M. Yannakakis. Node- and edge-deletion NP-complete problems. in Proc. 10th Ann. ACM Symp. on Theory of Computing, (1978) 253-264.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....has weights on its vertices and the goal is to nd a maximum weight induced subgraph with the property. For example, we might search for an induced subgraph of maximum size that is chordal, acyclic, without cycles of a speci ed length, without edges, of maximum degree r 1, bipartite or a clique [Yan78]. For exact de nitions of various NP hard problems in this paper, the reader is referred to Garey and Johnson s seminal book [GJ79] Yannakakis has shown that for many natural hereditary properties , MISP( is NP complete even when restricted to planar graphs [Yan78] Using the results of ....

.... 1, bipartite or a clique [Yan78] For exact de nitions of various NP hard problems in this paper, the reader is referred to Garey and Johnson s seminal book [GJ79] Yannakakis has shown that for many natural hereditary properties , MISP( is NP complete even when restricted to planar graphs [Yan78]. Using the results of Section 4, we obtain approximation algorithms for both maximization and minimization problems such as the maximum independent set problem, the minimum vertex cover problem and the minimum dominating set problem on single crossing minorfree graphs. Theorem 10. For G a ....

Mihalis Yannakakis. Node- and edge-deletion NP-complete problems. In Proceedings of the Tenth Annual ACM Symposium on Theory of Computing, pages 253-264, 1978. 29

....2 I, computes an ffl close solution for x in polynomial time. In the above definition, uniformity means that there is an algorithm that, given ffl, computes A ffl . Among NP optimization problems, we mainly focus on those problems which are also hereditary (see Definition 4) In fact, Yannakakis [Yan78] has shown that many natural hereditary problems are NPcomplete even when the graphs under consideration are planar graphs. Definition 4. Yan78] Property on graphs is called hereditary if, whenever holds for G, holds for all induced subgraphs of G. For hereditary property , the maximum ....

....computes A ffl . Among NP optimization problems, we mainly focus on those problems which are also hereditary (see Definition 4) In fact, Yannakakis [Yan78] has shown that many natural hereditary problems are NPcomplete even when the graphs under consideration are planar graphs. Definition 4. [Yan78] Property on graphs is called hereditary if, whenever holds for G, holds for all induced subgraphs of G. For hereditary property , the maximum induced subgraph problem associated with (MISP( is the problem of finding a maximum subset U of vertices of a graph G such that G[U ] has property . ....

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Mihalis Yannakakis. Node- and edge-deletion NP-complete problems. In Conference Record of the Tenth Annual ACM Symposium on Theory of Computing (San Diego, CA, 1978), pages 253--264. ACM press, New York, 1978. 20

....The maximum induced subgraph problem for hereditary property , is the following problem: Given a graph G = V; E) find a maximum subset of V that induces a subgraph satisfying . We call this problem MISP( A wide range of this type of problems has been shown to be NP complete by Yannakakis in [17]. There is a long series of results concerning fast approximation algorithms (serial or parallel) for such problems. An algorithm, that given an instance of MISP( always returns a solution that is of size at least a constant factor 1=ff, is called an approximation algorithm for MISP( with ....

M. Yannakakis. Node- and edge-deletion NP-complete problems. In Proceedings of the 10th Annual Symposium on Theory of Computing, pages 253--264, New York, 1978. ACM Press.

.... more general theorem: Theorem 8 ( HNRT01,Haj01] Given the clique sum series of an H minor free graph G, where H is a single crossing graph, there are PTASs with approximation ratio 1 1=k (or 1 2=k) running in n) time (c is a small constant) on graph G for hereditary maximization problems (see [Yan78] for exact de nitions) such as maximum independent set and other problems such as maximum triangle matching, maximum H matching, maximum tile salvage, minimum vertex cover, minimum dominating set, minimum edge dominating set, and subgraph isomorphism for a xed pattern. 8 Applying Theorem 5, we ....

Mihalis Yannakakis. Node- and edge-deletion NP-complete problems. In Conference Record of the Tenth Annual ACM Symposium on Theory of Computing (San Diego, CA, 1978), pages 253-264. ACM press, New York, 1978. 11

....an optimization problem has a fully polynomial time approximation scheme if there exists a PTAS A whose running time is bounded by a polynomial in x and 1 #. Among NP optimization problems, we mainly focus on those problems which are also hereditary (see Definition 2. 4) In fact, Yannakakis [Yan78] has shown that many natural hereditary problems are NP complete even when the graphs under consideration are planar graphs. Definition 2.4 [Yan78] Property # on graphs is called hereditary if, whenever # holds for G, # holds for all induced subgraphs of G. For hereditary property #, the maximum ....

.... x and 1 #. Among NP optimization problems, we mainly focus on those problems which are also hereditary (see Definition 2.4) In fact, Yannakakis [Yan78] has shown that many natural hereditary problems are NP complete even when the graphs under consideration are planar graphs. Definition 2. 4 [Yan78] Property # on graphs is called hereditary if, whenever # holds for G, # holds for all induced subgraphs of G. For hereditary property #, the maximum induced subgraph problem associated with # (MISP(#) is the problem of finding a maximum subset U of vertices of a graph G such that G[U ] has ....

[Article contains additional citation context not shown here]

Mihalis Yannakakis. Node- and edge-deletion NP-complete problems. In Conference Record of the Tenth Annual ACM Symposium on Theory of Computing (San Diego, CA,

....one correspondence between bicliques in the bipartite double of the graph and bicliques in the graph itself. If one defines a biclique as an induced complete bipartite subgraph (so A and B are independent sets in G) then the maximum vertex biclique problem for general graphs is NP complete (see [10]) A natural third variant is the so called maximum edge biclique problem (MBP) where the requirement is that A # B #k.Uptonowthe complexity of this problem was still undecided. In various papers the complexity of MBP is mentioned and guessed to be NP complete ( 1, 4, 3, 9] In [1] some ....

M. Yannakakis, Node- and edge-deletion NP-complete problems, Proc. 10th Ann. ACM Symp. on Theory of Computing, Association for Computing Machinery, New York, 253-264, (1978). 5

....to us by J. Kratochv il [12] We also give a short list of problems that are NP complete for graphs, but easy for switching classes. The second part of the paper is devoted to a number of problems that are hard for switching classes. We generalize to switching classes a result of Yannakakis [16] on graphs, which is then used to prove that the independence problem is NP complete for switching classes. This problem can be polynomially reduced to the embedding problem (given two graphs G and H, does there exist a graph in [G] in which H can be embedded) Hence, the latter problem is also ....

....hereditary properties of graphs that are also switch nontrivial: G is discrete, G is complete, G is bipartite, G is complete bipartite, G is acyclic, G is planar, G has chromatic number (G) k where k is a fixed integer, G is chordal, and G is a comparability graph. 2 Yannakakis proved in [16] (see also [8] the following general completeness result. Theorem 6.2 Let P be a nontrivial hereditary property of graphs. Then the problem for instances (G; k) with k jV (G)j whether G has an induced subgraph GjA such that jAj k and P(GjA ) is NP hard. Moreover, if P is in NP, then the ....

M. Yannakakis. Node- and edge-deletion NP-complete problems. In Proc. 10th Ann. ACM Symp. on theory of Computing, pages 253 -- 264, New York, 1978. ACM.

....ACM, 36:474 509. Lengauer, T. 1990) Combinatorial Algorithms for Integrated Circuit Layout. Applicable Theory in Computer Science. B. G. Teubner and John Wiley Sons. Leung, J. 1992) A new graph theoretic heuristic for facility layout. Management Science, 38(4) 594 605. Lewis, J. M. and Yannakakis, M. 1980) The node deletion problem for hereditary properties is NP complete. Journal of Computer and System Sciences, 20(2) 219 230. Liebers, A. 1996) Methods for planarizing graphs A survey and annotated bibliography. Technical Report Konstanzer Schriften in Mathematik und Informatik Nr. ....

Yannakakis, M. (1978). Node- and edge-deletion NP-complete problems. In Proceedings 10th Annual ACM Symposium on the Theory of Computing (STOC'78), pages 253-264.

....1 Introduction Given a graph it is easy to test whether or not the graph is bipartite. If the graph is not bipartite then it may be rendered so by deleting some of its edges or vertices. These are, respectively, the problems of edge deletion and vertex deletion. Both are NP complete problems [1, 2, 3]. In this paper we present an estimator to estimate the number of vertices that may have to be deleted to render a graph bipartite. Although such an estimator does not give us a solution to the problem, it is nevertheless useful because it does give us an estimate of the cost of the solution. This ....

M. Yannakakis, "Node and edge deletion np-complete problems," in Proceedings, 10th ACM Symp. on Theory of Computation, pp. 253--264, 1978.

....of finding a minimum k vertex connected or k edge connected spanning subgraph are NP hard for any fixed k 2. For the more relaxed problem of finding sparse but not necessarily minimal k edge connected and k vertex connected spanning subgraphs, linear time algorithms are known ( 18] Yannakakis ([24]; see also [15] showed that the related problem of deleting a minimum set of edges so that the resulting graph has a given property is NP hard for several graph properties (e.g. planar, outerplanar, transitive digraph) There is a natural sequential algorithm for finding a minimal spanning ....

M. Yannakakis, Node- and edge-deletion NP-complete problems, Proc. 10th Ann. ACM Symp. on Theory of Computing, New York, 1978, pp. 253-264. 41

....GreedyCut(G) Input: an undirected graph G with weights on edges Output: a cut of G 1. Start with both sides empty, so the initial value of the cut is zero. 2. For every node v in G do (a) Put v on the side which increases the current cut value more. Figure 2. 1: Algorithm GreedyCut cubic [123, 155], or quasi planar [14] Since this problem is so hard, researchers have been working along two different directions. The first direction is to look for algorithms for MAXCUT on special graphs. It is known that MAXCUT is polynomial time solvable on planar graphs [72, 114] weakly bipartite graphs ....

M. Yannakakis. Node- and edge-deletion NP-complete problems. In Conference Record of the Tenth Annual ACM Symposium on Theory of Computing, pages 253--264, San Diego, California, 1--3 May 1978.

....endpoints in W is at most k The k edge in subgraph problem is to find an induced subgraph on a maximum number of nodes and with at most k edges. For k = 0, this problem is equivalent to the maximum independent set problem. For k 0, the problem was proved to be strongly NP hard by Yannakakis [12]. For either k = 0 or k 0 the problems remain NP complete even if the input graph is cubic or planar [12] The k edge cut problem is trivial: for maximization, the optimal solution is W = V , and for minimization W = The versions of the k edge in and k edge incident subgraph where the ....

....number of nodes and with at most k edges. For k = 0, this problem is equivalent to the maximum independent set problem. For k 0, the problem was proved to be strongly NP hard by Yannakakis [12] For either k = 0 or k 0 the problems remain NP complete even if the input graph is cubic or planar [12]. The k edge cut problem is trivial: for maximization, the optimal solution is W = V , and for minimization W = The versions of the k edge in and k edge incident subgraph where the number of spanning edges is required to be exactly equal to k are easily shown to be NP complete by a ....

M. Yannakakis, "Node- and edge-deletion NP-complete problems" Proceedings of the 10th Annual ACM Symposium on the Theory of Computing (1978) 253--264. 11

....minimum weight (or minimum cardinality) set of vertices (arcs) whose deletion gives a graph satisfying a given property. There are different versions of feedback set problems, depending on whether the graph is directed or undirected and or the vertices (arcs) are weighted or unweighted. Yannakakis [88] has given a general NP hardness proof for almost all vertex and arc deletion problems Date: February 18, 1999, Revised April 2, 1999. AT T Labs Research Technical Report: 99.2.2. 1 2 P. FESTA, P. M. PARDALOS, AND M. G. C. RESENDE restricted to planar graphs. These results apply to the planar ....

....Hochbaum [40] to be NP complete. The picture is still less clear for the directed case, where despite the arduous efforts of several researchers, the best known approximation bound is still O lognloglogn , and no approximation algorithm with constant ratio bound has been reported. Yannakakis [88] conjectured that there is a gap between the approximation of the directed and the undirected cases for feedback vertex set. This remains the biggest open question in the approximation of feedback vertex set problems. Traditionally, the major tool used to attack feedback vertex set has been graph ....

M. Yannakakis, Node and edge-deletion NP-complete problems, in Proceedings of the 10 th Annual ACM Symposium on Theory of Computing (1978) pp. 253-264.

....that for any two vertex u; v 2 VG , u and v have no common neighbor in f(GI) if and only if (u; v) 2 EG . Hence there is a neighborhood independent set of size t in G if and only if there is an independent set of size t in GI . 2 Remark: On page 195 of Garey and Johnson [9] see also Yannakakis [16]) it is shown that all problems with Pi property are NP hard. The above problem in Lemma 5.4 satisfies a kind of strong Pi property (but not the exact Pi property) Theorem 5.5 It is NP complete to decide whether a given multicast graph is strongly t connected. Proof. It is clear that the ....

M. Yannakakis . Node- and edge-deletion NP-complete problems. In: Proc. ACM STOC '78, pages 253--264, ACM Press, 1978.

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M. Yannakakis. Node- and edge-deletion NP-complete problems. in Proc. 10th Ann. ACM Symp. on Theory of Computing, (1978) 253-264.

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Yannakakis, M.: Node- and Edge-Deletion NP-Complete Problems. Proc. 10th Annual ACM Symp. on the Theory of Comp. (STOC78), ACM New York, 1978, pp 253--264. 17

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Yannakakis, M.: Node- and Edge-Deletion NP-Complete Problems. Proc. 10th Annual ACM Symp. on the Theory of Comp. (STOC78), ACM New York, 1978, pp 253--264.

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M. Yannakakis, Node and edge{deletion NP{complete problems, in "Proceedings of the 10th Annual ACM Symposium on Theory of Computing ", 1978, 296-313. 17

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Mihalis Yannakakis. Node- and edge-deletion NP-complete problems. In Proceedings of the Tenth Annual ACM Symposium on Theory of Computing, pages 253-264, 1978. 30

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Mihalis Yannakakis. Node- and edge-deletion NP-complete problems. In Proceedings of the Tenth Annual ACM Symposium on Theory of Computing, pages 253--264, 1978. 30

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M. Yannakakis. Node and edge-deletion NP-complete problems. In Proceedings of the 10th Annual ACM Symposium on Theory of Computing, pages 253--264, May 1978. 18

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M. YANNAKAKIS, Node- and edge-deletion NP-complete problems, in Conference Record of the Tenth Annual ACM Symposium on Theory of Computing (San Diego, CA,

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Mihalis Yannakakis. Node- and edge-deletion NP-complete problems. In Conference Record of the Tenth Annual ACM Symposium on Theory of Computing (San Diego, CA, 1978), pages 253--264. ACM press, New York, 1978.

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Yannakakis, M.: Node- and Edge-Deletion NP-Complete Problems. Proc. 10th Annual ACM Symp. on the Theory of Comp. (STOC78), ACM New York, 1978, pp 253--264. 17

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M. Yannakakis . Node- and edge-deletion NP-complete problems. In: Proc. ACM STOC '78, pages 253--264, ACM Press, 1978. 14

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