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Facility Location using Linear Programming Duality Yinyu Ye Department if Management Science and Engineering Stanford University Facility Location Problem Input • A set of clients or cities D • A set of facilities F with facility cost fi • Connection cost Cij, (obey triangle inequality) Output • A subset of facilities F’ • An assignment of clients to facilities in F’ Objective • Minimize the total cost (facility + connection) Facility Location Problem location of a potential facility (opening cost) client (connection cost) Facility Location Problem location of a potential facility (opening cost) client (connection cost) min opening cost connection cost R-Approximate Solution and Algorithm An algorithm found a feasible (integral) solution of UFLP, with the total cost, Cost , that satisfies the following : Cost R Cost * for some constant R 1. Hardness Results NP-hard. Cornuejols, Nemhauser & Wolsey [1990]. 1.463 polynomial approximation algorithm implies NP =P. Guha & Khuller [1998], Sviridenko [1998]. ILP Formulation Min C iF jD ij xij f i yi iF s.t. x iF ij 1 jD xij yi j D, i F xij , yi {0,1} j D, i F •Each client should be assigned to one facility. •Clients can only be assigned to open facilities. LP Relaxation and its Dual Min C x f y iF jD ij ij iF i i Max jD j s.t. x ij 1 jD s.t. j ij cij j D, i F f iF iF xij yi j D, i F jD ij i xij 0 j D, i F ij 0 j D, i F Interpretation: clients share the cost to open a facility, and pay the connection cost. ij max{0, j cij } is the contributi on of client j to facility i. Bi-Factor Dual Fitting Suppose an algorithm found a feasible (integral) solution of FLP, with the total cost j , where j satisfies the following : jD (1) j ij Rc cij j D, i F (2) jD ij R f fi iF for some constant Rc , R f 1 and ij 0, then we have : F C j R f F * Rc C * . jD A bi-factor (Rf,Rc)-approximate algorithm is a max(Rf,Rc)-approximate algorithm Simple Greedy Algorithm Jain et al [2003] Introduce a notion of time, such that each event can be associated with the time at which it happened. The algorithm start at time 0. Initially, all facilities are closed; all clients are unconnected; all j set to 0. Let C=D While C , increase j simultaneously for all j C , until one of the following events occurs: (1). For some client j C , and a open facility i such that j cij , then connect client j to facility i and remove j from C; (2). For some closed facility i, max( 0, j cij ) f i , then open jC facility i, and connect client j C with j cij to facility i, and remove j from C. Time = 0 F1=3 F2=4 3 5 4 3 6 4 Time = 1 F1=3 F2=4 3 5 4 3 6 4 Time = 2 F1=3 F2=4 3 5 4 3 6 4 Time = 3 F1=3 F2=4 3 5 4 3 6 4 Time = 4 F1=3 F2=4 3 5 4 3 6 4 Time = 5 F1=3 F2=4 3 5 4 3 6 4 Time = 5 F1=3 F2=4 Open the facility on left, and connect clients “green” and “red” to it. 3 5 4 3 6 4 Time = 6 F1=3 F2=4 Continue increase the budget of client “blue” 3 5 4 3 6 4 Time = 6 F1=3 F2=4 The budget of “blue” now covers its connection cost to an opened facility; connect blue to it. 3 5 4 3 6 4 5 5 6 The Bi-Factor Revealing LP Jain et al [2003], Mahdian et al [2006] Given R f , Rc is bounded above by k j 1 j R f fi max k c j 1 ij Subject to: 1 2 | D| max( 0, l j j cil ) f i j l cij cil In particular, if R f 1.861, then Rc 1.861. We got a 1.861 - appr. alg. Approximation Results Ratio Reference Algorithm 1+ln(|D|) Hochbaum (1982) Greedy algorithm 3.16 Shmoys et.al (1997) LP rounding 2.408 Guha and Kuller (1998) LP rounding + Greedy augmentation 1.736 Chudak (1998) LP rounding 1.728 Charika and Guha (1999) LP + P-dual + Greedy augmentation 1.61 Jain et.al (2003) Greedy algorithm 1.517 Mahdian et.al (2006) Revised Greedy algorithm

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