Patricia Carstensen. Complexity of some parametric integer and network programming problems. In Mathematical Programming, volume 26, pages 64--75, 1983. 85 86  Home/Search   Document Not in Database   Summary   Related Articles   Check

....problem fast in parallel is optimal. The result holds under fairly stringent restrictions on the input, and can be 3 extended to randomized algorithms, and also to the case where the input graph is sparse. The proof is a substantial simplification and careful reworking of a proof of Carstensen [7, 8] which takes into account the size of the coe#cients that arise. The lower bound also extends to a weak lower bound on the problem of computing the maximum weight perfect matching in a bipartite graph. This was the problem that we had originally set out to study because it has no fast parallel ....

....and the precise statement of the lower bound can be found in Section 4.3. The main technical lemma is stated in Section 4.3.2. The inductive construction can be found in Section 4.4. Proofs are relegated to the end of the chapter in Section 4.5. The proof is based on a theorem of Carstensen [8, 7]. However, Carstensen s proof is very complex and does not take into account the issue of bit lengths. It is not possible to obtain a lower bound that is sensitive to bit lengths without obtaining good bounds on the bit lengths of the coe#cients of the edge weights. We give a simplified proof of ....

Patricia Carstensen. Complexity of some parametric integer and network programming problems. In Mathematical Programming, volume 26, pages 64--75, 1983. 85 86

....and #(n) the bit size of the parameters. Then the decision version of the problem cannot be solved in the PRAM model without bit operations in o p log #(n,#(n) time using 2 # log #(n,#(n) processors, even if we restrict every numeric parameter in the input to size O(#(n) Carstensen [4, 3] proved the following theorem: Theorem 1.2 (Carstensen) There is an explicit family of graphs Gn on n vertices with edge weights that are linear functions in a parameter #, such that the optimal cost graph of the shortest path between s and t has 2# (log 2 n) breakpoints. However, ....

Patricia Carstensen. Complexity of some parametric integer and network programming problems. In Mathematical Programming, volume 26, pages 64--75, 1983.

....[10] and transversality techniques [16] in conjunction with diophontine techniques over integer lattices. We also need lower bounds for the so called parametric complexity (section 3) of the mincost flow and max flow problems; these were proved earlier in some other contexts combinatorially [38, 14, 57, 7] (see also Theorem 3.8) Our proof should be contrasted with the combinatorial proofs of the lower bounds for, say, constant depth [12, 19, 52] or monotone circuits [42, 1, 5, 56] and also with the proofs of the lower bounds in the algebraic computation tree model [2, 46, 53] these are for ....

....result (Theorem 3.3) which says that if the so called parametric complexity of the problem is high, then it is hard to parallelize in our model. This implies our lower bounds for the mincost flow and max flow problems because their parametric complexity is exponential, as was shown by Carstensen [7, 8] and Zadeh [57] Our proof has two steps. In the first step we show that if the problem with high parametric complexity had a fast parallel algorithm in our model, then there would exist a certain low degree algebraic decomposition of a small subset of the three dimensional integer lattice. In the ....

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<F3.758e+05> P.<F3.811e+05> Carstensen,<F3.81e+05> Complexity of some parametric integer and network programming<F3.811e+05> problems, Math. Programming, 26 (1983), pp. 64--75.

....stated above. Related Work. Several researchers have considered parametric versions of combinatorial optimization problems. Murty [Mur80] showed that the number of breakpoints for parametric linear programming problems can be exponential in the number of variables. Subsequently, Carstensen [Car83] proved exponential lower bounds for, among other problems, parametric minimum cut and knapsack. Van Hoesel et al. vHKRW89] have compiled an extensive bibliography on parametric computing. The algorithmic approach followed in this paper was first used in [FeSl89] to analyze various optimization ....

....S v will have no breakpoints in (L ; 1) Once the output element v of B is resolved, we return 1 = L . 2 The restriction to graphs of bounded tree width seems to be important in achieving our bounds. Without it, some problems do indeed have an exponential number of breakpoints in the worst case [Car83]. However, the bound of Theorem 4.2 can probably be sharpened considerably in certain special cases. A natural candidate for further study is the maximum independent set problem. Improving the running times of the algorithms for the search problems described in Section 5 is another intriguing ....

P. Carstensen. Complexity of some parametric integer and network programming problems. Math. Programming, 26:64--75, 1983.

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