H. Kaplan, R. Shamir, and R. E. Tarjan. Tractability of parameterized completion problems on chordal and interval graphs. Found. Comput. Ser., pages 780 -- 791, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....in practice. See, for example, Weihe [Wei00] 13 There are many others. One notable example is the one we met earlier CHORDAL GRAPH COMPLETION. Give a graph G can one add k vertices to get a triangulated graph. Yannakakis proved that the general problem is NPcomplete. Kaplan, Shamir and Tarjan [KST94] used Kernelization to show that there is a O( k jV jjEj f(k) algorithm for suitably chosen computable f . Leizhen Cai [LeC96] also used the Kernelization method to give a O( 4 ( k 1) 3=2 ) jE(G)jjV (G)j jV (G)j ] algorithm. Time might be appropriate to re examine these for ....

H. Kaplan, R. Shamir and R. E. Tarjan, "Tractability of Parameterized Completion Problems on Chordal and Interval Graphs: Minimum Fill-In and DNA Physical Mapping," in: Proc. 35th Annual Symposium on the Foundations of Computer Science (FOCS), IEEE Press (1994), 780--791.

....of three vertices) Natanzon et al. 14] give a general constant factor approximation for deletion and editing problems on boundeddegree graphs with respect to properties (such as being a cluster graph) that can be characterized by a finite set of forbidden induced subgraphs. Kaplan et al. [10] and Mahajan and Raman [13] considered other special cases of edge modification problems with particular emphasis on fixed parameter tractability results. Khot and Raman [11] recently investigated the parameterized complexity of vertex deletion problems for finding subgraphs with hereditary ....

H. Kaplan, R. Shamir, and R. E. Tarjan. Tractability of parameterized completion problems on chordal, strongly chordal, and proper interval graphs. SIAM Journal on Computing, 28(5):1906--1922, 1999.

....gives an algorithm with running time O(jGj ) for VERTEX COVER. These simple strategies have been successfully applied to a wide variety of problems such as PLANAR DOMINATING SET, FEEDBACK VERTEX SET, FACE COVER NUMBER FOR PLANAR GRAPHS, see [DF98] MIMIMUM FILLIN (Kaplan, Tarjan and Shamir [KST94]) for search trees, and k LEAF SPANNING TREE, DF98] various Pi i;j;k GRAPH MODIFICATION PROBLEMS (Leizhen Cai [LeC95] SET BASIS, UNIQUE HITTING SET ( DF98] and various phylogenetic tree metric problems (Allen [Al98] 2.4 Why This Is Interesting: The Deal with the Devil The above ....

H. Kaplan, R. Shamir and R. E. Tarjan, "Tractability of Parameterized Completion Problems on Chordal and Interval Graphs: Minimum Fill-In and DNA Physical Mapping," in: Proc. 35th Annual Symposium on the Foundations of Computer Science (FOCS), IEEE Press (1994), 780--791.

....into a corresponding variety of heuristics. We begin by describing a new FPT algorithm for Vertex Cover that is currently the best known. It is based on two simple, but standard methods, reduction to a problem kernel, and search trees. More examples and discussion of these methods can be found in [DF95c, DF98, KST94, MR98]. Algorithm 4.1. An Improved Direct FPT Algorithm for Vertex Cover. The algorithm proceeds in two phases. In the first phase we compute the kernel of the given instance (G; k) or answer no . The kernel is an instance (G ) where jG k such that G has a vertex cover of size k ....

H. Kaplan, R. Shamir and R. E. Tarjan, "Tractability of Parameterized Completion Problems on Chordal and Interval Graphs: Minimum Fill-In and DNA Physical Mapping," in: Proc. 35th Annual Symposium on the Foundations of Computer Science (FOCS), IEEE Press (1994), 780--791.

....in which the clique size of the interval graph may be at most four. Another closely related problem is Unit Intervalizing Sandwich Graphs, which asks whether, for a given sandwich graph S = V; E 1 ; E 2 ) there is a graph G = V; E) such that G is a unit interval graph and E 1 E E 2 . In [KS93, KST94], it is shown that this problem is NP complete, polynomial for a fixed maximum clique size of G, hard for W [1] but solvable in O(n ) time if k is the maximum clique size of G. This paper is organized as follows. In Section 2, necessary preliminary definitions and results are given, and the ....

....case of Unit Intervalizing Sandwich Graphs (UISG) is the problem Unit Intervalizing Colored Graph (USCG) which asks whether there exists a supergraph G of a given graph G, such that G is a unit interval graph, and is properly colored by a given coloring c for G. The O(n ) algorithm of [KS93, KST94] for UISG with maximum clique size k can also be used for UICG with k colors. For k = 3, this gives an O(n ) time algorithm. We expect that our algorithm for ICG can be used to obtain a linear time algorithm for this problem with k = 3. ....

H. Kaplan, R. Shamir, and R. E. Tarjan. Tractability of parameterized completion problems on chordal and interval graphs: Minimum fill-in and physical mapping. In Proceedings of the 35th annual symposium on Foundations of Computer Science (FOCS), pages 780--791. IEEE Computer Science Press, 1994.

....Parameter: jAj FPT (solvable in O(2 jAj n) time [55, 54] Minimum Fill in Instance: A graph G = V; E) a positive integer k. Question: Can we add no more than k edges to G and cause G to become chordal Parameter: k FPT (solvable in time O(c k jEj) and O(k 5 jEj jV j f(k) [82]) Minimum Quartet Inconsistency Instance: A set S of n taxa; a set Q S of quartet topologies such that there is exactly one topology for every quartet set corresponding to S; an integer k. Question: Is there an evolutionary tree T for S such that the set of quartet topologies induced by T di ers ....

....Graph With Bounded Clique Size Instance: A graph G = V; E) a set E 0 V V E of prohibited edges; a positive integer k. Question: Is there a G 0 G which is a proper interval graph and has clique size at most k, and G 0 has no edges from E 0 Parameter: k W[t] hard for all t ([81, 82]; it remains W[t] hard even when E 0 = 81] Note: See Proper Intervalizing Colored Graphs for a restricted version of this problem. Restricted Valence Isomorphism Instance: Two graphs G = V; E) and H = V 0 ; E 0 ) a positive integer k. Question: Are G and H isomorphic graphs such ....

H. Kaplan, R. Shamir, and R. E. Tarjan. Tractability of parameterized completion problems on chordal and interval graphs: Minimum Fill-in and DNA Physical Mapping. In Proc. 35th Annual Symposium on Foundations of Computer Science, pages 780-791, Santa Fe, New Mexico, 20-22 Nov. 1994. IEEE Computer Society. 88

....to determine the most likely orderings of the clones from the overlap data. In reality Physical Mapping is a complicated process which includes stochastic methods. Several idealized combinatorial versions of the Physical Mapping Problem have been investigated from the viewpoint of complexity in [1,3,6,9,11,14,15]. Alizadeh et al. 1] have shown that it is an NP complete problem to find the most likely order of the endpoints of the clones with respect to various goal functions (see [12] for the theory of NP completeness) Since it is a common belief that NP complete problems do not allow efficient ....

.... from different parts of the DNA molecule may be concatenated to form one clone; chimerism is of fundamental concern as chimerisms have been estimated to occur in 40 60 of the clones for the two most widely used human YAC libraries [10] Several variants of (1) and (2) have been studied in [3,6,11,14,15] some of which have been shown to be polynomial time solvable for fixed parameters. Goldberg et al. 9] leave open the complexity of (4) with a sparse probe clone matrix (i.e. each clone contains at most a constant number of probes and each probe is contained in at most a constant number of ....

H. Kaplan, R. Shamir, R. E. Tarjan, Tractability of parameterized completion problems on chordal and interval graphs: Minimum Fill-in and Physical Mapping, in: Proc. FOCS '94 (1994) 780-791.

....the exponent c is typically small. One can equivalently define fixed parameter tractability to mean solvability in time O(f(k) n c ) that is, with only an additive contribution from the parameter [CCDF97] There is a rich collection of distinctive techniques for devising FPT algorithms (see [DF95c, DF97b, KST94, LeC97, Ste92]) Contrasting complexity behavior is exhibited by the naturally parameterized problems such as Clique, Dominating Set, and Bandwidth, for which the best known algorithms have running times O(n ck ) These problems have been shown to be complete or hard for the various levels of the W hierarchy ....

H. Kaplan, R. Shamir, and R.E.Tarjan, Tractability of parameterized completion problems on chordal and interval graphs: minimum fill-in and DNA physical mapping, in Proc. 35-th Annual Symp. Found. Computer Science, (1994), 780--891.

....research was for the design of algorithms for the problems to determine whether a given colored graph is contained in a properly colored chordal or interval graph, for small number of colors, larger than three. While the complexity of these problems is now more or less well understood (see e.g. [11, 6, 5]) the result of this paper may be perhaps useful as a step for solving special cases. 2 Definitions The graphs in this paper are considered to be simple and undirected. We say that a set S separates vertices x and y in a graph G, if every path from x to y in G uses a vertex in S. The subgraph ....

H. Kaplan, R. Shamir, and R. E. Tarjan. Tractability of parameterized completion problems on chordal and interval graphs. Found. Comput. Ser., pages 780 -- 791, 1994.

....an integer k. Question: Is there an interval graph G 0 = V; E 0 ) such that E E 0 and jE 0 n Ej k is NP complete even when G is stipulated to be an edge graph (see [14] Problem GT35) Interval Graph Completion arises in computational biology (see, e.g. 4] and is known to be FPT [7, 17]. 9 From Theorem 3 it follows immediately that the problem of Search Cost: deciding given a graph G and an integer k, whether fl(G) k or not, is NPcomplete even for edge graphs and that finding the search cost is FPT for a fixed k. An O(n 1:722 ) time algorithm was given in [19] for the ....

H. Kaplan, R. Shamir, and R. E. Tarjan, Tractability of parameterized completion problems on chordal and interval graphs: Minimum fill-in and physical mapping, in Proceedings of the 35th annual symposium on Foundations of Computer Science (FOCS), IEEE Computer Science Press, 1994, pp. 780--791.

....of the size of the graph. For instance, in the sequencing of the yeast genome, a typical working parameter value of k = 8 is reported in [18] Hence, the complexity of ICG when parameterized by k is of importance. More on intervalizing graphs and its application to physical mapping can be found in [33, 42]. A related problem, where G 0 is requested to be a proper interval graph was shown to be W [1] hard by Kaplan and Shamir [41] Lemma 4 Let G = V; E) be a graph with a vertex coloring c : V f1; 2; kg, that is a subgraph of a properly colored interval graph G 0 . Then the pathwidth ....

H. Kaplan, R. Shamir and R. E. Tarjan. Tractability of Parameterized Completion Problems on Chordal and Interval Graphs. FOCS 1994, 780-791.

....even for trees [ADS98a] The problem that we will study here is another restriction of the above problems, the Proper interval colored graph (picg) problem: Given a colored graph, can one add edges to obtain a proper interval graph that is still colored by the initial coloring. In [KS96] and [KST94] it was shown that the parameterized version of the picg problem, with parameter the number of colors, is W [1] hard, this implies the NP completeness of picg. They also gave a polynomial time algorithm for constant number of colors. For another simpler NPcompleteness proof for picg see [GGKS95] ....

H. Kaplan, R. Shamir, and R. E. Tarjan. Tractability of parameterized completion problems on chordal and interval graphs: Minimum fill-in and physical mapping. In 35th Annual Symposium on Foundations of Computer Science (FOCS'94), pages 780--791. IEEE Computer Society Press, 1994.

....into a P k n . Hence, the Bandwidth minimization problem for a graph G is equivalent to finding an embedding of G into a proper interval graph with the smallest maximumclique size. Recently, new applications of the Bandwidth problem arose in physical mapping of DNA in molecular biology, compare [16]. In this area, the embedding formulation is the natural one. The treewidth tw(G) resp. pathwidth pw(G) of a graph G is known to correspond to the smallest maximum clique size of all chordal resp. interval supergraphs of G, minus one. Observe that interval graphs are chordal. Hence, in this ....

H. Kaplan, R. Shamir, and R. E. Tarjan. Tractability of parameterized completion problems on chordal and interval graphs: Minimum fill-in and physical mapping. Proc. 35th FOCS, pages 780--791, 1994.

....a minimal triangulation of a graph is in general NP hard [48] but for the graphs we will consider, even simple heuristics work effectively and efficiently. Furthermore, it was recently shown that exact solutions can be obtained efficiently when the number of additional edges needed is small [10, 29]. Lemma 3.2 (from [24] Every triangulated graph G has a perfect elimination scheme, v 1 ; v 2 ; vn ; this is an ordering of the nodes so that the set X i = fv j : j i and (v i ; v j ) 2 Eg forms a clique (i.e. for all fv k ; v l g X i ; v k ; v l ) 2 E) The maximal cliques ....

H. Kaplan, R. Shamir, and R.E. Tarjan. Tractability of parameterized completion problems on chordal and interval graphs: minimum fill-in and physical mapping. In Proceedings of the 35th Symposium on Foundations of Computer Science, pages 780-791. IEEE Computer Science Press, Los Alamitos, California, 1994. To appear, SIAM J. Computing.

....in which the clique size of the interval graph may be at most four. Another closely related problem is Unit Intervalizing Sandwich Graphs, which asks whether, for a given sandwich graph S = V; E 1 ; E 2 ) there is a graph G = V; E) such that G is a unit interval graph and E 1 E E 2 . In [KS93, KST94], it is shown that this problem is NP complete, polynomial for a fixed maximum clique size of G, hard for W [1] but solvable in O(n k Gamma1 ) time if k is the maximum clique size of G. This paper is organized as follows. In Section 2, necessary preliminary definitions and results are given, ....

....Unit Intervalizing Sandwich Graphs (UISG) is the problem Unit Intervalizing Colored Graph (USCG) which asks whether there exists a supergraph G 0 of a given graph G, such that G 0 is a unit interval graph, and is properly colored by a given coloring c for G. The O(n k Gamma1 ) algorithm of [KS93, KST94] for UISG with maximum clique size k can also be used for UICG with k colors. For k = 3, this gives an O(n 2 ) time algorithm. We expect that our algorithm for ICG can be used to obtain a linear time algorithm for this problem with k = 3. ....

H. Kaplan, R. Shamir, and R. E. Tarjan. Tractability of parameterized completion problems on chordal and interval graphs: Minimum fill-in and physical mapping. In Proceedings of the 35th annual symposium on Foundations of Computer Science (FOCS), pages 780--791. IEEE Computer Science Press, 1994.

....choice for elucidating inherent problem difficulties is parameterized complexity analysis. Several recent papers have applied this theory to problems in biological computing (Bodlaender et al. 1992; Fellows et al. 1993; Kaplan and Shamir, 1993; Bodlaender et al. 1994a; Bodlaender et al. 1994b, Kaplan et al. 1994). We wish to make the point that the theory is potentially of very wide applicability in computational biology. In x2 the basics of parameterized complexity theory are briefly reviewed. In x3 we apply this theory to the Longest common subsequence (LCS) problem when the the fixed parameter is the ....

.... been derived for versions of these problems that have been restricted to fragments of unit length, fragments that are not allowed to be properly included inside each other, and solution interval graphs of bounded clique size (Goldberg et al. 1993; Kaplan and Shamir, 1993; Golumbic et al. 1994; Kaplan et al. 1994). Note that these problems can also be stated on colored graphs, yielding alternate (and more biologically useful) versions of the problem Intervalizing colored graphs studied in Fellows et al. 1993) and Bodlaender et al. 1994b) 4.3 3 D Biopolymer Folding All known approaches to determining ....

Kaplan, H., Shamir, R. and Tarjan, R. E. (1994) Tractability of parameterized completion problems on chordal and interval graphs: minimum fill-in and physical mapping. To appear, Proceedings of the 35th Annual IEEE Conference on the Foundations of Computer Science.

....is a new graph theoretic parameter closely related to pathwidth. An unexpected useful consequence is the equivalence of this parameter to the bandwidth of the graph. Portions of this paper were presented at the 34th Annual IEEE Symp. on the Foundations of Computer Science, Santa Fe, NM 1994 [25] Research supported in part by a grant from the Ministry of Science and Technology, Israel. email: shamir math.tau.ac.il 1 Introduction This paper studies the following graph theoretic questions: ffl Problem A: Given a graph G and a constant k, does there exist a supergraph G which is a ....

Haim Kaplan, Ron Shamir, and Robert E. Tarjan. Tractability of parameterized completion problems on chordal and interval graphs: Minimum fill-in and physical mapping (extended abstract). In Proc. FOCS '94, pages 780--791. IEEE Computer Science Press, 1994.

....complexity will be polynomial. In [13] sparsity was modeled by bounding the clique size of the sandwich graph, for the case of equal length clones. It was shown that the unit interval sandwich problem is polynomial whenever the clique size in the resulting realization is bounded by a constant [13, 15]. Here we deal with clones of arbitrary lengths. The problem discussed in Section 3.1 models sparsity by bounding the maximum clique size of the sandwich graph, and additionally, by bounding the maximumdegree in the input graph. The first requirement means that the largest set of mutually ....

H. Kaplan, R. Shamir, and R. E. Tarjan. Tractability of parameterized completion problems on chordal and interval graphs: Minimum fill-in and physical mapping. In Proceedings of the 35th Symposium on Foundations of Computer Science, pages 780-- 791. IEEE Computer Science Press, Los Alamitos, California, 1994.

....and details. Thus, for example, VERTEX COVER and PATHWIDTH are in FPT [3, 11, 24] but INDEPENDENT SET is W [1] complete [1] and BANDWIDTH is W[t] hard for all t [4] Portions of this paper were presented at the 34th Annual IEEE Symp. on the Foundations of Computer Science, Santa Fe, NM 1994 [21]. AT T Labs, 180 Park Ave, Florham Park, NJ 07932 USA. hkl research.att.com Department of Computer Science, Sackler Faculty of Exact Sciences, Tel Aviv University, TelAviv 69978 ISRAEL. Research supported in part by a grant from the Ministry of Science and the Arts, Israel. ....

....graphs which have such length restrictions. Their results, together with the characterizations of [25] imply that the obstruction size is O(c) and thus for this case too the search tree technique applies and the k completion problem is FPT. After a preliminary version of this paper appeared in [21], L. Cai published a paper [6] that rediscovers our simple search tree based algorithm for CHORDAL COMPLETION (k) see Section 2.1) Using a better known bound on the l th Catalan number, namely c l = O(4 =l ) and a lemma showing that c i 1 c j 1 c i j 1 , Cai proves that our algorithm in ....

H. Kaplan, R. Shamir, and R. E. Tarjan, Tractability of parameterized completion problems on chordal and interval graphs: Minimum fill-in and physical mapping, in Proceedings of the 35th Symposium on Foundations of Computer Science, IEEE Computer Science Press, Los Alamitos, California, 1994, pp. 780--791.

....is superior. Parameterized complexity theory, initiated by Downey and Fellows (cf. 6] studies the complexity of such problems. Parameterized problems 2 that have algorithms of complexity O(f(k)n ff ) with ff a constant) are called fixed parameter tractable. Kaplan, Shamir and Tarjan [16] (henceforth KST) and independently Cai [3] proved that the minimum fill in problem is fixed parameter tractable, by giving an algorithm of complexity O(exp(k)m) for the problem. KST also gave a more efficient O(exp(k) k 2 nm) time algorithm (henceforth KST algorithm) In this paper we give ....

....by giving an algorithm of complexity O(exp(k)m) for the problem. KST also gave a more efficient O(exp(k) k 2 nm) time algorithm (henceforth KST algorithm) In this paper we give the first polynomial approximation algorithm for the minimum fill in problem. Our algorithm builds on ideas from [16]. For an input graph G with minimum fill in of size k, our algorithm produces a triangulation of size at most 8k 2 within a factor of 8k of optimal. The approximation is achieved by identifying in G a kernel set of vertices A of size at most 4k, such that one can triangulate G by adding edges ....

[Article contains additional citation context not shown here]

H. KAPLAN, R. SHAMIR, AND R. E. TARJAN, Tractability of parameterized completion problems on chordal and interval graphs: Minimum fill-in and physical mapping, in Proceedings of the 35th Symposium on Foundations of Computer Science, IEEE Computer Science Press, Los Alamitos, California, 1994, pp. 780--791. To appear in SIAM J. Computing.

....is a new graph theoretic parameter closely related to pathwidth. An unexpected useful consequence is the equivalence of this parameter to the bandwidth of the graph. Portions of this paper were presented at the 34th Annual IEEE Symp. on the Foundations of Computer Science, Santa Fe, NM 1994 [25] y Research supported in part by a grant from the Ministry of Science and Technology, Israel. email: shamir math.tau.ac.il 1 Introduction This paper studies the following graph theoretic questions: ffl Problem A: Given a graph G and a constant k, does there exist a supergraph G 0 of G which ....

Haim Kaplan, Ron Shamir, and Robert E. Tarjan. Tractability of parameterized completion problems on chordal and interval graphs: Minimum fill-in and physical mapping (extended abstract). In Proc. FOCS '94, pages 780--791. IEEE Computer Science Press, 1994.

....complexity, chordal graphs, proper interval graphs, strongly chordal graphs, minimum fill in, physical mapping of DNA. Abbreviated title: Parameterized Completion. Portions of this paper were presented at the 34th Annual IEEE Symp. on the Foundations of Computer Science, Santa Fe, NM 1994 [20]. y Department of Computer Science, Princeton University, Princeton, NJ 08544 USA. Research at Princeton University partially supported by the NSF, Grant No. CCR 8920505, and the Office of Naval Research, Contract No. N00014 91 J 1463. hkl cs.princeton.edu. z Department of Computer Science, ....

H. Kaplan, R. Shamir, and R. E. Tarjan. Tractability of parameterized completion problems on chordal and interval graphs: Minimum fill-in and physical mapping. In Proceedings of the 35th Symposium on Foundations of Computer Science, pages 780--791. IEEE Computer Science Press, Los Alamitos, California, 1994.

....complexity will be polynomial. In [13] sparsity was modeled by bounding the clique size of the sandwich graph, for the case of equal length clones. It was shown that the unit interval sandwich problem is polynomial whenever the clique size in the resulting realization is bounded by a constant [13, 15]. Here we deal with clones of arbitrary lengths. The problem discussed in Section 3.1 models sparsity by bounding the maximum clique size of the sandwich graph, and additionally, by bounding the maximumdegree in the input graph. The first requirement means that the largest set of mutually ....

H. Kaplan, R. Shamir, and R. E. Tarjan. Tractability of parameterized completion problems on chordal and interval graphs: Minimum fill-in and physical mapping. In Proceedings of the 35th Symposium on Foundations of Computer Science, pages 780-- 791. IEEE Computer Science Press, Los Alamitos, California, 1994.

....complexity will be polynomial. In [15] sparseness was modeled by bounding the clique size of the sandwich graph, for the case of equal length clones. It was shown that the unit interval sandwich problem is polynomial whenever the clique size in the resulting realization is bounded by a constant [15, 16]. Here we deal with clones of arbitrary lengths. The problem discussed in Section 3.1 models sparseness by bounding the maximum clique size of the sandwich graph, and additionally, by bounding the maximum degree in the input graph. The first requirement means that the largest set of mutually ....

H. Kaplan, R. Shamir, and R. E. Tarjan. Tractability of parameterized completion problems on chordal and interval graphs: Minimum fill-in and physical mapping. In Proceedings of the 35th Symposium on Foundations of Computer Science, pages 780--791. IEEE Computer Science Press, Los Alamitos, California, 1994.

....problems, seek a supergraph satisfying a given property which does not include (pre defined) forbidden edges. Polynomial algorithms or NP hardness results are known for many sandwich problems [16, 15, 18, 21] Several results on the parametric complexity of completion problems were also obtained [22, 7]. Approximation algorithms exist for several problems. In [28] an 8k approximation algorithm is given for the minimum fill in problem, where k denotes the size of an optimum solution. In [1] an O(m 1=4 log 3:5 n) approximation algorithm is given for the minimum chordal supergraph problem ....

H. Kaplan, R. Shamir, and R. E. Tarjan. Tractability of parameterized completion problems on chordal and interval graphs: Minimum fill-in and physical mapping. In Proceedings of the 35th Symposium on Foundations of Computer Science, pages 780--791. IEEE Computer Science Press, Los Alamitos, California, 1994. to appear in SIAM J. Computing.

....is superior. Parameterized complexity theory, initiated by Downey and Fellows (cf. 6] studies the complexity of such problems. Parameterized problems that have algorithms of complexity O(f(k)n ff ) with ff a constant) are called fixed parameter tractable. Kaplan, Shamir and Tarjan [16] (henceforth KST) and independently Cai [3] proved that the minimum fill in problem is fixed parameter tractable, by giving an algorithm of complexity O(exp(k)m) for the problem. KST also gave a more efficient O(exp(k) k 2 nm) time algorithm (henceforth KST algorithm) In this paper we give ....

....by giving an algorithm of complexity O(exp(k)m) for the problem. KST also gave a more efficient O(exp(k) k 2 nm) time algorithm (henceforth KST algorithm) In this paper we give the first polynomial approximation algorithm for the minimum fill in problem. Our algorithm builds on ideas from [16]. For an input graph G with minimum fill in of size k, our algorithm produces a triangulation of size at most 8k 2 within a factor of 8k of optimal. The approximation is achieved by identifying in G a kernel set of vertices A of size at most 4k, such that one can triangulate G by adding edges ....

[Article contains additional citation context not shown here]

H. Kaplan, R. Shamir, and R. E. Tarjan. Tractability of parameterized completion problems on chordal and interval graphs: Minimum fill-in and physical mapping. In Proceedings of the 35th Symposium on Foundations of Computer Science, pages 780--791. IEEE Computer Science Press, Los Alamitos, California, 1994. To appear in SIAM J. Computing.

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H. Kaplan, R. Shamir, and R. E. Tarjan. Tractability of parameterized completion problems on chordal and interval graphs. Found. Comput. Ser., pages 780 -- 791, 1994.

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H. Kaplan, R. Shamir, and R. E. Tarjan. Tractability of parameterized completion problems on chordal, strongly chordal, and proper interval graphs. SIAM Journal on Computing, 28(5):1906--1922, 1999. 10

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H. Kaplan, R. Shamir, and R. E. Tarjan. Tractability of parameterized completion problems on chordal, strongly chordal, and proper interval graphs. SIAM J. Comput., 28(5):1906--1922, 1999.

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Haim Kaplan, Ron Shamir, and Robert E. Tarjan. Tractability of parameterized completion problems on chordal, strongly chordal, and proper interval graphs. SIAM Journal on Computing, 28(5):1906--1922, 1999. 20

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Haim Kaplan, Ron Shamir, and Robert E. Tarjan. Tractability of parameterized completion problems on chordal, strongly chordal, and proper interval graphs. SIAM J. Comput., 28(5):1906--1922, 1999.

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H. Kaplan, R. Shamir, and R. E. Tarjan. Tractability of parameterized completion problems on chordal, strongly chordal, and proper interval graphs. SIAM Journal on Computing, 28(5):1906.

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Haim Kaplan, Ron Shamir, and Robert E. Tarjan. Tractability of parameterized completion problems on chordal, strongly chordal, and proper interval graphs. SIAM Journal on Computing, 28(5):1906--1922, 1999. 20

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Haim Kaplan, Ron Shamir, and Robert E. Tarjan. Tractability of parameterized completion problems on chordal, strongly chordal, and proper interval graphs. SIAM J. Comput., 28(5):1906-1922, 1999.

No context found.

H. Kaplan, R. Shamir and R. E. Tarjan. Tractability of parameterized completion problems on chordal and interval graphs: minimum fill-in and DNA physical mapping (extended abstract). In 35th Ann. Proc. of the Foundations of Computer Science (FOCS '94), (1994), 780--891.

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