G. Brightwell and D. Winkler. Counting linear extensions is #Pcomplete. In Proc. 23rd Annual ACM Symposium on Theory of Computing, pages 175-181. Association of Computing Machinery, Academic Press, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....with k or more leaves Parameter: k FPT (Downey and Fellows [53] correcting [48] also Bodlaender [13] also in LOGSPACE advice [34] Linear Extension Count Instance: A poset (P; a positive integer k. Question: Does P have at least k linear extensions Parameter: k randomized FPT ([27, 28]) Linear Inequalities Instance: A system of linear inequalities; a positive integer k. Question: Can we delete k of the inequalities and get a system that is consistent over the rationals Parameter: k W[P] complete (hardness: reduction from Weighted Monotone Circuit Satisfiability [2, 3] ....

G. Brightwell and D. Winkler. Counting linear extensions is #Pcomplete. In Proc. 23rd Annual ACM Symposium on Theory of Computing, pages 175-181. Association of Computing Machinery, Academic Press, 1991.

....of distinct endpoint realizations. On the other hand, the consistent instances for the restricted domain problem on Delta = fOE; OEg are precisely the partial orders, and Brightwell and Winkler have recently shown that computing the number of linear extensions of a partial order is #P complete [10]. This implies that computing the number of solutions (and the number of realizations) is #P complete for Delta = fOE; OEg, and thus also for A 3 and the larger algebras. 6 Conclusion In this paper we have dealt with the consistency of assertions about the relations of intervals. We have ....

G. Brightwell and P. Winkler. "Counting linear extensions is #P-complete". Proc. 23rd Annual ACM Symp. on Theory of Computing (1991) 175--181.

.... #P complete [Sta] Even counting the number of linear extensions of a poset is an open problem for some specific posets, for example, the Boolean lattice [SK87] Brightwell and Winkler have recently shown that the problem of counting the number of linear extensions of a given poset is # P complete [BW92]. On the brighter side, Pruesse and Ruskey [PR94] have found a CAT algorithm for listing linear extensions so that successive extensions differ by one or two adjacent transpositions and Canfield and Williamson [CW95] have shown 26 how to make it loop free. In [PR93] Pruesse and Ruskey consider ....

G. Brightwell and P. Winkler. Counting linear extensions is #P-complete. Order, 8:225-- 242, 1992.

....= S 2E(OE) S . An application of (1) shows easily that vol n (S ) 1=n always, so vol n (P (OE) e(OE) n , as required. It was conjectured that e(OE) was #P hard, but this issue, though of considerable interest, remained open for some years. Recently, however, Brightwell and Winkler [6] have finally settled this conjecture in the affirmative. Their proof is a little too complicated to sketch here, but their result implies, in particular, that polyhedral volume computation is strongly #P hard, even for this natural application. We will return to this application in Section 5.2 ....

G. Brightwell and P. Winkler, Counting linear extensions is #P-complete, DIMACS Technical Report 90-49, 1990.

....posets. Given a poset P , two questions naturally arise. The generation question asks whether the linear extensions, E(P) of P be efficiently generated. The counting question asks whether e(P) the size of the set E(P) can be efficiently determined. The recent result of Brightwell and Winkler [4] that the counting question is #Pcomplete indicates that the counting question may be no easier than the generation question. We give the best possible answer to the generation question in the sense that our algorithm generates E(P) in time complexity O(e(P) aside from a small amount of ....

G. Brightwell and P. Winkler, Counting linear extensions is #P-complete, Order, 8 (1992), pp. 225--242.

....to select among the M total orderings consistent with the DAG these total orderings are known as the linear extensions of the DAG. Hence, L (s) 1 = log K K(K Gamma 1) 2 Gamma log M: 4) Unfortunately, counting the number of linear extensions of a DAG is known to be an NPhard problem [3]. There are efficient means of producing an upper bound to M [12] and we are investigating its use to provide an estimate for the value of M . In the meantime, we calculate M by brute force, with the understanding that this technique will be applicable only to modestly sized causal models. 0.0 ....

Graham Brightwell and Peter Winkler. Counting linear extensions is #p-complete. In Proc. of the 23rd Annual ACM Symposium on the Theory of Computing, New Orleans, LA, 1991.

....As the number of orderings increase, the number of different final schedules increase, allowing a scheduler more opportunities to create good schedules. A difficulty with reducing the number of edges is the resultant increase in the size of the search space. Recent work by Brightwell and Winkler [BW90] has shown that determining the actual number of total orders in a dag, given a partial ordering, is #P complete. That is, as Garey and Johnson state [GJ79] the problem is at least as hard as finding all the Hamiltonian circuits that exist in a graph. #P complete enumeration problems are ....

Graham Brightwell and Peter Winkler. "Counting linear extensions is #p-complete". DIMACS Technical Report 90-49, Bellcore, 445 South Street, Morristown, New Jersey, 07960, July 1990.

..... q n s . By the Aleksandrov Fenchel inequality (applied to in the case s = 1) it follows that N(i) 2 # N(i 1)N(i 1) for i = 1, n 1 and, hence, the sequence N(1) N(n) is unimodal. Observe that the evaluation of N(i 1 , i 2 . i r ) is #P complete even when s = 0, [BW92]; in this case, N is the number of linear extensions of the poset. It follows that computing the volume of H polytopes is #P hard in the strong sense. 4.5. An application of mixed volumes in algebraic geometry. Let S 1 , S 2 , S n be subsets of Z n , and consider a system F = f 1 , ....

G. BRIGHTWELL AND P. WINKLER, Counting linear extensions is #P -complete, Order, 8 (1992), pp. 225--242.

....by a National Defense Science and Engineering Graduate Fellowship. explanation for the apparent difficulty of enumeration, counting has held an important place in theoretical computer science. Although several researchers continued Valiant s work by adding to the list of #P complete problems [PB83, DF89, BW90, Rot96, Gol93, BD97, MW92, Lin86, JVW90], our understanding of the complexity of counting still pales in comparison to our understanding of decision problems. This is unfortunate, for counting, aside from being mathematically interesting, is closely related to important practical problems. For instance, reliability problems are often ....

Graham Brightwell and Peter Winkler. Counting linear extensions is #P-complete. Technical Report 90--49, DIMACS, July 1990.

.... and 1 represents unordered service: m(PO) log e(PO) log N (1) Using a metric based on e(PO) presents some difficulties since computing e(PO) for an arbitrary partial order is #P Gamma Complete, and it is therefore highly unlikely that any polynomial algorithm exists for this computation [3, 4]. 3.2 A metric based on precedence constraints: density The density of a PO is defined as follows [9] Let a partial order PO be represented as a transitively closed 0 1 matrix of size N by N , where a i;j = 1 iff i OE j in PO. In this representation, a i;i = 0, for all i. Let D be defined as ....

G. Brightwell and P. Winkler. Counting Linear Extensions is #P-Complete. In Proceeedings of the 23rd ACM Symposium on the Theory of Computing, 175--181, 1991.

....only one machine, the calculation of all the possible permutations is prohibitively expensive. For this reason, sequencers are usually heuristically driven, producing a sequence that is reasoned to be superior to the others possible. Scheduling is also a difficult problem. Brightwell and Winkler [BW90] proved that producing all of the total orders from a given partial order is #P complete, showing that exhaustive searching by a scheduler for an optimal answer is also not advisable. For this reason, schedulers are often deterministic such that given any one sequence of jobs, the scheduler will ....

Graham Brightwell and Peter Winkler. "Counting linear extensions is #p-complete". DIMACS Technical Report 90-49, Bellcore, 445 South Street, Morristown, New Jersey, 07960, July 1990.

....as the probability that two randomly chosen elements are ranked differently by two randomly chosen linear extensions. Computing this probability is closely related to the problem of computing the number of linear extensions of a finite partial order. That problem is known to be #P complete in (Brightwell Winkler 1991). We use the approximation algorithm of (R.Bubley M.Dyer 1998) which reduces the problem of counting linear extensions to one of sampling and uses a Markov chain algorithm to do the sampling. 3 http: www.rulequest.com Determining the User s Initial Preferences As mentioned in Section 1, we ....

Brightwell, G., and Winkler, P. 1991. Counting linear extensions is #p-complete. In Proceedings of the twenty-third annual ACM symposium on Theory of computing, 175--181.

....no losses are permitted Answering this question allows one to quantify and compare two or more R PO services. Unfortunately there is currently no known formula for calculating e(P ) for an arbitrary partial order. Recently it has been shown that the problem of computing e(P ) is #P complete 2 [9]. There is an O(N 5 ) algorithm, where N is the number of objects, for computing e(P ) for partial orders that form a tree when all edges are considered undirected [7] Similarly, there is an O(N 8 ) algorithm for computing e(P ) for any graph (and therefore for any partial order) where if the ....

G. Brightwell and P. Winkler. Counting linear extensions is #P-complete. In Proc 23rd ACM Symp on the Theory of Computing, 175--181, 1991.

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