A lagrangean decomposition for the maximum independent set problem applied to map labeling

Abstract

The Maximum Independent Set (MIS) problem is a well-known problem where the aim is to find the maximum cardinality independent set in an associated graph. Map labeling problems can often be modeled as a MIS problem in a conflict graph, where labels are selected to be placed near graphical features not allowing overlaps (conflicts) between labels or between labels and features. However, the MIS problem is NP-hard and exact techniques present difficulties for solving some instances. Thus, this paper presents a Lagrangean decomposition to solve a map labeling problem, the Point-Feature Label Placement problem. We treated the problem by a conflict graph that is partitioned into small sub-problems and copies of some decision variables are done. These copies are used in the subproblems constraints and in some constraints to ensure the equality between the original variables and their copies. After these steps, we relax these copy constraints in a Lagrangean way. Using instances proposed in the literature, our approach was able to prove the optimality for all of them, except one, and the results were better than the ones provided by a commercial solver.

Beyond Maximum Independent Set: An Extended Integer Programming Formulation for Point Labeling

“…We start by recalling an ILP for the basic map-label problem, which was also used by Ribeiro et al [6]. This ILP forms the basis for our extensions with respect to additional cartographic criteria (see Sections 3.3-3.5).…”

“…In order to speed up the computation of optimal or near-optimal solutions, they suggested to employ so-called cutting plane techniques. Ribeiro et al [6] presented a heuristic approach based on a relaxation of an ILP for the basic map-labeling problem. Other heuristic approaches to finding a feasible labeling of maximum total weight were based on simulated annealing [19] or used a small set of rules [20].…”

“…3 Let S = ∅ be an empty set of labels. 4 for i = 1 to n do 5 if S ∪ { i } is a feasible solution then 6 Set S = S ∪ { i } and set the x-variable corresponding to i to 1.…”

“…For the above basic problem, highly sophisticated optimization methods have been developed, ranging from exact approaches based on mathematical programming [4], via approximation algorithms [5], to specialized heuristics [6]. We argue, however, that the basic map-labeling problem lacks expressiveness.…”

A Labeling Model Based on the Region of Movability for Point-Feature Label Placement

“…The label placement is usually divided into three tasks: labeling points, labeling lines, and labeling areas. Because labeling points is a more fundamental problem, point-feature label placement (PFLP) has been extensively studied in recent years [6][7][8][9].…”