Proc. 8th ALENEX-06, pp. 86-94, SIAM, 2006

Data , Exact, and Heuristic for Cover

Jens Gramm∗ Jiong Guo† Falk H¨uffner† Rolf Niedermeier†

Abstract Our study problem Clique Cover, also known To cover the edges of a graph with a minimum number of as Keyword Conflict problem [12] or Covering cliques is an NP-complete problem with many applications. by Cliques (GT17) or Intersection Graph Basis The state-of-the-art solving is a polynomial-time (GT59) [9], has applications in diverse fields such as heuristic from the 1970’s. We present an improvement compiler optimization [19], computational geometry [2], of this heuristic. Our main contribution, however, is the and applied statistics [11, 18]. Thus, it is not surprising development of efficient and effective polynomial-time data that there has been substantial work on (polynomial- reduction rules that, combined with a search tree algorithm, time) heuristic algorithms for Clique Cover [12, 14, allow for exact problem solutions in competitive time. This 19, 18, 11]. In this paper, we extend and complement is confirmed by experiments with real-world and synthetic this work, particularly introducing new data reduction data. Moreover, we prove the fixed-parameter tractability techniques. of covering edges by cliques. Formally, as a (parameterized) decision problem, Clique Cover is defined as follows: 1 Introduction Clique Cover Data reduction techniques for exactly solving NP-hard Input: An undirected graph G = (V, E) and combinatorial optimization problems have proven use- an integer k ≥ 0. ful in many studies [1, 3, 10, 16]. The point is that by Question: Is there a set of at most k cliques polynomial-time executable reduction rules many input in G such that each edge in E has both instances of hard combinatorial problems can be signifi- its endpoints in at least one of the selected cantly shrunk and/or simplified, without sacrificing the cliques? possibility of finding an optimal solution to the given problem. For such reduced instances then often exhaus- Clique Cover is hard to approximate in polyno- tive search algorithms can be applied to efficiently find mial time and nothing better than only a polynomial optimal solutions in reasonable time. Hence data reduc- approximation factor is known [5]. tion techniques are considered as a “must” when try- We examine Clique Cover in the context of ing to cope with computational intractability. Studying parameterized complexity [7, 17]. An instance of a the NP-complete problem to cover the edges of a graph parameterized problem consists of a problem instance I with a minimum number of cliques ((Edge) Clique and a parameter k. A parameterized problem is fixed- Cover)1, we add a new example to the success story parameter tractable if it can be solved in f(k) · |I|O(1) of data reduction, presenting both empirical as well as time, where f is a computable function solely depending theoretical findings. on the parameter k, not on the input size |I|. Our contributions are as follows. We start with ∗Wilhelm-Schickard-Institut f¨ur Informatik, Univer- a thorough mathematical analysis of the heuristic of sit¨at T¨ubingen, Sand 13, D-72076 T¨ubingen, Germany, Kellerman [12] and the postprocessing of Kou et al. [14], [email protected]. Supported by DFG which is state of the art [19, 11]. For an n- and project OPAL, NI 369/2. m-edge graph, we improve the runtime from O(nm2) † Institut f¨ur Informatik, Friedrich-Schiller-Universit¨at to O(nm). Afterwards, as our main algorithmic contri- Jena, Ernst-Abbe-Platz 2, D-07743 Jena, Germany, {guo|hueffner|niedermr}@minet.uni-jena.de. Jiong Guo bution, we introduce and analyze data reduction tech- and Falk H¨uffner supported by DFG Emmy Noether re