A greedy approximation algorithm for the uniform metric labeling problem analyzed by a primal-dual technique

Abstract

We consider the uniform metric labeling problem. This NP-hard problem considers how to assign objects to labels respecting assignment and separation costs. The known approximation algorithms are based on solutions of large linear programs and are impractical for moderate-and large-size instances. We present an 8 log n-approximation algorithm that can be applied to large-size instances. The algorithm is greedy and is analyzed by a primal-dual technique. We implemented the presented algorithm and two known approximation algorithms and compared them at randomized instances. The gain of time was considerable with small error ratios. We also show that the analysis is tight, up to a constant factor.

Real-Time Multi-Criteria Social Graph Partitioning

“…Even though both LP methods provide a 2-approximation algorithm for UML, their respective complexities, O (|E| + k · |V |) 3.5 and O (k · (|V | + k · |E|) 3.5 ) , are still prohibitive for practical applications. To avoid linear programming, Bracht et al [5] propose a greedy approach that runs in O(k · |V | 3.6 ), but guarantees a much looser approximation ratio 8 log |V |. Furthermore, the algorithm requires extensive graph transformations; i.e., for each class it generates a new graph that connects the class to all nodes.…”

“…Initially, RM GP b assigns every player to a random class and computes the maximum social cost maxSCv, assuming that all friends of v are assigned to a different class. Then, it starts the best-response procedure (Lines [4][5][6][7][8][9][10][11][12][13][14]. Each iteration of the repeat-loop corresponds to a round.…”

“…(ii) The linear programming UML algorithm (U M L lp ) [13], which guarantees an approximation ratio 2. (iii) The faster greedy UML approach (U M Lgr) [5], which guarantees a looser approximation ratio 8 log |V |.…”

Semidefinite Programming Based Algorithms for the Sparsest Cut Problem

“…The Sparsest Cut problem has applications in image segmentation [19], the metric labeling problem [3], and a natural application in graph conductance. It is a known NP-hard problem.…”