Abstract
Knapsack median is a generalization of the classic k-median problem in which we replace the cardinality constraint with a knapsack constraint. It is currently known to be 32-approximable. We improve on the best known algorithms in several ways, including adding randomization and applying sparsification as a preprocessing step. The latter improvement produces the first LP for this problem with bounded integrality gap. The new algorithm obtains an approximation factor of 17.46. We also give a 3.05 approximation with small budget violation.
Highlights
Preliminaries(our algorithm extends to the general case.) For a client j, the connection cost of j, denoted as cost ( j), is the distance from j to the nearest open facility in our solution
A natural generalization of k-median is knapsack median (KM), in which we assign nonnegative weight wi to each facility i ∈ F, and instead of opening k facilities, we require that the sum of the open facility weights be within some budget B
For the ease of analysis, we assume that each client has unit demand. For a client j, the connection cost of j, denoted as cost ( j), is the distance from j to the nearest open facility in our solution
Summary
(our algorithm extends to the general case.) For a client j, the connection cost of j, denoted as cost ( j), is the distance from j to the nearest open facility in our solution. The goal is to open a subset S ⊆ F of facilities such that the total connection cost is minimized, subject to the knapsack constraint i∈S wi ≤ B. The natural LP relaxation of this problem is as follows In this LP, xi j and yi are indicator variables for the event client j is connected to facility i and facility i is open, respectively. Given a KM instance I = (B, F , C, c, w), let OPTI and OPT f be the cost of an optimal integral solution and the optimal value of the LP relaxation, respectively. Let us fix any optimal integral solution of the instance for the analysis
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