Automated Generation of Search Tree Algorithms for Hard Graph Modification Problems

Abstract

We present a framework for an automated generation of exact search tree algorithms for NP-hard problems. The purpose of our approach is twofold-rapid development and improved upper bounds. Many search tree algorithms for various problems in the literature are based on complicated case distinctions. Our approach may lead to a much simpler process of developing and analyzing these algorithms. Moreover, using the sheer computing power of machines it may also lead to improved upper bounds on search tree sizes (i.e., faster exact solving algorithms) in comparison with previously developed "hand-made" search trees. Among others, such an example is given with the NP-complete CLUSTER EDITING problem (also known as CORRELATION CLUSTERING on complete unweighted graphs), which asks for the minimum number of edge additions and deletions to create a graph which is a disjoint union of cliques. The hand-made search tree for CLUSTER EDITING had worst-case size O(2.27 k ), which now is improved to O(1.92 k ) due to our new method. (Herein, k denotes the number of edge modifications allowed.) Key Words. NP-hard problems, Graph modification, Search tree algorithms, Exact algorithms, Automated development and analysis of algorithms, Algorithm engineering. 1. Introduction. In the field of exactly solving NP-hard problems [1], [15], [40], the developed algorithms often employ exhaustive search based on a clever search tree (also called splitting) strategy. For instance, search tree based algorithms have been developed for SATISFIABILITY [20], [25], MAXIMUM SATISFIABILITY [3], [6], [19], [31], EXACT SATISFIABILITY [12], [21], [24], INDEPENDENT SET [9], [35], [36], VERTEX COVER [7], [33], CONSTRAINT BIPARTITE VERTEX COVER [16], 3-HITTING SET [32], and numerous other problems. Moreover, most of these algorithms have undergone some kind of "evolution" toward better and better exponential-time bounds. The improved upper bounds on the running times, however, usually come at the cost of distinguishing between more and more combinatorial cases which makes the development and the correctness proofs a tedious and error-prone task. For example, in a series of papers the upper bound on the search tree size for an algorithm solving MAXIMUM SATISFIABILITY was improved from 1.62 K [28] to 1.38 K [31] to 1.34 K [3] to recently 1.32 K [6], where K denotes the number of clauses in the given formula in conjunctive normal form.

Error Compensation in Leaf Power Problems

“…Encouraging results for the "simpler" but still NP-complete Closest 2-Leaf Power problem are obtained in [15,16] (where the problem is referred to as Cluster Editing).…”

Generalized Graph Clustering: Recognizing (p,q)-Cluster Graphs

“…Cluster Editing, parameterized by the number k of false positives and negatives, is FPT [3,9,10]. Furthermore, it has a polynomial-time 4-approximation algorithm but it does not admit a PTAS unless P = NP [14].…”

Fixed-Parameter Algorithms for Cluster Vertex Deletion

“…Parameterized complexity studies for Cluster Editing were initiated by Gramm et al [21] and have been further pursued in a series of papers [5,6,10,13,20,22,32,33]. A previously shown bound of O(1.92 k + n 3 ) for an n-vertex graph [20] can be improved by combining a linear-time problem kernelization algorithm [13] that yields an instance with O(k 2 ) vertices with the currently best claimed running time of O(1.82 k +n 3 ) [6] to get an algorithm with running time O(1.82 k +n+m), where m is the number of edges in the graph. Moreover, problem kernels, based on efficient data reduction rules, with only O(k) vertices are known [13,22], the best upper bound currently being 4k [22].…”

“…3 Using this characterization, one directly obtains fixed-parameter tractability [7] as well as a factor-3 polynomial-time approximation algorithm. Gramm et al [20] used an elaborate case distinction found with computer help to derive a search tree algorithm running in O(2.26 k m) time for an m-edge graph. This can be improved to O(2.08 k + n 3 ), n denoting the number of vertices, by using a straightforward reduction of unweighted Cluster Vertex Deletion to the 3-Hitting Set problem (transforming each induced P 3 into a three-element set) and employing a sophisticated algorithm for 3-Hitting Set [37].…”