Complexity classification of some edge modification problems

Natanzon, Assaf; Shamir, Ron; Sharan, Roded

doi:10.1016/s0166-218x(00)00391-7

Cited by 159 publications

(80 citation statements)

References 33 publications

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“…These are referred to in the tables to follow. [30], [38]. We consider three of these in our context.…”

Section: Applications and Resultsmentioning

confidence: 99%

“…Then the user may think of additional problem-specific rules to improve this situation, obtain a better bound, and repeat this process. 4 The automated generation of search tree algorithms in this paper is restricted to the class of graph modification problems [4], [27], [30], although the basic ideas appear to be generalizable to other graph and even nongraph problems. In particular, we study the following NP-complete edge modification problem CLUSTER EDITING, which is motivated by, e.g., data clustering applications in computational biology [37], [38] and correlation analysis in machine learning [2]:…”

Section: Introductionmentioning

confidence: 99%

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Automated Generation of Search Tree Algorithms for Hard Graph Modification Problems

et al. 2004

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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.

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“…These are referred to in the tables to follow. [30], [38]. We consider three of these in our context.…”

Section: Applications and Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Automated Generation of Search Tree Algorithms for Hard Graph Modification Problems

et al. 2004

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“…• the maximum-weight triangulated subgraph problem is N P-hard [11], and it clearly makes no sense to employ an exact solution approach (such as Branch&Bound) in this application; • even if it were computationally feasible to exactly (or approximately, with some tight a-priori ratio) find a maximum-weight triangulated subgraph of G, using it as S would not necessarily result in an efficient approach, due to the above-mentioned delicate balance between the extra cost of finding and factoring a "larger" preconditioner M S and the decrease in PCG iterations [7]; • once S has been determined, some work still has to be done to find the "good" ordering of the nodes, i.e., a n × n permutation matrix P n such that the reordered matrix P n M S P T n has a Cholesky factorization without fill-in. For the case of trees, P n corresponds to any permutation P (such as that given by a reverse BreadthFirst Search) of the nodes such that if (i, j ) is an arc of S with i father of j , then row j precedes row i in P, and therefore is already implicitly given by, e.g., the description of the tree in terms of the predecessor function Pred [·]; conversely, for the general case P n has to be explicitly computed [19].…”

Section: Support-graph Preconditionersmentioning

confidence: 99%

Prim-based support-graph preconditioners for min-cost flow problems

Frangioni

Gentile

2007

Comput Optim Appl

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Support-graph preconditioners have been shown to be a valuable tool for the iterative solution, via a Preconditioned Conjugate Gradient method, of the KKT systems that must be solved at each iteration of an Interior Point algorithm for the solution of Min Cost Flow problems. These preconditioners extract a proper triangulated subgraph, with "large" weight, of the original graph: in practice, trees and Brother-Connected Trees (BCTs) of depth two have been shown to be the most computationally efficient families of subgraphs. In the literature, approximate versions of the Kruskal algorithm for maximum-weight spanning trees have most often been used for choosing the subgraphs; Prim-based approaches have been used for trees, but no comparison have ever been reported. We propose Prim-based heuristics for BCTs, which require nontrivial modifications w.r.t. the previously proposed Kruskal-based approaches, and present a computational comparison of the different approaches, which shows that Prim-based heuristics are most often preferable to Kruskal-based ones.

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“…Computing a minimum completion of an arbitrary graph into a specific graph class is an important and well studied problem with applications in molecular biology, numerical algebra, and more generally areas involving graph modeling with missing edges due to lacking data [2][3][4]. Unfortunately minimum completions into most interesting graph classes, including cographs, are NP-hard to compute [5][6][7]3,8].…”

Section: Introductionmentioning

confidence: 99%

“…Unfortunately minimum completions into most interesting graph classes, including cographs, are NP-hard to compute [5][6][7]3,8]. This fact encouraged researchers to focus on various alternatives that are computationally more efficient, at the cost of optimality or generality.…”

Section: Introductionmentioning

confidence: 99%

Characterizing and computing minimal cograph completions

Lokshtanov

Mancini

Papadopoulos

2010

Discrete Applied Mathematics

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a b s t r a c tA cograph completion of an arbitrary graph G is a cograph supergraph of G on the same vertex set. Such a completion is called minimal if the set of edges added to G is inclusion minimal. In this paper we present two results on minimal cograph completions. The first is a characterization that allows us to check in linear time whether a given cograph completion is minimal. The second result is a vertex incremental algorithm to compute a minimal cograph completion H of an arbitrary input graph G in O(|V (H)| + |E(H)|) time.An extended abstract of the result has been already presented at

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Complexity classification of some edge modification problems

Cited by 159 publications

References 33 publications

Automated Generation of Search Tree Algorithms for Hard Graph Modification Problems

Automated Generation of Search Tree Algorithms for Hard Graph Modification Problems

Prim-based support-graph preconditioners for min-cost flow problems

Characterizing and computing minimal cograph completions

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