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CS880: Approximations Algorithms
Scribe: Matthew Darnall Lecturer: Shuchi Chawla Topic: Primal/Dual Method: Facility Location/Cut Date: 3/
17.1 Facility Location
Recall from last time the formulation of the facility location problem as an LP with primal and dual. We have costs cij of routing a person j to facility i. There is a cost fi of opening facility i. We have a variable xi for each facility i that is the amount that the facility is open. We have a variable yij for the amount that person j is routed to facility i. We wish to minimize
∑
i
fixi +
i,j
yij cij
suject to the constraints that for each j yij ≤ xi
and for each j (^) ∑
i
bij ≥ 1
The dual problem then has variables αj for every person j and βij for every customer/facility pair. The dual problem is to maximize (^) ∑
j
αj
subject to the constraints that for any i, j
αj − βij ≤ cij
and for any i (^) ∑
j
βij ≤ fi
as well as the usual condition that all variables are non-negative.
17.2 Algorithm
We shall construct an integral solution to the primal LP as well as a feasible solution to the dual LP such that the integral primal is within a factor of three of the dual solution. Thus, our final solution shall be a three approximation for the facility location problem, since any dual feasible solution is an upper bound on a primal solution. As we usually do, we shall ensure that one of the complementary slackness conditions remains tight, while relaxing the other. This time, we shall relax the primal slackness condition. We shall do the following to get a set I of possible facilities to open. We continue until all the people are assigned a facility.
For all unassigned customers, raise the corresponding αj uniformly. If an equaltiy of the form αj = cij is reached the corresponding βij must be raised also to ensure that the constraint αj − βij ≤ cij isn’t violated. Remember that we want this to be a feasible dual solution. For such customers, αj − βij is fixed to be cij and that constraint is tight. Once a constraint of the form
j βij^ ≤^ fi^ is reached for some^ i, we include this^ i^ in our set^ I^ of possible facilities to open. Also, we consider the j that have nonzero bij to be assigned to i and freeze the values of βij and αi. Then we continue the process.
At this point, we have a set of facilities I that we wish to open. We call a pair i, j tight if αj − βij = cij at the time i was included in the set I. We shall decide the final facilities to open by dong the following. There is a natural ordering on the facilities by the time at which they were included in I. We call a pair i, j tight if αj = cij. Now, we recursively do the following. Select the first facility in I that is not thrown out or already selected. Now, throw out all the facilities that share a tight customer with the selected facility. Continue until all facilities are selected or thrown out. The selected facilities shall be the ones we open, call this set of facilities S. We shall route each customer completely to a tight facility if one exists. If one doesn’t exist, we route the customer to a facility that caused one of the tight facilities to the customer to be thrown out. By the metric propery, we know that the cost of routing to the open facility shall be bounded appropriately.
17.3 Analysis
For each customer, we break down the αj into two portions, the portion payng for facility opening
αfj and a portion for routing αrj. If j is assigned to a facility with which it is tight, then αj = βij +cij
for that facility. We call αfj = βij and αrj = cij. Else, αrj = αj and the facility portion for this αj is zero. Let∑ Si be the set of customers assigned to facility i. We notice that for any facility in S,
j∈Si α
f j =^
j∈Si βij^ =^ fi. Since these^ Si^ are disjoint,^
j α
f j is precisely the opening cost of the facilities in S. Now, for the routing costs we have that for j with a tight facility αrj is exactly the cost of routing j to that facility. For the other j, we don’t have a tight facility to route to. But, the facility that j is routed to must share a customer, j′, with a tight facility, i′, for j. By the order in which we chose the facilities to include, the routing cost from i to j′, from i′^ to j′^ and from i′^ to j are all bounded by αj , since each of these pairs is tight. The metric property gives us that the cost paid to route j is then at most 3αj. So, we have that:
3
j
αj ≥
i∈S
fi +
i,j
cij
Since the RHS is the objective function of the original problem and the optimal value for facility location is bounded by the sum of the αj , we get a 3 approximation to facility location. This idea was first seen in [1].
subject to the constraints that for any u, v, w:
d(u, v) + d(v, w) − d(u, w) ≥ 0 (17.5.3) d(u, v) − d(v, u) = 0 (17.5.4) d(u, u) ≥ 0 (17.5.5) d(s, t) ≥ 1 (17.5.6)
Notice that the first three of these constraints are perfectly encapsulates what it means for d to be a metric! Also, notice that the metric induced by any feasible cut satisfies the constraints. If the cut is optimal, then
(u,v)∈E c(u,v)d(u, v) =^
e cexe^ since we can lower the value of^ x(u,v)^ down to d(u, v) without violating any of the contraints. Thus, finding the solution to Min Cut is equivalent to finding the best integral solution to the LP above.
As we will see in future lectures, phrasing some problems as a metric LP and using metric embed- ding techniques can lead to good approximation factors.
References
[1] K. Jain, V. Vazarani. Primal-Dual Algorithms for Mteric Facility Location andk-Median Prob- lems. In 40th annual Symposium on the Foundations of Computer Science, 1999.
