv = {S ⊆ V : f (S) = 1 and f (T ) = 0 for all S ( T ⊆ V } say that moats S1 and S2 collide at time τ if 1. S1 and S2 are moats at some time τ ′ < τ , and be the set of maximal f i -violated vertex-sets. Use S¯ as a 2. τ is the first time during the execution of the algorithm short for the complement V \ S of S ⊆ V , and define the at which forest F¯ τ contains a connected component death-time of terminal v as containing the vertices of both S1 and S2 . ¯ c(v, S) d(v) = max (9) If this happens, we add the edges on a shortest S1 , S2 -path to 2 S∈Sv F τ and continue. The algorithm terminates when no active
A
An improved mechanism for downwards monotone games
¯ is the minimum cost of any v, S¯ path. For ter- terminal remains. where c(v, S) Algorithm KLS is in itself an adaptation of a primal-dual ¯ minal v and S ∈ Sv , let PvS be an arbitrary v, S-path of cost algorithm for the Steiner forest problem due to Agrawal et ¯ We refer to the endpoint v¯ 6= v of PvS as the mate of v c(v, S). al. [1]. In the following main technical result of [1] we let FQ with respect to S. be the forest produced by KLS. We further use CQ to denote i the connected components (i.e., the trees) of FQ . For a tree T Corollary 6. Let F be a forest that is infeasible for f , i.e., i of F Q , we use d(T ) for the maximum death-time among all there is a tree T in F for which f (T ) = 1. Then there must i terminals in RQ ∩V (T ). Let y be the dual solution produced exist a terminal v ∈ R ∩ T that has one of its mates outside by the algorithm. T. Theorem 6 (see [1]). The forest FQ produced by KLS has Proof. Let S ⊆ V be a maximal f i -violated set that contains cost at most 2 ∑S⊆V yS − 2 ∑T ∈CQ d(T ). T . Downwards monotonicity implies that player i has at least one terminal v in T . The mate v¯ of v with respect to Computing cost shares. The cost shares for players in Q S is not in T . are computed just as before: the algorithm computes a cost Q Now suppose we want to compute cost shares for players share for each terminal v ∈ R . The cost share of v is 0 at Q ⊆ N. The algorithm differs from that presented in Section time τ = 0. At all times during the algorithm, the growth of 3 and has two phases. In the first phase we run algorithm variable yS for some active moat S is shared evenly among KLS from [12]. The algorithm produces a forest FQ and cost all active terminals in S. Formally, let AτQ (v) be the connected component in F¯ τ shares ξQ (i) for all i ∈ Q. Forest FQ may not be feasible for Q f and hence we patch up FQ in a second phase. that contains v ∈ RQ . Also let aτQ (v) be the number of active terminals in AτQ (v). We then define the cost share of terminal vertex v ∈ RQ as A.1 Phase I: Algorithm KLS Just like GW and GW+ , algorithm KLS is a primal-dual algorithm. As such it attempts to produce a feasible solution for (IP) with cut requirement function f Q in place of f and at the same time it produces a dual lower-bound. A dual solution once again assigns a non-negative value yS to every set S ⊆ V and its value is ∑S yS . The main difference between the new algorithm and GW and GW+ is that it is lazy: edges are not added when they become tight but only if a tight path between two active connected components appears. Because of this more careful way of adding edges, the algorithm does not need a reverse-delete step. Once again the algorithm is best viewed as a process over time. Let F τ and yτ be the partial forest and the dual lowerbound at time τ . Initially, F 0 is the empty forest and y0S = 0 11
ξQ+ (v) =
Z d(v)
1
τ τ =0 aQ (v)
dτ
and we let ξQ+ (i) = ∑v∈Ri ξQ+ (v) for all i ∈ Q. As the dual growth process is the same as that of GW+ it can easily be seen that ξ + is a cross-monotonic cost-sharing method (see also [12]). Lemma 11. The cost shares ξ + computed by KLS are crossmonotonic. It is easy to construct example instances for which the above run of KLS with the death-times defined in (9) does not produce a feasible solution for f Q . We therefore add a second patching-phase in which paths are added to FQ in order to obtain a feasible solution.
A.2 Phase II: Patching FQ
A.3 Analyzing the budget balance of ξ +
We describe an iterative procedure for adding paths to FQ in order to obtain a feasible solution for f Q . Initially, let F0 = FQ . Let T0 be the set of all violated trees in F0 and give each tree T ∈ T0 a budget b0 (T ) of 2d(T ). In iteration i of the procedure, we start with an infeasible forest Fi whose violated trees are Ti . In the iteration, we will find a path Pi to add to Fi in order to produce Fi+1 . Inductively we maintain the following invariants:
We first prove that Fk has cost at most twice the total dual value computed by KLS.
i−1
∑ c(Pj ) + ∑
j=0
bi (T ) ≤
∑
b0 (T )
d(v) ≤ bi (T )/2 ∀T ∈ Ti , ∀v ∈ T
Proof. Theorem 6 together with invariant (I1) in iteration k imply k−1
c(Fk ) ≤ c(FQ ) + ∑ c(Pi ) i=0
(I1)
T ∈T0
T ∈Ti
Lemma 12. The final forest Fk has cost at most 2 ∑i∈Q ξQ (i).
(I2)
The two invariants clearly hold for i = 0. We describe iteration i ≥ 0. Choose a violated component Ti ∈ Ti of smallest death time d(Ti ). Corollary 6 guarantees the existence of a v ∈ RQ ∩ Ti and S ∈ Sv such that v’s mate with respect to S is not in Ti . Let Pi = PvS and obtain
≤2
∑ yS − 2 ∑
S⊆V
≤2
T ∈CQ
∑ yS − ∑
S⊆V
T ∈T0
k−1
d(T ) + ∑ c(Pi ) i=0
k−1
b0 (T )) + ∑ c(Pi ) ≤ 2 i=0
∑ yS
S⊆V
where the second inequality uses Theorem 6, the third inequality uses the definition of b0 (T ) and that of T0 , and the c(Pi ) ≤ 2d(v) ≤ bi (Ti ) (10) last inequality uses invariant (I1). The lemma follows from the fact that ∑S yS = ∑i∈Q ξQ (i). where the first inequality uses the definition of death time in (9), and the inequality uses (I2). It remains to show that the sum of the cost shares of Let T1′ , . . . , Tp′ be the set of trees in Fi that contain vertices players in Q is at most optQ . We first develop an intuof path Pi . W.l.o.g., assume that v ∈ T1 and thus v¯ 6∈ T1 . itive view of feasible solutions for downwards monotone Adding Pi to Fi and removing cycles creates a new forest cut-requirement functions in the following lemma. Fi+1 which contains a tree Tvv¯ whose vertex set contains the Lemma 13. Let f : 2V → {0, 1} be a downwards monotone vertices of T1′ , . . . , Tp′ . We let cut-requirement function. Then there exists a family S of p vertex subsets such that A ⊆ E is feasible for f iff for all bi+1 (Tvv¯ ) = ∑ bi (T j ) S ∈ S and for all v ∈ S, A contains a v, u-path for some j=2 u 6∈ S. be the budget of the new tree. All other budgets remain the same. Proof. We choose S to be the family of inclusion-wise Inequality (10) immediately implies maximal sets S with f (S) = 1. First assume that A is feasible for f . Consider an arbii trary set S ∈ S and some vertex v ∈ S. Let C ⊆ S be a ∑ c(Pj ) + ∑ bi+1(T ) ≤ ∑ b0(T ) ¯ j=1 T ∈T0 T ∈Ti+1 v, S-cut (where S¯ = V \ S is the complement of S). Using the fact that f (S) = 1 together with downwards monotonicity and invariant (I1) continues to hold in iteration i + 1. Observe first that downwards monotonicity entails that implies that f (C) = 1 and hence degA (C) ≥ 1. A standard ¯ f Q (Tvv¯ ) = 0 if f Q (T j′ ) = 0 for any 1 ≤ j ≤ p. So, if T j′ 6∈ Ti maxflow-mincut argument now implies that there is a v, Sfor some 1 ≤ j ≤ p, then invariant (I2) holds trivially by in- path in A. Now assume that we are given an edge set A ⊆ E s.t. there duction. Therefore, we assume from now on that T j′ ∈ Ti for ¯ is a v, S-path in A for all S ∈ S and for all v ∈ S. Let S′ ⊆ V all 1 ≤ j ≤ p. Notice that our choice of Ti implies that be a vertex set with f (S′ ) = 1. There must be a set S ∈ S d(Ti ) = d(T1′ ) ≤ min d(T j′ ). ¯ that contains S′ . Let v be a vertex in S′ . As A has a v, S-path 2≤ j≤p ′ we clearly must have degA (S ) ≥ 1. Hence, for all terminals v ∈ Tvv¯ we have Lemma 14. ∑i∈Q ξQ (i) ≤ optQ . d(v) ≤ d(T ) ≤ b (T ) vv¯
i+1
vv¯
for all v ∈ Tvv¯ . Thus, invariant (I2) also continues to hold. Finally notice that each iteration of the patching phase reduces the number of violated trees: Either T j′ 6∈ Ti for some 1 ≤ j ≤ p in which case Tvv¯ 6∈ Ti+1 , or at least two violated trees are combined into at most one. The procedure terminates in the first iteration k for which Fk has no violated trees left. 12
Proof. Order the terminals in RQ by non-decreasing deathtime such that d(v) < d(u) for u, v ∈ RQ implies v ≺ u. Let AτQ (v) be the moat containing terminal v ∈ RQ at time τ ≥ 0 in KLS(Q). Recall that we reserve the term moat for connected components of F¯ τ . Now call a terminal v ∈ RQ responsible for moat AτQ (v) if there is no terminal u ∈ AτQ (v), v 6= u, such that v ≺ u. It can be shown (see [12]) that each
terminal v ∈ RQ is responsible in some interval [0, r(v)] and where the last inequality uses (11). Path P has length at least nowhere outside. We refer to r(v) as v’s responsibility time. 2d(v p) and this is at least r(v p ). Therefore we have By definition, we have
∑ ξQ (i) =
i∈Q
∑
p
∑ r(vi ) ≤ c(T )
r(v)
i=1
v∈RQ
and hence it is sufficient to bound the sum on the right. Consider an optimum solution FQ∗ for f Q . Let T be an arbitrary tree in FQ∗ and let v1 , . . . , v p be the terminals in RQ ∩ V (T ). W.l.o.g., assume that r(v1 ) ≤ . . . ≤ r(v p ). First assume that all moats for which terminals v1 , . . . , v p are responsible intersect T . It is not hard to see that this is true if δ (AτQ (v p )) ∩ T 6= 0/ for all 0 ≤ τ ≤ r(v p ). In this case, we can charge the total growth of moats owned by v1 , . . . , v p to the cost of T : p
∑ r(vi ) ≤ c(T )
i=1
(see [12] for a more formal argument). Now let τ0 be the earliest time during the execution of KLS(Q) where AτQ (v p ) does not intersect T . The moat AτQ (v p ) contains terminals v1 , . . . , v p−1 for all τ ≥ τ0 and thus v p is the highest ranked terminal spanned by T . Lemma 13 shows that T has a v p , u-path P of length at least 2d(v p ) for some vertex u. For all 1 ≤ i ≤ p and for all 1 ≤ τ ≤ r(vi ) let degτP (vi ) = |δ (AτQ (vi )) ∩ P| be the number of edges of P that cross vi ’s moat at time τ . Notice that v p ’s moat intersects P at least once at all times 0 ≤ τ ≤ τ0 . Since P is a path with one end in v p , there can be at most one other active moat intersecting P exactly once at all times 0 ≤ τ ≤ τ0 . Let l1 ≤ τ0 be the total load of such moats on P. For 0 ≤ τ ≤ τ0 , let M τ be the set of those terminals in RQ that are responsible for a moat that intersects P at least twice. Feasibility implies that
τ0 + l1 +
Z τ0 0
∑ τ degτP (v) d τ ≤ c(P)
v∈M
Let sl(P) be the slack in the above inequality. Since degτP (v) ≥ 2 for all τ and for all v ∈ M τ , and since τ0 ≥ l1 , this implies sl(P) + τ0 +
Z τ0 0
∑ τ (degτP(v) − 1) d τ ≥
v∈M
c(P) . 2
(11)
Before time τ0 , all moats are intersecting T . Hence we obtain p−1
c(T ) ≥
∑ r(vi ) + τ0 + sl(P) +
i=1
p−1
≥
∑ r(vi ) +
i=1
Z τ0 0
∑
v∈M τ
(degτP (v) − 1) d τ
c(P) 2
13
and summing over all trees T of FQ∗ yields the lemma. Lemmas 11, 12 and 14 together imply Theorem 5.
