Algorithms for Maximum Independent Set Applied to Map Labelling
نویسندگان
چکیده
We consider the following map labelling problem: given distinct points p,p, . . . ,pn in the plane, and given σ, find a maximum cardinality set of pairwise disjoint axis-parallel σ× σ squares Q1, Q2, . . . , Qr. This problem reduces to that of finding a maximum cardinality independent set in an associated graph called the conflict graph. We describe several heuristics for the maximum cardinality independent set problem, some of which use an LP solution as input. Also, we describe a branch-and-cut algorithm to solve it to optimality. The standard independent set formulation has an inequality for each edge in the conflict graph which ensures that only one of its endpoints can belong to an independent set. To obtain good starting points for our LP-based heuristics and good upper bounds on the optimal value for our branch-and-cut algorithm we replace this set of inequalities by the set of inequalities describing all maximal cliques in the conflict graph. For this strengthened formulation we also generate lifted odd hole inequalities and mod-k inequalities. We present a comprehensive computational study of solving map labelling instances for sizes up to n = 950 to optimality. Previously, optimal solutions to instances of size n ≤ 300 have been reported on in the literature. By comparing against these optimal solutions we show that our heuristics are capable of producing near-optimal solutions for large-scale instances.
منابع مشابه
A note on greedy algorithms for the maximum weighted independent set problem
In this paper, we consider three simple and natural greedy algorithms for the Maximum Weighted Independent Set problem. We show that two of them output an independent set of weight P v2V (G) W (v) d(v)+1 and the other outputs an independent set of weight P v2V (G) W (v) 2 P u2N + G (v) W (u) . Those results are generalization of Tur an's theorem.
متن کاملAchieving Simultaneous Spectrum Utilization and Revenue Improvements in Practical Wireless Spectrum Auctions
Spectrum is a valuable, scarce and finite natural resource that is needed for many different applications, so efficient use of the scarce radio spectrum is important for accommodating the rapid growth of wireless communications. Spectrum auctions are one of the best-known market-based solutions to improve the efficiency of spectrum use. However, Spectrum auctions are fundamentally differen...
متن کاملUsing Combinatorics to Prune Search Trees: Independent and Dominating Set
We introduce a surprisingly simple technique to design and analyze algorithms based on search trees, that significantly improves many existing results in the area of exact algorithms. The technique is based on measuring the progress of Branch & Bound algorithms by making use of a combinatorial relation between the average and maximum dual degrees of a graph. By dual degree of a vertex, we mean ...
متن کاملAlgorithms for clique-independent sets on subclasses of circular-arc graphs
A circular-arc graph is the intersection graph of arcs on a circle. A Helly circular-arc graph is a circular-arc graph admitting a model whose arcs satisfy the Helly property. A clique-independent set of a graph is a set of pairwise disjoint cliques of the graph. It is NP-hard to compute the maximum cardinality of a clique-independent set for a general graph. In the present paper, we propose po...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2000