P. L. Hammer, T. Ibaraki, and U. N. Peled. Threshold numbers and threshold completions. In P. Hansen, editor, Studies on Graphs and Discrete Programming, pages 125--145. North-Holland, 1981.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of relaxation of the recognition problem: Certain edges must definitely be included in the graph, and certain edges are disallowed, but there is freedom in deciding to include any subset of the (possibly many) other edges. Sandwich problems have been studied explicitly in [20] and implicitly in [21], 2] and [33] Below we give several examples of important sandwich problems arising in practice. Definitions of the graph families mentioned in the examples are given in later sections. Physical Mapping of DNA [5] In molecular biology, information on intersection or non intersection of pairs ....

P. L. Hammer, T. Ibaraki, and U. N. Peled. Threshold numbers and threshold completions. In P. Hansen, editor, Studies on Graphs and Discrete Programming, pages 125--145. North-Holland, 1981.

....exist a polynomialtime membership and equivalence query algorithm for learning unions (or by a dual argument, intersections) of k halfspaces, for any fixed k 3. Towards extending the above result, note that a monotone function is a union of k halfspaces iff it is a union of k monotone halfspaces [30]. Also, a monotone DNF formula whose terms all have two variables represents the union of k halfspaces iff it represents the union of k graphic halfspaces [58] also follows from [24, 28] By combining these facts with Lemma 8.2 and with Corollaries 6.3 and 6.4, we get the following: Corollary ....

P. L. Hammer, T. Ibaraki, and U. N. Peled, Threshold Numbers and Threshold Completions, pages 125--145, North Holland, Amsterdam, 1981.

....the coloring (see [5] and references thereof) Other biologically motivated problems, called sandwich 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 [15, 18, 20, 23]. Results on the parametric complexity of several completion problems were also obtained [8, 24] Approximation algorithms exist for several problems. In [31] 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 ....

P. L. HAMMER, T. IBARAKI, AND U. N. PELED, Threshold numbers and threshold completions, in Studies on Graphs and Discrete Programming, P. Hansen, ed., North-Holland, 1981, pp. 125--145.

....the coloring (see [5] and references thereof) Other biologically motivated problems, called sandwich 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 ....

P. L. Hammer, T. Ibaraki, and U. N. Peled. Threshold numbers and threshold completions. In P. Hansen, editor, Studies on Graphs and Discrete Programming, pages 125--145. North-Holland, 1981.

....( no threshold completion exists ) 7) fi; 8) T T [ f vw 2 E j w 2 Gamma GW (v) g; 9) W V (S Gamma T ) 10) od Algorithm 2.1 Fact 2: A set S E has a threshold completion (in G) if and only if G does not contain an alternating cycle relative to S. Proof. Fact 2 is Corollary 4 in [11]. Another proof can be found in the full paper. 2 In the light of Fact 1 and 2 the threshold cover problem looks like a coloring problem: Find the smallest integer k such that an AC 2l free partition E = E 0 : E k Gamma1 exists, i.e. no AC 2l , l 2, is contained in any of the E i , i ....

....no set in the bipartition contains an AC 2l , l 2 Theorem 2.5 Let G = V; E) be a graph with (G ) 2. If there is an AC 2l , l 3, relative to one of the color classes of G then an AC 6 relative to one of the color classes of G must also exist. Proof. Theorem 2. 5 is Corollary 6 in [11]. Another proof will be given in the full paper. 2 As mentioned before there are two possibilities of an AC 6 . Now let us have a closer look at them in respect of the color classes of G . Definition 2.6 Let v 0 ; v 5 be an AC 6 relative to one of the color classes of G . Then the ....

P.L.Hammer, T.Ibaraki, U.N.Peled, Threshold numbers and threshold completions, Ann. Discrete Math. 11 (1981), 125-145.

....k 2 n Gamma1 , there is a Boolean function having threshold number k, with the parity function attaining the maximum value. Chv atal and Hammer [1] proved that computing the threshold number is NPhard. The first explicit definition of the threshold number appears in later work by Hammer et al. [3]. They established bounds for the threshold number of positive Boolean functions. A Boolean function is positive (sometimes called monotone) if it satisfies f(x) f(y) for any pair of binary vectors x; y with x y. Here x y means that x i y i for all i = 1; n. A binary vector p is said ....

....means that x i y i for all i = 1; n. A binary vector p is said to be a prime implicant of a Boolean function f if f(p) 1 holds and p is minimal with respect to having this property. Every positive Boolean function is known to have a unique collection of prime implicants. Hammer et al. [3] show in particular that the threshold number of such a function is less than i n bn=2c j . Zuev and Lipkin [18] proved that almost all Boolean functions have a threshold number that satisfies the bounds Omega Gamman n =n) and O( ln n) Delta 2 n =n) In [9] see also [10] Maass and ....

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P. L. Hammer, T. Ibaraki and U. N. Peled, Threshold numbers and threshold completions, in: Studies on Graphs and Discrete Programming, ed. P. Hansen, Annals of Discrete Mathematics 11, (Mathematics Studies 59), North-Holland, Amsterdam, 1981, pp. 125--145.

....10.4 in [8] another proof is given in [23] Now assume that a graphic function f is a union of l halfspaces, i.e. let f = f 1 Delta Delta Delta f l , where the f i s are halfspaces. As f is a monotone function, without loss of generality the f i s can be also assumed to be monotone [9]. Then we have f = f (gr) 1 Delta Delta Delta f (gr) l , where f (gr) 1 ; f (gr) l are graphic halfspaces, which completes the proof of the claim. It remains to show that for any fixed k 3 the UNION OF k HALFSPACES problem with instances in monotone 2 DNF is in NP. Using ....

....result follows in both cases from Theorem 3.1, as the considered classes have the projection property. The NP hardness result for part (ii) follows from Theorem 4.11. Using the fact that for any l 1, the threshold number of the function y 1 y 2 y 3 y 4 . y 2l Gamma1 y 2l 2 F 2l is exactly l [6, 9], all hardness results can be shown to remain valid even when the problems are restricted to instances in 3 DNF. The inclusions in Sigma p 2 are proved in all cases by the algorithm that guesses existentially a threshold decomposition, and then verifies universally whether the ....

[Article contains additional citation context not shown here]

P. L. Hammer, T. Ibaraki and U. N. Peled, "Threshold Numbers and Threshold Completions ", in: Studies on Graphs and Discrete Mathematics (P. Hansen, ed.), North-Holland, Amsterdam, 1981, pp. 125--145.

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