3-Approximation Algorithm for Uncapacited Facilty Location

Chapter 24 in the text.

Problem

We are given a bipartite graph G=(F,C;E) where F is the set of facilities and C is the set of cities (aka customers). Let f(i) be a cost of opening the facility i and c(i,j) be the cost of connecting facility i to city j. The connection costs satisfy the triangle inequality. The problem is to find a subset I ⊆ F of facilities that should be opened, and a function φ:C -> I assigning (connecting) cities to open facilities in such a way that a total cost of opening the facilities and connecting the cities is minimized.

LP formulation

Let y(i) be an indicator variable equal to 1 iff facility i is open and 0 otherwise. Let x(i,j) be an indicator variable equal to 1 iff facility i is connected to city j and 0 otherwise. Consider the following integer program for the problem

IP

minimize ∑_i,j c(i,j)x(i,j) + ∑_i f(i)y(i)

∑_{i ∈ F} x(i,j) ≥ 1 j∈ C

y(i)-x(i,j) ≥ 0 i ∈ F, j∈ C

x(i,j),y(i) ∈ {0,1} i ∈ F, j∈ C

Primal

The LP-relaxation of the program is

minimize ∑_i,j c(i,j)x(i,j) + ∑_i f(i)y(i)

∑_{i ∈ F} x(i,j) ≥ 1 j∈ C

y(i)-x(i,j) ≥ 0 i ∈ F, j∈ C

x(i,j),y(i) ≥ 0 i ∈ F, j∈ C

Dual

The dual program is

maxizmize ∑_{j ∈ C} α_j

α_j - β_ij ≤ c(i,j) i ∈ F, j∈ C

∑_{j ∈ C} β_ij ≤ f(i) i ∈ F

α_j, β_ij ≥ 0 j∈ C, i ∈ F

Algorithm

Algorithm runs in 2 phases. In the first phase we open some facilities temporarily; in the second phase we shut some of them down.

Phase 1.

We want the dual solution to be as large as possible. This motivates the following approach. We start from α_i, β_ij = 0 for all i,j, and raise the variables α_j gradually. When α_j = c(i,j) for some edge (i,j), the algorithm will declare this edge to be tight. From now on to keep the solution feasible we will have to raise variable β_ij. We will say that an edge (i,j) is special if β_ij > 0.

Facility i is paid for if ∑_j β_ij=f(i). If a facilities i is paid for, it is opened temporarily, and unconnected cities with tight edges are connected (assigned) to i. The facility i is a connecting witness for those cities. In the future if an edge connecting a temporarily open facility i and a city j gets tight we connect facility i to j, and i becomes the connecting witness (note that in this case β_ij = 0 and the edge is not special). The first phase terminates when all cities get connected.

Phase 2.

Note that in the first phase, a city might have paid for opening a facility it is not connected to. In the next phase we ensure that a city only pays for opening facility it is eventually connected to. (Can you find an example that shows that the second phase is really needed - that is - the algorithm is not a 3-approximation algorithm w/o the second phase. How bad can the approximation ratio get without the second phase?)

Let F_t denote the set of facilities that are temporarily opened and T denote the subgraph of G with special edges. Let H be the subgraph of T² induced on F_t. Find maximal independent set in H, say I. The facilities in I are opened.

Now we need to connect cities to the opened facilities. For a city j, define M_j={i ∈ F_t: (i,j) is special}. For a city j if there is an opened facility in M_j we connect j to it. We call j directly connected. Otherwise, consider a city j and a facility i such that i was a connecting witness for j. If i ∈ I we connect i to j (direct connection) again (note that in this case the edge (i,j) is not special). In the remaining case let i' be any neighbor of i in graph H such that i' ∈ I. Connect i' to j and declare j indirectly connected.

Analysis

The algorithm constructs a feasible solution (x,y) to the primal, and a feasible solution (α, β) to the dual. It is enough to show that cost(x,y) ≤ 3cost(α, β) , this will imply that the approximation ration of the algorithm is 3.

If a city j is connected directly to facilty i, then set α^f_j = β_ij and α^e_j = c(i,j) . If a city j is connected indirectly then set α^f_j = 0 and α^e_j = α_j . Clearly, for all cities α_j = α^f_j + α^e_j .

First we show that if i ∈ I, then ∑_j:φ(j)=i α^f_j =f_i. Indeed, since the facility i must be fully paid for, we have ∑_{j:(i,j) is special} β_j =f_i. Since only cities connected directly to j pay for it, and for these cities α^f_j = β_j , the claim follows. The claim immediatelly implies that ∑_{j ∈ C} α^f_j =∑_{i ∈ I} f_i.

If a city is connected directly then c(i,j) = α^e_j . We will now show that for an indirectly connected city j to i c(i,j) ≤ 3 α^e_j . Let j be an indirectly connected city, and let i' be the facility j is connected to (we use the notation from the algorithm). Let j' <> j be any city connected to i' (j' must exists, since j does not pay for the facility) and let i be the facility j was connected to in the first phase. It is enought to show that α^e_j ≥ c(i, j), c(i, j'), c(i', j') . First observe that α^e_j > c(i, j) and that α^e_j' > c(i, j'), c(i',j) since all these edges are special. It remains to show that α^e_j > α^e_j' . This inequality holds because city j' was connected to i' before j was connected to i. (Since otherwise j' would be connected to i, not i'.)

  i' *---* j'
        /
       /
      /
     /
  i *---* j

Using the results from the two last paragraphs, we have

∑_{i ∈ F, j ∈ C} c(i,j)x(i,j) + ∑_{i ∈ F} f(i)y(i) ≤ ∑_{i ∈ F, j ∈ C} c(i,j)x(i,j) + 3∑_{i ∈ F} f(i)y(i) ≤ ∑_{i,j: j = φ(i)} c(i,j) + 3 ∑_{i ∈ I} f(i) ≤ 3 ∑_{i,j: j = φ(i)} α^e_j + 3 ∑_{j ∈ C} α^f_j ≤ 3 ∑_j α_j

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References:

• Primal-Dual Approximation Algorithms for Metric Facility Location and k-Median Problems. http://citeseer.nj.nec.com/3752.html