The Index Coding Problem: A Game-Theoretical Perspective

Transcription

1 The Index Coding Problem: A Game-Theoretical Perspective Yu-Pin Hsu, I-Hong Hou, and Alex Sprintson Department of Electrical and Computer Engineering Texas A&M University {yupinhsu, ihou, spalex}@tamu.edu Abstract Recently, the Index Coding problem has attracted a significant interest from the research community. In this problem, a server needs to deliver packets to a set of wireless clients over a noiseless broadcast channel. Each client needs to obtain a certain subset of packets and can take advantage of another subset available to it as side information. The objective is to satisfy the demands of all clients with the minimum number of transmissions. In this paper, we study the Index Coding problem from the game-theoretic perspective. We assume that each client is selfish and has a hidden private value for each packet it requests. The objective of server is to maximize the social welfare that captures the trade-off between the values of the received packets and the transmission cost incurred by the server. The transmission process is decided through an auction in which the clients are required to submit bids to the server. Each bid implies an obligation of a client to pay for the packets it can obtain from the channel. Our goal is to design a truthful auction scheme that provides an incentive for each client to bid the true value of the packets. We propose a truthful mechanism to achieve the maximum social welfare. The key component of our scheme is to determine the encoding functions of the transmitted packets, which is NP-hard. We present an efficient approximation algorithm that guarantees both the approximation ratio and the truthfulness properties. I. INTRODUCTION The Index Coding problem is one of the basic problems in wireless network coding. Recently, this problem has attracted a significant interest from the research community (see e.g., [4, 5]). An instance of the Index Coding problem includes a server, a set of wireless clients, and a set P = {p,...,p m } of m packets that need to be delivered to clients. Each client is interested in a certain subset of packets available at the server and has a (different) subset of packets available to it as side information. Server can transmit the packets or encoding thereof to clients via a noiseless broadcast channel. The goal is to find a transmission scheme that requires the minimum number of transmissions to satisfy the requests of all clients. Fig. depicts an instance of the Index Coding problem, where a server needs to deliver four packets P = {p,,p 4 } to four clients. It can be verified that the demands of all clients can be satisfied by broadcasting three packets: p +, +p 3, and p 4 (all operations are over GF (2)). Note that the traditional approach (without coding) requires the transmissions of all four packets p,,p 4. The prior works on the Index Coding problem have focused on developing algorithms, establishing rate bounds, and analyzing the computational complexity of the problem [6 ]. In contrast to the original Index Coding problem where the goal of the server is to satisfy the demands of all clients, we focus on settings where the server s decisions are driven by the values of the transmitted packets to the clients and by the transmission cost. This is a new problem in the intersection of the coding theory and game theory and belongs to the general framework of mechanism design with social choices [2]. Intuitively, since each transmission at server incurs a certain cost, the server may decide to transmit a packet only when the packet is important enough for the clients. Thus, our goal is to maximize the social welfare, i.e., the total value of the packets the clients are able to decode minus the total cost of transmitting the packets by the server. The social welfare reflects the positive externalities in terms of clients valuations of packets and the negative externalities related by the transmission costs of server. To calculate the social welfare, the server needs to know, for all clients in the system, the client valuation of the packet it requests. This information is usually private and selfish clients could not be expected to reveal this information to the server because the server might decide not to transmit packets with low valuation. Accordingly, we aim to design a tractable auction mechanism that provides an incentive for each client to reveal the true valuation of its packets. In the auction, the clients submit bids to the server, and the server decides whether to accept or reject the bid and charges clients accordingly. More specifically, the auction mechanism includes two main components, namely the coding algorithm and payment function. The coding algorithm determines which packets are transmitted over the channel and how they are encoded. The payment function determines the amount of money the client will pay to the server for each packet it is able to decode. Both coding algorithm and payment functions are known to the clients in advance. A classical solution for the problem of maximizing social welfare in the presence of selfish clients is to use the Vickery- Clarke-Groves (VCG) mechanism [ 3]. Such mechanism was shown to be truthful, i.e., in the sense that each client maximizes its own utility by bidding its true valuations of packets. However, the VCG mechanism cannot be used for our

2 v =0.2 b =0.8 v 3 =0.5 b 3 =0.5 v 2 =0.9 p desired packet b 2 =0.9 p 3 c c 2 Server p p 3 p 4 p c 3 c 4 p 3 p 4 p 4 side information p 3 v 4 =0.6 b 4 =0.6 Fig.. An instance of the Index Coding problem. Client c i,i =,...,4 requests packet p i. The side information sets of clients c,...,c 4 are {p 3 }, {p }, {,p 4 }, and {p 3 }, respectively; v i and b i denote the value and bid of packet p i, respectively. problem because it involves solving an NP-hard optimization problem. Accordingly, in this paper we design a novel auction mechanism. We prove that this mechanism is truthful, computationally efficient, and provides a provable approximation guarantee with the respect to the optimization function. Contributions. First, we present a truthful scheme, based on the Vickrey-Clarke-Groves (VCG) mechanism [ 3], that maximizes the social welfare of the system. It turns out that finding VCG based encoding functions for this scheme is an NP-hard optimization problem. Next, to mitigate the intractability of the this problem, we present an approximation algorithm with provable performance guarantees. We show, through a counter-example, that a VCG based algorithm that relies on an approximation solution is no longer truthful. Finally, we present a non-vcg based payment scheme that achieves the truthfulness property in the presence of the approximation solution for the problem. II. MODEL We begin with the definition of the Index Coding problem. The input to the problem includes a server S, a set of n wireless clients = {c,,c n }, and a noiseless wireless broadcast channel. The server has a set of m packets P = {p,,p m } that need to be distributed to clients in. We assume, without loss of generality, that each client requests a single packet in P and has access to a subset of packets in P as a side information. Indeed, a client that wants more than one packet can be substituted by multiple clients that share the same side information set. In particular, each client c i 2 is characterized by the ordered pair (w i,h i ), where w i 2 P is the packet requested by c i and H i P is the set of packets available to c i as a side information. We assume that each packet p i represents an element of the Galois field GF (q). We assume that each client c i 2 has the internal (private) value v i for the packet w i. The transmission process includes an auction, in which each client submits a bid b i for packet w i. We denote by V = {v,,v n } and B = {b,,b n } the arrays that include the internal valuations and bids of the clients. Based on the bid vectors, the server identifies linear combinations that will be transmitted over the channel. In this paper, we focus on the scalar-linear coding schemes in which each packet q i = P m j= g i,jp j transmitted by server at the iteration i, apple i apple, is a linear combination of packets in P. Here, is the total number of transmissions made by the server and g i = {g i,,,g i,m }2GF (q) m is the encoding vector for the iteration i. We denote by G =[g i ] the encoding matrix, rows of G correspond to encoding vectors of the packets transmitted over the channel. A sparse code is the general linear code in which each transmitted packet from the server is a linear combination of at most two packets in P. The transmission of a packet by the server incurs the cost C T. Without lost of generality, we assume that C T =, i.e., the total cost incurred by the server is equal to. The purpose of the server is to maximize the social welfare, defined as P n i= v i i, where i is the indicator function that specifies if client c i is able to decode the required packet w i. In particular, i = if there exists a linear decoding function r i such that w i = r i (q,,q,h i ); otherwise, i = 0. We denote by = {,, n } the vector of indicator functions. For example, in the setting depicted in Fig., the social welfare of broadcasting the solution to the Index Coding problem, i.e. {p + ; + p 3 ; p 4 }, is = 0.8, while the higher social welfare of the sequence consisting of a single combination {p 3 +p 4 } is equal to =0.. Thus, transmitting a single combination p 3 +p 4 would be more desirable than transmitting three packets {p + ; +p 3 ; p 4 } from the server s point of view. The payment function is an important part of the auction mechanism. Server determines the amount of payment i that each client needs to pay for the obtained packet. We denote by ={,, n} the payment vector for all clients. The value of i is a function of B and. The auction mechanism, including the algorithm for determining the encoding function and the payment function is known to all the parties. We assume that every client c i is selfish and chooses its best bidding policy b i that maximizes its utility defined by u i (B) = v i i i. We say that a mechanism (i.e., an encoding matrix associated with a payment function) is truthful if u i v i,b i u i ˆbi,B i for all ˆb i and B i, where B i = B \{b i }. That is, regardless of the bids submitted by other clients, the utility of client c i is maximized when b i = v i. In this paper, we design the encoding matrix and the payment function that satisfies the following two conditions: (i) every client is incentivized to report its true valuation of the desired packet (i.e., truthful property) and (ii) the social welfare is maximized. III. VCG-BASED MECHANISM DESIGN In this section, we propose a truthful scheme based on the celebrated Vickrey-Clarke-Groves (VCG) approach [ 3] to mechanism design. Note that the indicator functions and the number of transmissions are functions of the encoding matrix G. In particular, it is easy to determine and for a given

3 matrix G. We define the social welfare function w(b,g) as w(b,g) = P n i= b i i. Note that the social welfare function is defined similarly to the social welfare, with the value of bids B substituted by packet valuations V. In addition, we define w i (B,G) = P j6=i b j j. In our mechanism, we choose the encoding matrix that maximizes the the value of social welfare function for the given bids B. VCG-coding scheme: Choose the encoding matrix G that maximizes the value of w(b,g), i.e., G = arg max w(b,g). () G If there are multiple encoding functions that maximize the social welfare functions, we choose one that satisfies the larger number of clients. Note that without loss of generality, we can assume that v i 2 [0, ], since client c i is assured to get the desired packet when submitting the bid b i = C T =. Our scheme uses the following payment function i : VCG-payment function: The client c i is charged as follows. i = max G i w(b i,g i ) w i (B,G ). (2) Based on the property of the VCG mechanism [2], we conclude as follows. The following proposition follows from the basic properties of the VCG mechanism. Proposition. (i) The VCG-coding scheme associated with the VCG-payment function is a truthful mechanism, i.e. B = V. Moreover, the pricing function in (2) is non-negative, and the utilities of all clients are non-negative, i apple v i. (ii) The VCG-coding scheme associated with the VCG-payment function maximize the social welfare. Proof: The proof follows from the properties of the VCG mechanism [2]. The proof is omitted due to the space constraints. The problem of finding an optimal VCG encoding matrix G is an interesting combinatorial problem by itself. Unfortunately, this problem is computationally intractable. More specifically, for the instance of the problem in which the bid of each client b i is equal to one, the problem if maximizing the value of the social welfare function w(b,g) is equivalent to the problem of minimizing the required number of transmissions to satisfy all clients. Indeed, in this case there exists an optimal solution that satisfies all clients, since the client bid is equal to the cost of transmitting a single client. IV. APPROXIMATION ALGORITHM In this section, we present an approximation algorithm for the special case of multiple unicast. In the multiple unicast environment, every packet is required by exactly one client. This is in contrast to the multiple multicast scenario in which a packet can be required by multiple clients. We also considering a special case of sparse code, where each packet is a combination of at most two packets in P. With sparse c 0.5 c3 Fig. 2. Weight dependency graph for the instance of the multiple unicast problem in Fig. coding, both encoders and decoders can be implemented in a simpler manner which has a significant advantage for practical applications. Our scheme uses the notion of a weighted dependency graph described below. Definition 2. The weighted dependency graph is a directed graph G(V,A) defined as follows: For each client c i 2 C, there is a corresponding vertex v i in V. For any two clients c i and c j such that w i 2 H j, there is a directed arc (v i,v j ) 2 A. The weight of arc (v i,v j ) is i,j = b i. Fig. 2 illustrates the weight dependency graph corresponding to the instance of the social welfare minimization problem depicted in Fig.. For each vertex-disjoint cycle in the weight dependency graph, the server can save one transmission. That is, for a cycle C of length C, all of the clients that correspond to its nodes can be satisfied using C transmissions. The clients belonging to cycle C share the transmission cost C, resulting in the higher social welfare. Indeed, in Fig. 2 we have a cycle (c,c 2,c 3 ) that involves three clients. It is easy to verify that the three clients can be satisfied by two transmissions: p + and + p 3. Note that if there is no cycle in G(V,A) then sparse code can not improve the value of social welfare function. Given a cycle C =(c,c 2,,c k ), server encodes along cycle C means that server would transmit the sequence of k packets {p + ; ; p k + p k }.We first investigate the hardness of the VCG-coding scheme. Lemma 3. In multiple unicast scenario with the restriction to the sparse code, the social welfare minimization problem is NP-hard. Proof: This is shown via a reduction from the cycle packing problem in directed graphs. We propose the greedy-vcg-coding scheme in Alg. as an approximation algorithm for the PCIC problem in this setting. Next, we present a corresponding pricing scheme that motivates all clients to reveal the true value of the packets. P We define the weight of cycle C in G(V,A) as (C) = a2c a ( C ), where the first term is the sum of all bids along the cycle, and the second one is the total transmission cost of all packets that correspond to this cycle. Let C be a P set of vertex-disjoint cycles in G(V,A). Then w(b,g) = C2C (C), where server encodes along the cycles in C as the encoding matrix G c2 c4

4 Algorithm : Greedy-VCG-coding scheme input : Bids vector B and side information H i for all clients output: Encoding matrix G Create the weight dependency graph G(V,A); 2 Define the cost a of arc a 2 A by a = a; 3 By (C) = P a2c a, we define the cost of cycle C; 4 S, S 2 ;; // S, S 2 will include the clients that are served 5 while in G(V,A), there is a cycle with the cost less than or equal to one do 6 Find the cycle C in G(V,A) with the smallest cost; 7 Server encodes along cycle C; 8 S S [{c v : v 2 V,v 2 C} ; 9 G(V,A) G(V,A)\{vertices along C, edges along or incidnet to C}; 0 end for i to n do 2 if b i =but c i /2 S then 3 Server will transmit w i ; 4 S 2 S 2 [{c i }; 5 end 6 end The idea of the greedy-vcg-coding scheme is iteratively to identify the cycle of the maximum weight and then to remove it. However, this scheme cannot be implemented directly since it is NP-hard to find a maximum weight P cycle. We note that (C) can be revised as (C) = a2c ( a). Thus, in this case, a maximum weight cycle with respect to weight (C) corresponds to the minimum cost cycle with respect to cost P a2c ( a). Therefore, in lines 2 and 3 of Alg., we define the cost of arc a 2 A by a = a and the cost of cycle C by (C) = P a2c a. Then, the algorithm finds the minimum cost cycle in Line 6. To this end, we can use wellknown polynomial time algorithms such as Floyd-Warshall algorithm. After such a cycle is identified, it is removed from the graph in Line 9. The condition in Line 5 guarantees that the maximum weight cycle in the remaining graph G(V,A) (i.e., the minimum cost cycle) has a non-negative weight. In Line 3, the clients with the bids equal to the transmission cost are served. Such clients are then included in set S 2. In addition, set S contains the clients associated with the cycles chosen in Line 6. Note that all procedures in Alg. require a polynomial number of steps, hence Alg. can be executed in polynomial time. Unfortunately, Alg. cannot be used with the VCGpayment functions specified by (2) (i.e to replace G with the approximate solution). In particular, the following example shows that such combination might not have the truthfulness property. Example 4. Consider the instance of the problem depicted in Fig. 3. Given B fixed, we will show that client c has the potential to lie. Suppose that client c submits the bid b = 0.55 equal to its internal value of packet p. In this case, Alg. will identify the cycle (c 2,c 3 ) and output the corresponding solution + p 3. In this case, the utility u of c i is equal to zero. Now, suppose that c submits bid b =0.7. In this case, Alg. will transmit the sequence of packets {p + ; p 3 +p 4 } p c c 3 v =0.55 b =0.55 Server p c c 2 c 3 c 4 p 3 p 4 v 2 =0.6 b 2 =0.6 p p 3 c 2 c 4 p 3 v 3 =0.6 v 4 =0.55 p 4 p 4 b 3 =0.6 b 4 =0.55 p (a) 0.6 Fig. 3. (a) Counter-example of the truthful property for the greedy-vcgcoding scheme accompanied with the VCG-payment function (b) The weight dependency graph (b) that corresponds to cycles (c,c 2 ) and (c 3,c 4 ), respectively. According to (2), the payment of client c is equal to = ( ) ( ) = In this case, the utility of client c is equal to 0.25, which is bigger than the case in which the client is truthful. J Accordingly, we present the greedy-vcg-payment scheme implemented by Alg. 2 to charge the clients c i. This scheme results in a truthful auction Algorithm 2: Greedy-VCG-payment function input : Bids vector B and side information H i for all clients output: Payment i for client c i Clients that are not in S and S 2 do not be charged, return i =0; 2 The payment for the clients belonging to S 2 is, return i =; // Following deals with the clients in S 3 Create the weight dependency graph G(V,A); 4 The cost is defined as follows; for arc (i, j) 2 A, (i,j) = for arc a 6= (i, j), which is not outgoing from vertex i, a = a 5 By (C) = P a2c a, we define the cost of cycle C; 6 P ;; // set P will include the possible payments 8 while in G(V,A), there is a cycle of the cost less than or equal to one, as well as there exists a cycle that traverses vertex i do 9 Find the cycle C in G(V,A) with the smallest cost, say the cost ; 0 Find the cycle C 2 in G(V,A) that traverses vertex i and has the smallest cost among these that go through vertex i, say the cost ; if =0then 2 return i =0 3 end 4 else 5 P P[( ); 6 G(V,A) G(V,A)\{vertices along C, edges along or incident to C }; 7 end 8 end 9 Find the cycle C 2 in G(V,A) (if any) that traverses vertex i and has the smallest cost among these that go through vertex i, say the cost ; 20 P P[( ); 2 return i =minp

5 It can be easily verified that Alg. 2 has polynomial complexity. Note that in Line 4 of Alg. 2, the cost function of G(V,A) is different from that in Alg.. To compute the payment for client c i 2 S, we assign the unit cost for every outgoing arc (i, j) of vertex i (i.e. set i,j =0and b i =0). For each iteration of Line 8, we calculate the difference of the cost between the cycles C and C 2 in G(V,A), where C has the minimum global cost while C 2 is the local optimal cycle that traverses vertex i. We note that when the cycle C last containing vertex i is selected in the last iteration of Alg., the while loop in Alg. 2 might not capture C last because the cost of arc (i, j) is set to be one. Line 20 deals with this case. When client c i submits b i =, the weight of cycle C last in G(V,A) is ( )+( ) = 0, where the first term is the weight when b i =0while the second term is the increment when b i =. Therefore, the cost of C last is one, which satisfies the condition of Line 5 of Alg.. The set P includes the result of each iteration. If the difference is zero, client c i will not be charged; otherwise, it will be charged the value equal to the minimum of an element in set P. Given the bids of all clients, the coding algorithm determines the encoding matrix G, which, in turn, determines the indicator vector. We say that a coding algorithm is monotone if, for any B i, there exists a single critical value b i such that c i gets the desired packet (i.e. i =) when b i b i ; otherwise, c i can not. Theorem 5 (Theorem [3]). The mechanism is truthful if and only if the coding algorithm is monotone and the payment scheme is based on the critical value. Theorem 6. The Algorithms and 2 result in a truthful mechanism. Proof: According to Theorem 5, it is sufficient to show that Alg. is monotone, as well as Alg. 2 is based on the critical value. Suppose client c i is able to recover the desired packet. Then, a cycle containing vertex v i will be chosen by Alg.. Alg. prefers the cycle of the maximum weight (i.e., minimum cost) in weight dependency graph G(V,A). When c i increases the bid, so does the weight of the cycles that pass through vertex v i, and therefore a cycle containing vertex v i will be selected by Alg.. Hence, the greedy-vcg-coding scheme is monotone. Given B i, for each iteration in Alg. 2, we calculate the lowest bid b i for client c i such that a cycle containing vertex v i would be selected by Alg.. Since the minimum of these values are adopted as the payment for client c i 2 S, client c i 2 S is charged by the critical value. Other cases, i.e. client c i /2 S, follow immediately. Let OPT be the optimal solution of the VCG-coding scheme, i.e. OPT = w(b,g ). Assume G greedy is the encoding matrix from the greedy-vcg-coding scheme. We define the approximation ratio of Alg. by OP T/AP X, where AP X = w(b,g greedy ). b = b5 = b2 = b6 = p c c3 p3 p5 p5 p2 p p3 p6 p2 c5 c7 p7 p Server p7 p4 p3 p p8 p4 c5 c7 c c3 p2 p3 p4 p5 p6 p7 p8 b3 = b7 = b4 = b8 = (a) (b) Fig. 4. (a) Tight example of greedy-vcg-coding scheme (b) The weight dependency graph Theorem 7. The approximation ratio of Alg. scheme is equal to the maximum length of the cycle in the weight dependency graph. Proof: The optimal solution G in () corresponds to the maximum weight vertex-disjoint cycle packing, say C, in the weight dependency graph G(V,A). The total weight of these cycles in C is OPT, i.e. w(b,g )= P C2C (C) Given a cycle C in the weight dependency graph, let C \C be the set of the cycles in C that have a shared vertex with C. For each iteration k of Alg., we encode along the cycle C k of the minimum cost in G k (V,A), where G k (V,A) is the remaining graph in iteration k. Let C k be the set of cycles in C that belongs to G k (V,A). We define AP X k = (C k ) and OPT k = P C2(C k \C k ) (C). Because C k is the maximum weight cycle in this iteration, the weight of the cycles in C k \ C k is smaller than AP X k. Moreover, since there are C k vertices in cycle C k, there are at most C k cycles in C k \C k. Hence, OPT k apple AP X k C k, which results in the approximation ratio of the maximum cycle length in the weight dependency graph. In the next example, we show that the approximation ratio proven in Theorem 7 is tight. Example 8 (Tight example of Alg. ). Consider the setting depicted in Fig. 4(a). For this setting, Alg. results in the following transmissions p +, + p 3, and p 3 + p 4. Note that these transmissions correspond to cycle (c,c 2,c 3,c 4 ) in Fig. 4(b). In contrast, the optimal solutions is p +p 5, +p 6, p 3 + p 7, and p 4 + p 8. Therefore, the approximation ratio is 4( )/ for any >0. J c2 c4 V. CONCLUSION In this paper, we focus on an interesting problem that lies in the intersection of game theory and coding theory. Based p2 c2 c4 p4 c8 c6 p6 c6 c8 p8

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