An Efficient LP Rounding Scheme for Replica Placement Zhihui Du Sen Zhang Department of Computer Science Department of Mathematics, Computer Sciences and Statistics New Jersey Institute of Technology, Nework, New Jersey, US State University of New York, College at Oneonta [email protected] [email protected] David A. Bader Jingkun Hu Department of Computer Science Worldmoney Blockchain Management Limited New Jersey Institute of Technology, Nework, New Jersey, US Hong Kong [email protected] [email protected] Abstract—Large fault-tolerant network systems with high to obtain sub-optimal integer solution to meet constraints of Quality of Service (QoS) guarantee are critical in many real world the original problems. applications and entail diverse replica placement problems. In In this paper we present a novel LP rounding scheme this paper, the replica placement problem in terms of minimizing the replica placement cost subject to both QoS and fault-tolerant to efficiently solve the replica placement problem that aims constraints is formulated as a binary integer linear programming to minimize the total replica cost subject to both QoS and problem first and then relaxed as a linear programming problem. FT constraints simultaneously. Furthermore, we identify the Given the optimal fractional linear programming solution, we condition in which the proposed rounding algorithms can propose a two-step rounding algorithm to obtain its integer achieve provable constant approximation ratio 2. solution. In the first step, a half rounding algorithm is used to simplify the problem. In the second step, a cheapest amortized The major contributions of this paper are as follows. cost rounding algorithm uses a novel metric, named amortized 1) Propose a two-step LP rounding scheme to significantly cost, to make locally optimal rounding decision for the remaining simplify the problem into different solvable steps. vertices independently. Furthermore, a conflict resolution algo- rithm is presented to tackle the situations when different vertices 2) Develop a novel cheapest amortized cost rounding and make conflicting rounding decisions. Finally, we prove that the conflict resolution algorithm to generate optimized inte- proposed two-step rounding algorithm has a 2-approximation ger programming solution. ratio when the additional conflict cost meets a given constraint. 3) Prove the proposed rounding algorithm has a 2- Index Terms—Replica Placement, Quality of Service, Fault approximation ratio when the conflict amortized cost is Tolerance, Rounding Algorithm, Approximation Ratio no more than the cheapest amortized cost. I. INTRODUCTION II. PROBLEM DESCRIPTION High Quality of Service (QoS) and fault-tolerance (FT) In this paper we attack the replica placement optimization become much more important in emerging applications. When problem that considers both QoS and FT constraints at the widely employed replica placement [5] technology is com- same time [6] by developing a novel rounding scheme. This bined with both QoS and FT constraints, the problem becomes section serves as a recap of the problem description, to both NP-hard [6]. make this paper self-contained and facilitate the presentation Approximation algorithms [10], [11] are important because of the proposed LP rounding scheme and approximation ratio they can run in polynomial time and provide guarantee for proof. the quality of solution even under the worst case. Considering The application concerns a group of geographically dis- replica placement technology has been widely used in many tributed servers that can provide service directly (with replica) practical fields such as Content Delivery/Distribution Network or indirectly (without replica but it can relay the request to (CDN) [8], cloud [4] and edge computing environment [2], it other servers). Furthermore, each server is associated with a is necessary to explore approximation algorithms for replica potential cost if being placed with a replica. Each link between placement problems to limit the cost boundary. a pair of servers is associated with a distance weight. All When a combinatorial optimization problem can be modeled servers are subject to both QoS requirement, which is defined as a binary integer linear programming (BILP) problem, the with respect to a desired reachable distance threshold, and BILP could be relaxed to a linear programming (LP) problem fault-tolerance, which is a desired number of replica servers to first get the fractional solution. Proper LP rounding [9] in its close vicinity. The object is to determine a replica methods then could be used convert the fractional solutions placement strategy that minimizes the total cost yet meets both 978-1-7281-9219-2/20/$31.00 ©2020 IEEE Authorized licensed use limited to: New Jersey Institute of Technology. Downloaded on January 06,2021 at 00:53:33 UTC from IEEE Xplore. Restrictions apply. the above QoS and FT constraints which are allowed to be constraints of QoS and fault tolerance of every vertex at the customized with respect to individual servers. same time. Here, IX is a vector of 0/1 values, where ones refer to a set of vertices which are placed replicas. Formally, A. Graph Model the above problem can be modeled as the following a binary We model the above replicated network topology using a integer linear programming problem, whose objective is to connected undirected weighted (both vertices and edges) graph minimize the total cost of the network subject to the Q and G = (V; E). Let V = fv0; v1; : : : ; vN−1g is the set of vertices M constraints: whose cardinality is N, each representing a site that can be Min : Cost(IX) = IX · ST (1) placed replica. E = f(u; v)ju; v 2 V g is a set of edges built X subject to IXj ≥ mi; 8vi 2 V (2) over V . Let L =< l0; l1; : : : ; ljE|−1 > be the weights associ- j2IG(vi) ated with each e 2 E. Let S =< s0; s1; : : : ; sN−1 > be the IXi 2 f0; 1g (3) cost vector with each element representing the cost associated III.ROUNDING SCHEME with each v 2 V . Let Q =< q0; q1; : : : ; qN−1 > stand for the QoS requirements on V with qi corresponding to the QoS In this section, we describe a rounding scheme to solve the requirement of vi. Let M =< m0; m1; : : : ; mN−1 > stand for proposed replica placement problem. V m the fault-tolerant requirements on with i corresponding to A. Linear Programming Relaxation as a Pre-Process the fault-tolerant level of vertex vi. We relax the f0; 1g constraints of IX in Eq.(3) to the B. Two Auxiliary Sets range of [0; 1], consequently, convert the original binary integer The following two critical auxiliary data structures [6] linear programming optimization problem into a continuous will be used to facilitate the algorithm presentation in the linear programming optimization problem which can be ef- subsequent section. ficiently solved by LP solvers. The optimal result computed Influencing set IG: 8v 2 V , the influencing set of v is from this relaxed linear programming optimization problem the set of vertices u whose distances to v are within the QoS then will be subsequently used as the input of our rounding requirement of v. algorithms. According to Eq.(2), the number of constraints of our IG(v) = fujd(v; u) ≤ q(v)g: problem is a linear function of number of variables, it is thus Influenced set ID: 8v 2 V , the influenced set of v is easy to see that a linear programming solver will be able to run the set of vertices u whose distances to v are within the QoS in polynomial time with respect to the numbers of the variables requirement of u. and the constraints to produce fractional coefficients. ID(v) = fujd(v; u) ≤ q(u)g: B. Two-Step Rounding Scheme We present an efficient rounding scheme to obtain an d(u; v) Here is used to denote the shortest distance among integer solution from the optimal fractional linear solution. It v u all paths between and . Since we assume the graph is includes two rounding algorithms: half rounding and cheapest d(u; v) connected, always exists and can be found by any amortized cost rounding. The pseudo-codes of the two-step well-known all-pairs shortest path algorithms. rounding scheme is given in Alg. 1. From the perspective of QoS, IG(v) denotes a set of vertices that can meet the QoS requirement of a given v, which Algorithm 1: Two-Step Rounding Algorithm means if any member in IG(v) is placed replica, it can serve input : M; S; LX the request of vertex v. ID(v) denotes a set of vertices whose output: integer solution IX 1 IX = 0; QoS requirements can be met by v, which means if v is placed 2 forall LXi 2 LX do 3 if (LXi ≥ 0:5) then a replica, those vertices in ID(v) will be served. From fault 4 IXi = 1 5 end tolerance perspective, IG(v) denotes a set of vertices whose 6 end 7 if (IX can meet constraint M) then failures may influence vertex v; while ID(v) denotes a set 8 return IX 9 end of vertices whose service may be influenced by the failure of 10 else 11 Call the cheapest amortized cost rounding algorithm Alg. 2 to update IX specific replica on given vertex v. 12 return IX Note that the general problem configuration is very flexible 13 end with respect to both edges and vertexes and the elements in S and Q vector can take different values. Let LX =< LX0; :::; LXN−1 > be the fractional co- efficients of linear programming solution. Let IX =< C. Problem Formulation IX0; :::; IXN−1 > be initialized to be all zeros and be Given a graph G = (V; E) with each vertex annotated by holding the binary integer values obtained from the rounding s, q and m values, and each edge annotated by l, if any algorithms.