M. Garey and D. Johnson. Computers and Intractibility. Freeman and Co, 1979.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....shortly, our proof provides insight into the inherent di#culty of the grooming problem. Theorem 3.1 The decision version of the tra#c grooming problem in unidirectional path networks (bifurcated routing of tra#c not allowed) is NP Complete. Proof. The reduction is from the Subsets Sum problem [9]. An instance of the Subsets Sum problem consists of n elements of size s i , n , and a goal sum B. The question is whether there exists a subset of elements whose sizes total to B. Let B 1 = max B, B . For the purpose of the Subsets Sum problem, posing the instance with B or B 1 is ....

....that the optical routing at the hub be Q or more, where Q = i,j=1 F . In the proof below, we use Q rather than F for notational convenience. Theorem 3.6 The decision version of tra#c grooming in star networks is NP Complete. Proof. We reduce the decision version of the Knapsack problem [9] to the grooming problem. An instance of the Knapsack problem is given by a finite set U of cardinality n, for each element u i U a weight , and a value v i , n , a target weight B , and a target value K . The problem asks whether there exists a binary vector X = x 1 ....

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M. R. Garey and D. S. Johnson. Computers and Intractability. W. H. Freeman and Co., New York, 1979.

....complexity of SPADA leads to the notorious trade off between expressiveness and efficiency in first order representations. Indeed, it is well known that a simple matching of two expressions with commutative and associative operators (such as the logical OR of atoms in a clause) is NP complete [12]. Therefore, any known algorithm that checks the coverage of an atom set or that equivalently evaluates a query with respect to a relational database has an exponential complexity. Nevertheless, it has also been proved that queries with up to k atoms, where each atom contains at most j terms, can ....

Garey, M. R. and Johnson, D. S., Computers and intractability, W. H. Freeman and Co., San Francisco, California, 1979.

....: mg with A i 1 A i 2 A i k = 0. Theorem 8 Given a set F of m 3 3 matrices with rational entries and an integer K 1 m=2, the decision problem Is F K length mortal is NP hard. Proof. Via the reduction of [20] or the proof of Proposition 1) and the NP completeness of Bounded PCP [7]. 2 Observe that [2] proves that this result remains true whenever the matrices are assumed to have entries in f0; 1g. 4.4.2 Mortality without repetition When repetitions of matrices are not allowed, the problem also becomes clearly decidable. A multi set F = fA 1 ; Am g of d d ....

....that A i 1 A i 2 A i k = 0 and i j1 6= i j2 for all j 1 6= j 2 . Theorem 9 Given a nite multi set F of m 2 2 matrices, and an integer K, the decision problem Is F K length mortal without repetition is NP hard in the strong sense. 14 The proof uses a reduction from subset product [7]. We restate this problem here. Proposition 2 (Subset Product (Yao) Given a nite set A, a size s(a) 2 N for each a 2 A, and a positive integer B, the decision problem Is there a subset A A such that the product of the sizes of the elements in A is exactly B is NP complete in the ....

M. R. Garey and D. S. Johnson. Computers and Intractability. W. H. Freeman and Co., New York, 1979.

....work. I describe these approaches here, and show that they can be viewed as special cases of multi fidelity algorithms (Figure 3.2) Approximation algorithms produce results that are provably within some bound of the true result. A subset of this class polynomial time approximation schemes [39] have a tuning parameter analogous to fidelity: the error bound . Lower values of lead to longer running times, corresponding to higher fidelity requiring a higher resource consumption. Approximation algorithms are typically of interest for intractable (NP hard) problems, and concentrate on ....

M.R. Garey and D.S. Johnson. Computers and Intractability. Freeman and Co., New York, NY, 1979.

....if edge belongs to the graph connecting all the masters (28) Constraints (29) and (30) guarantee that loop free routes are established. C. Remarks It can be shown that problem is at least as complex as the Geometric Connected Dominating Set problem, which is proven to be NP complete [9]; hence is NP complete. By solving , we obtain the optimal BT WPAN topology, which minimizes the traffic load of the most congested node, i.e. its energy consumption, while guaranteeing the desired throughput. Then, for any source destination pair (u , we can verify whether an ....

M. R. Garey and D. S. Johnson, Computer and Intractability, W. H. Freeman and Co., 1979.

....this is a minimal set de ning this upper envelope. No point in this set is a convex combination of the others. We remark that the followings follow from the de nition of a convex optimal interval caching scheme. This is the problem called Knapsack which is known to be NP complete [13] 1. The set of points f(c B (s; V ) p B (s; V ) g generated by an optimal interval caching scheme is unique. 2. If B is convex optimal then every interval caching algorithm B(V ) is optimal. The following lemma bounds the performance of the optimal cache replacement algorithm by the ....

M. R. Garey and D. S. Johnson. Computers and Intractability. W. H. Freeman and Co., San Francisco, CA, 1979.

....we have f(di) f(dj) or infer that no such assignment ex ists. Theorem 1 The zero conflict memory assignment problem is computationally equivalent to the graph k colorability problem. Proof. Straightforward reduction. Graph k colorability is among the most difficult intractable problems [5, 11, 20]. Thus, Theorem 1 implies that it is very unlikely to find a provably good algorithm for the memory assignment problem. 4 Overview of our method for memory assignment In this section we present an overview of our two phase approach to dealing with the memory assignment problem. In the first ....

M. R. Garey and D. S. Johnson. Computc,s and Int,actability. W. H. Freeman and Co., San Francisco, 1979.

....several natural and more general formulations whic hcapture a cost vs. performance tradeoff (e.g. minimizing congestion subject to a delay constrain t) Such formulations were sho wn to be NP hard and could in fact be view edas instances of the classical shortest weigh t constrained path problem [3]. A pseudopolynomial algorithm for such formulations was also presented, which finds the set of all source to sink paths that lie on the cost vs. dela y tradeoff curve (a similar algorithm appears in [4] This w ork w as supported b y the Design Automation Conference Scholarship Program. ss ....

....with every edge. We are not limited for example to using Elmore delay or considering only the total capacitance as our cost measure. Moreover, this graph model points out the intimate relationship between the context aware buffer insertion problem and the shortest weight constrained problem in [3]. As stated earlier, a path in such a graph represents not only the wiring route to be taken, but by virtue of the vertices on the path, the buffers to be inserted. Figure 5 presents a complete example of a set of buffer stations, the corresponding graph model, and two s # t paths in the ....

M.R.Garey and D.S.Johnson, Computers and Intractability, (Problem ND30) W.H.Freeman and Co.,1979.

....routing. Ideally, we would like to find the set of nodes Nt to be designated as transparent nodes, composed of pairwise non adjacent interior nodes) such that pcR(T(P) T(P) is maximized. However, this is equivalent to finding a maximal independent set in a graph, which is NP complete [14]. An efficient way to pick Nt is to utilize the level ordering of the tree T. Designate any interior node r as the root of the tree. We partition the interior nodes of the tree into two sets, No and N, such than No contains all interior nodes which are at even depth of the tree from the root r ....

M. R. Garey and D. S. Johnson, Computers and Intractability, W. H. Freeman and Co., New York, 1979.

....routing. Ideally, we would like to nd the set of nodes N t to be designated as transparent nodes, composed of pairwise non adjacent interior nodes) such that p2R ( T (p) T (p) is maximized. However, this is equivalent to nding a maximal independent set in a graph, which is NP complete [14]. An ecient way to pick N t is to utilize the level ordering of the tree T . Designate any interior node r as the root of the tree. We partition the interior nodes of the tree into two sets, N 0 and N 1 , such than N 0 contains all interior nodes which are at even depth of the tree from the root r ....

M. R. Garey and D. S. Johnson, Computers and Intractability, W. H. Freeman and Co., New York, 1979.

....edge # # # # # belongs to the graph connecting all the masters (28) Constraints (29) and (30) guarantee that loop free routes are established. C. Remarks It can be shown that problem is at least as complex as the Geometric Connected Dominating Set problem, which is proven to be NP complete [9]; hence # is NP complete. By solving # , we obtain the optimal BT WPAN topology, which minimizes the traffic load of the most congested node, i.e. its energy consumption, while guaranteeing the desired throughput. Then, for any source destination pair ,we can verify whether an alternative ....

M. R. Garey and D. S. Johnson, Computer and Intractability, W. H. Freeman and Co., 1979.

....is generated. All transitions that are covered by the chosen transition (using the variable hierarchy) are removed from the BDD. Step 3 does not necessarily produce a minimum number of tests. Finding the minimal transition coverage is NP Complete. The easiest reduction is to subset sum [14] which can be done by putting all the variables into the Care category. T v v , T T T = T T T v v , Test Generation Efficiency Since the counter example mechanism [12] is heavily used during test generation, we have improved it in order to make test generation faster. ....

M. S. Garey and D. S. Johnson. Computers and Intractability. W. H. Freeman and Co., New York, 1979.

....the outer cycle after all inner cycles become stable. Although there is no direct relation between the number of iterations and the number of feedback edges, fewer feedback edges may give fewer possible value changes. However, nding the smallest number of feedbacks is NP hard on a general graph [8]. feedback edges ###### ## ### #### ####### ## ######## ##### One drawback of the above approach is that, since all the feedbacks are gate inputs, changed values on them always need to be propagated. Studying the interactions in our system, we nd that our system is a heterogeneous system. That ....

M. R. Garey and D. S. Johnson. ######### ### ##############.W.H.Freeman and Co., 1979.

....Number problem asks for a partition into the least number of independent sets. Even the fixed parameter version of this problem, vertex k coloring, where we ask for the existence of a partition into k independent sets, is intractable (NP complete) for any k 3, while it is easy for smaller k [2]. Similarly, the K k cover problem asking for a partition into k perfect codes is NP complete starting from k 4 [5] whereas the partition into perfect matchings problem is NP complete already for k 2 [7] These problems can all be defined as Partitions Into Generalized Dominating Sets. ....

....dominating sets, introduced by Telle in [8] and defined formally in the next section, are parameterized by two sets oe and ae of nonnegative integers. Many well studied vertex subset properties with applications in facility location and network communication can be expressed as (oe; ae) sets [8, 1, 2, 3], see Table 1. In this paper we present a systematic study of the complexity of partitioning vertices of a given graph G into k (oe; ae) sets, for varying values of the parameters k; oe; ae. Our focus has been on the vertex subset properties found in the literature, and on generalizations of ....

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M. R. Garey and D. S. Johnson, Computers and Intractability, W. H. Freeman and Co., 1978.

....q partition exists in the input graph. We call these partition minimization and partition maximization problems. To illustrate and give weight to this formalism, we express some well known problems 1 in the 1 [GTx] as a citation refers to the Graph Theory problem number x in Garey and Johnson [12] 7 terminology of vertex partitioning and also define new vertex partitioning problems as generalizations of old problems. In each case, correctness of the vertex partitioning formulation follows immediately from Definition 4.1. 4.1. Vertex subset problems. Many domination type problems can be ....

M.R. Garey and D.S. Johnson, Computers and Intractability, W.H.Freeman and Co., 1978.

....a graph theoretic algorithm that finds a set of center nodes to minimize the maximum distance between a node and its closest center. Given this definition, the min K center problem is relevant only in the case of optimization condition O(M; 1) The min K center problem is known to be NP complete [7], however a 2 approximate algorithm exists [8] With the 2 approximate algorithm, the maximum distance between a node and its nearest center is no worse than twice the maximum in the optimal case. For ease of reference, we include here our summary of the 2 approximate algorithm presented in [6] ....

Michael R. Garey and David S. Johnson, Computers and Intractability, NY, NY: W.H. Freeman and Co., 1979.

....can be computed in polynomial time. Furthermore, given an excess free subset Q # , we have shown how to verify in polynomial time whether or not Q # defines some binary tree. Consequently, the problem is in NP. We use a simple reduction from the NP complete problem, DIRECTED HAMILTONIAN PATH [13]. By a caterpil lar tree we mean a binary tree in which each non leaf vertex is adjacent to at least one leaf, that is, a binary tree with exactly two pairs of twins. Given a digraph G = V,A) choose two new vertices x, y # V and construct the set of quartet trees Q = xa by : a, b) ....

M.R. Garey and D.S. Johnson, Computers and intractability, W.H. Freeman and Co. San Francisco, 1979. TREE AMALGAMATION 21

....(a) The gadget for variable x i . b) The gadget for clause C j = fx 1 ; x 2 ; x 3 g. c) Tree T built for the instance = x 1 x 2 x 3 ) x 1 x 2 x 3 ) that is, C 1 = fx 1 ; x 2 ; x 3 g and C 2 = fx 1 ; x 2 ; x 3 g. Proof. The reduction is from 3 SAT, a well known NP complete problem [8]. Consider an instance of 3 SAT, that is, a set of m clauses C 1 ; C 2 ; Cm on n variables x 1 ; x 2 ; xn , each clause with exactly three literals. Let us construct an instance of Edge Multicut: a binary tree T and a set of pairs of distinct vertices of T . The tree T is built ....

M. R. Garey and D. S. Johnson, Computers and Intractability, W. H. Freeman and Co., 1979.

....the performance metric . 10 may no longer be satisfied for below a certain number. 3.2. 2 Minimum Center The placement of a given number of centers such that the maximum distance from a node to the nearest center is minimized, known as the minimum 3 center problem, is NP complete [43]. However, if we are willing to tolerate inaccuracies within a factor of 2 (2 approximate) i.e. the maximum distance between a node and the nearest center being no worse than twice the maximum in the optimal case, the problem is solvable in O [41] In contrast to the HST ....

M.R. Garey and D.S. Johnson, Computers and Intractability, NY, NY: W.H. Freeman and Co., 1979.

....algorithm that finds a set of center nodes to minimize the maximum distance between a node and its closest center. Given this definition, the min center problem is relevant only in the case of optimization condition ( A B, The min center problem is known to be NP complete [7], however a 2 approximate algorithm exists [8] With the 2 approximate algorithm, the maximum distance between a node and its nearest center is no worse than twice the maximum in the optimal case. For ease of reference, we include here our summary of the 2 approximate algorithm presented in [6] ....

Michael R. Garey and David S. Johnson, Computers and Intractability, NY, NY: W.H. Freeman and Co., 1979.

....the performance metric may no longer be satisfied for below a certain number. B. 2 Minimum Center The placement of a given number of centers such that the maximum distance from a node to the nearest center is minimized, known as the minimum center problem, is NPcomplete [20]. However, if we are willing to tolerate inaccuracies within a factor of 2 (2 approximate) i.e. the maximum distance between a node and the nearest center being no worse than twice the maximum in the optimal case, the problem is solvable in O A, 18] In contrast to the HST ....

M.R. Garey and D.S. Johnson, Computers and Intractability, NY, NY: W.H. Freeman and Co., 1979.

....j is the set of K centers Figure 5: Two approximate algorithm for the minimum K center problem. 3.2. 2 Minimum K Center The placement of a given number of centers such that the maximum distance from a node to the nearest center is minimized, known as the minimum K center problem, is NP complete [44]. However, if we are willing to tolerate inaccuracies within a factor of 2 (2 approximate) i.e. the maximum distance between a node and the nearest center being no worse than twice the maximum in the optimal case, the problem is solvable in O(N jEj) 42] In contrast to the k HST algorithm, one ....

M.R. Garey and D.S. Johnson, Computers and Intractability, NY, NY: W.H. Freeman and Co., 1979.

....: mg with A i 1 A i 2 A i k = 0. 10 Theorem 8 Given a set F of m 3 3 matrices with rational entries and an integer K 1 m=2 the decision problem Is F K length mortal is NP hard. Proof: Via the reduction of [19] or the proof of proposition 1) and the NPcompleteness of Bounded PCP [6]. 2 Observe that [2] proves that this result remains true whenever the matrices are assumed to have entries in f0; 1g. 5.2 Mortality without repetition When repetitions of matrices are not allowed, the problem becomes clearly also decidable: a set F = fA 1 ; Am g of d d matrices is ....

....: mg such that A i 1 A i 2 A i k = 0 and i j1 6= i j2 for all j 1 6= j 2 . Theorem 9 Given a nite set F of m 2 2 matrices, and an integer K, the decision problem Is F K length mortal without repetition is NP hard in the strong sense. The proof uses a reduction from subset product [6]. We restate this problem here: Proposition 2 (Subset Product (Yao) Given a nite set A, a size s(a) 2 N for each a 2 A, and a positive integer B, the decision problem Is there a subset A 0 A such that the product of the sizes of the elements in A 0 is exactly B is NP complete in the ....

M. R. Garey and D. S. Johnson. Computers and Intractability. W. H. Freeman and Co, 1979.

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M. Garey and D. Johnson. Computers and Intractibility. Freeman and Co, 1979.

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M. R. Garey and D. S. Johnson. Computers and Intractability.W.H. Freeman and Co., New York, 1979.

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M. Garey and D. Johnson. Computers and Intractability. W.H. Freeman and Co., New York, NY, 1979.

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M. R. Garey and D. S. Johnson. Computers and Intractability. W. H. Freeman and Co., New York, 1979.

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Garey, M. R., and Johnson, D. S. Computers and Intractability. W. H. Freeman and Co., New York, 1979.

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M. Garey and D. Johnson. Computers and Intractability. W. H. Freeman and Co., New York, 1979.

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M. R. Garey and D. S. Johnson. Computers and Intractability. W. H. Freeman and Co., New York, 1979.

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M. R. Garey and D. S. Johnson, Computers and Intractability, W. H. Freeman and Co., 1979. 147

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M. Garey and D. Johnson. Computers and Intractability. W. H. Freeman and Co., San Francisco, CA, 1979.

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M.R.Garey and D.S.Johnson, Computer and Intractability, W. H. Freeman and Co., 1979.

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M. R. Garey and D. S. Johnson. Computers and Intractability. W. H. Freeman and Co., New York.

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Garey, M. R., and Johnson, D. S. Computers and Intractability. W. H. Freeman and Co., New York, 1979.

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M. R. Garey and D. S. Johnson, Computers and Intractability. New York: W. H. Freeman and Co., 1979.

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M. Garey and D. Johnson, Computers and Intractability, NY, NY: W.H. Freeman and Co., 1979.

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M. R. Garey and D. S. Johnson, Computers and Intractability. W. H. Freeman and Co., 1979.

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M. R. Garey and D. S. Johnson. Computers and Intractability.W.H. Freeman and Co., New York, 1979.

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M. R. Garey and D. S. Johnson, Computers and Intractability, W.H. Freeman and Co., NY, 1979.

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M.R. Garey and D.S. Johnson, Computers and Intractability, W.H. Freeman and Co., 1979.

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M.R. Garey and D.S. Johnson, Computers and Intractability, W. H. Freeman and Co., New York, 1979.

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M. S. Garey and D. S. Johnson. Computers and Intractability. W. H. Freeman and Co., New York, 1979.

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M. R. Garey and D. S. Johnson. Computers and Intractability. W. H. Freeman and Co., 1979.

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M. R. Garey and D. S. Johnson. Computers and Intractability. W. H. Freeman and Co., New York, 1979.

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M. R. Garey and D. S. Johnson. Computers and Intractability. W. H. Freeman and Co., New York, 1979.

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M.R. Garey and D.S. Johnson, Computers and Intractability, W.H.Freeman and Co., 1978;

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M. Garey and D. Johnson. Computers and intractability. W. H. Freeman and Co., 1979.

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M.R. Garey and D.S. Johnson, Computers and Intractability, W.H.Freeman and Co., 1978

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