We present an algorithm for linear programming which requires O(((m+n)n 2+(m+n)1.5 n)L) arithmetic operations wherem is the number of constraints, andn is the number of variables. Each operation is performed to a prec...
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We present an algorithm for linear programming which requires O(((m+n)n 2+(m+n)1.5 n)L) arithmetic operations wherem is the number of constraints, andn is the number of variables. Each operation is performed to a precision of O(L) bits.L is bounded by the number of bits in the input. The worst-case running time of the algorithm is better than that of Karmarkar's algorithm by a factor of $\sqrt {m + n} $ .
We show that every bridgeless graph of order n and minimum degree delta has a strongly connected orientation of diameter at most 11/delta+1 n+9 and such an orientation can be found in polynomialtime. (C) 2015 Elsevie...
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We show that every bridgeless graph of order n and minimum degree delta has a strongly connected orientation of diameter at most 11/delta+1 n+9 and such an orientation can be found in polynomialtime. (C) 2015 Elsevier Ltd. All rights reserved.
We study some basic algorithmic problems concerning the intersection of tropical hypersurfaces in general dimension: deciding whether this intersection is nonempty, whether it is a tropical variety, and whether it is ...
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We study some basic algorithmic problems concerning the intersection of tropical hypersurfaces in general dimension: deciding whether this intersection is nonempty, whether it is a tropical variety, and whether it is connected, as well as counting the number of connected components. We characterize the borderline between tractable and hard computations by proving NP-hardness and #P-hardness results under various strong restrictions of the input data, as well as providing polynomial time algorithms for various other restrictions. (c) 2006 Elsevier Ltd. All rights reserved.
It is well known (cf. Krajicek and Pudlak ['Propositional proof systems, the consistency of first order theories and the complexity of computations', J. Symbolic Logic 54 (1989) 1063-1079]) that a polynomial t...
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It is well known (cf. Krajicek and Pudlak ['Propositional proof systems, the consistency of first order theories and the complexity of computations', J. Symbolic Logic 54 (1989) 1063-1079]) that a polynomialtime algorithm finding tautologies hard for a propositional proof system P exists if and only if P is not optimal. Such an algorithm takes 1((k)) and outputs a tautology tau(k) of size at least k such that P is not p-bounded on the set of all formulas tau(k). We consider two more general search problems involving finding a hard formula, Cert and Find, motivated by two hypothetical situations: that one can prove that NP not equal coNP and that no optimal proof system exists. In Cert one is asked to find a witness that a given non-deterministic circuit with k inputs does not define TAUT boolean AND{0, 1}(k). In Find, given 1((k)) and a tautology alpha of size at most k(0)(c), one should output a size k tautology beta that has no size k(1)(c) P-proof from substitution instances of alpha. We will prove, assuming the existence of an exponentially hard one-way permutation, that Cert cannot be solved by a time 2(O(k)) algorithm. Using a stronger hypothesis about the proof complexity of the Nisan-Wigderson generator, we show that both problems Cert and Find are actually only partially defined for infinitely many k (that is, there are inputs corresponding to k for which the problem has no solution). The results are based on interpreting the Nisan-Wigderson generator as a proof system.
It was conjectured by Hoffmann-Ostenhof that the edge set of every cubic graph can be decomposed into a spanning tree, a matching and a family of cycles. We prove the conjecture for 3-connected cubic plane graphs and ...
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It was conjectured by Hoffmann-Ostenhof that the edge set of every cubic graph can be decomposed into a spanning tree, a matching and a family of cycles. We prove the conjecture for 3-connected cubic plane graphs and 3-connected cubic graphs on the projective plane. Our proof provides a polynomialtime algorithm to find the decomposition for 3-connected cubic plane graphs. (C) 2015 Elsevier Ltd. All rights reserved.
A transshipment problem with demands that exceed network capacity can be solved by sending flow in several waves. How can this be done in the minimum number of waves? This is the question tackled in the quickest trans...
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A transshipment problem with demands that exceed network capacity can be solved by sending flow in several waves. How can this be done in the minimum number of waves? This is the question tackled in the quickest transshipment problem. Hoppe and Tardos [Math. Oper. Res., 25 (2000), pp. 36-62] describe the only known polynomialtime algorithm to solve this problem. They actually solve the significantly harder problem in which it takes a prespecified amount of time for flow to travel from one end of an arc to the other. Their algorithm repeatedly calls an oracle for submodular function minimization. We present an algorithm that finds a quickest transshipment with a polynomial number of maximum flow computations, and a faster algorithm that also uses minimum cost flow computations. When there is only one sink, we show how the algorithm can be sped up to return a solution using O(k) maximum flow computations, where k is the number of sources. Hajek and Ogier [Networks, 14 (1984), pp. 457-487] describe an algorithm that finds a fractional solution to the single sink quickest transshipment problem on a network with n nodes and m arcs using O(n) maximum flow computations. They actually solve the universally quickest transshipment a flow over time that minimizes the amount of supply left in the network at every moment of time. In this paper, we show how to solve the universally quickest transshipment in O(mn log(n(2)/m)) time, the same asymptotic time as a push-relabel maximum flow computation.
We give an explicit family of polynomial maps called centre unstable Henon-like maps and prove that they exhibit blenders for some parameter values. Using this family, we also prove the occurrence of blenders near cer...
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We give an explicit family of polynomial maps called centre unstable Henon-like maps and prove that they exhibit blenders for some parameter values. Using this family, we also prove the occurrence of blenders near certain non-transverse heterodimensional cycles under high regularity assumptions. The proof involves a renormalization scheme along heteroclinic orbits. We also investigate the connection between the blender and the original heterodimensional cycle.
We consider the problem of identifying group centre and group median of a tree. algorithms are presented that are better than the available algorithms which solve these problems on general networks. For the group medi...
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We consider the problem of identifying group centre and group median of a tree. algorithms are presented that are better than the available algorithms which solve these problems on general networks. For the group median problem, an alternate approach is suggested which runs in linear time on specially structured problems. Properties of the set of all optimal solutions of these problems are also investigated.
We consider several two-agent scheduling problems with resource consumption on a single machine, where each of the agents wants to minimize a measure dependent on its own jobs. The starting time of each job of the fir...
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We consider several two-agent scheduling problems with resource consumption on a single machine, where each of the agents wants to minimize a measure dependent on its own jobs. The starting time of each job of the first agent is related to the amount of resource consumed. The objective is to minimize the total amount of resource consumption of the first agent with the restriction that the makespan or the total completion time of the second agent cannot exceed a given bound U. The optimal properties and the optimal polynomial time algorithms are proposed to solve the scheduling problems.
Let H = (N, E, w) be a hypergraph with a node set N = {0, 1, n - 1}, a hyperedge set E subset of 2(N), and real edge-weights w(e) for e E E. Given a convex n-gon P in the plane with vertices x(0), x(1),..., x(n-1) whi...
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Let H = (N, E, w) be a hypergraph with a node set N = {0, 1, n - 1}, a hyperedge set E subset of 2(N), and real edge-weights w(e) for e E E. Given a convex n-gon P in the plane with vertices x(0), x(1),..., x(n-1) which are arranged in this order clockwisely, let each node i E N correspond to the vertex xi and define the area A(P)(H) of H on P by the sum of the weighted areas of convex hulls for all hyperedges in H. For 0 <= i < J < k <= n - 1, a convex three-cut C(i,j, k) of N is {(i,..., j - 1}, {j,..., k - 1}, (k,..., n - 1, 0,..., i - 1)) and its size c(H)(i,j, k) in H is defined as the sum of weights of edges e is an element of E such that e contains at least one node from each of {i,..., j - 1}, {j, j - 1} and {k, n - 1, 0, i - 1}. We show that the following two conditions are equivalent: A(P)(H) <= A(P)(H') for all convex n-gons P. c(H)(i, j, k) <= c(H)'(i, J, k) for all convex three-cuts C(i, j, k). From this property, a polynomialtime algorithm for determining whether or not given weighted hypergraphs H and H' satisfy "A(P)(H) <= A(P)(H') for all convex n-gons P" is immediately obtained. (c) 2006 Elsevier B.V. All rights reserved.
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