We consider polynomials f(x1, & mldr;, xn) over a finite filed that satisfy the following condition: the set of solutions of the equation f(x1, & mldr;, xn) = b, where b is some element of the field, coincides...
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We consider polynomials f(x1, & mldr;, xn) over a finite filed that satisfy the following condition: the set of solutions of the equation f(x1, & mldr;, xn) = b, where b is some element of the field, coincides with the set of solutions of some system of linear equations over this field. Such polynomials are said to be multiaffine with the right-hand side b (or with respect to b). We describe a number of properties of multiaffine polynomials. Then on the basis of these properties we propose a polynomial algorithm that takes a polynomial over a finite field and an element of the field as an input and decides whether the polynomial is multiaffine with respect to this element. In case of the positive answer the algorithm also outputs a system of linear equations that corresponds to this polynomial. The complexity of the proposed algorithm is the smallest in comparison with other known algorithms that solve this problem.
We give a transparent combinatorial characterization of the identities satisfied by the Kauffman monoid Our characterization leads to a polynomial time algorithm to check whether a given identity holds in K-3.
We give a transparent combinatorial characterization of the identities satisfied by the Kauffman monoid Our characterization leads to a polynomial time algorithm to check whether a given identity holds in K-3.
A class of polynomially solvable problems of combinatorial optimization is illustrated by the example of the traveling salesman problem. It is proved that this class includes problems for which the structure of initia...
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A class of polynomially solvable problems of combinatorial optimization is illustrated by the example of the traveling salesman problem. It is proved that this class includes problems for which the structure of initial data is specially modeled.
It is shown that there does not exist a polynomial algorithm to derive the optimal solution of a set cover problem that differs from the original problem in one position of the constraint matrix if the optimal solutio...
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It is shown that there does not exist a polynomial algorithm to derive the optimal solution of a set cover problem that differs from the original problem in one position of the constraint matrix if the optimal solution of the original problem is known and P not equal NP. A similar result holds for the knapsack problem.
Let xi := (xi(k))(k is an element of Z) be i.i.d. with P(xi(k) = 0) = P(xi(k) = 1) = 1/2, and let S := (S-k)(k is an element of N0) be a symmetric random walk with holding on Z, independent of. We consider the scenery...
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Let xi := (xi(k))(k is an element of Z) be i.i.d. with P(xi(k) = 0) = P(xi(k) = 1) = 1/2, and let S := (S-k)(k is an element of N0) be a symmetric random walk with holding on Z, independent of. We consider the scenery observed along the random walk path S, namely, the process (X-k := xi s(k))(k is an element of N0) With high probability, we reconstruct the color and the length of block", a block in of length >= it close to the origin, given only the observations (X-k)(k is an element of[0,2.33n]). We find stopping times that stop the random walker with high probability at particular places of the scenery, namely on block" and in the interval [-3(n), 3(n)]. Moreover, we reconstruct with high probability a piece of of length of the order 3,0,2 around block", given only 3[n(0.3)] observations collected by the random walker starting on the boundary of block". (c) 2005 Wiley Periodicals, Inc.
We prove a necessary condition for polynomial solvability of the jump number problem in classes of bipartite graphs characterized by a finite set of forbidden induced bipartite subgraphs. For some classes satisfying t...
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We prove a necessary condition for polynomial solvability of the jump number problem in classes of bipartite graphs characterized by a finite set of forbidden induced bipartite subgraphs. For some classes satisfying this condition, we propose polynomial algorithms to solve the jump number problem.
The vehicle routing with pickups and deliveries (VRPD) problem is defined over a graph G=(V,E). Some vertices in G represent delivery customers who expect deliveries from a depot, and other vertices in G represent pic...
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The vehicle routing with pickups and deliveries (VRPD) problem is defined over a graph G=(V,E). Some vertices in G represent delivery customers who expect deliveries from a depot, and other vertices in G represent pick-up customers who have available supply to be picked up and transported to a depot. The objective is to find a minimum length tour for a capacitated vehicle, which starts at a depot and travels in G while satisfying all the requests by the delivery and pickup customers, without violating the vehicle capacity constraint, and returns to a depot. We study the VRPD problem on some special graphs, including trees, cycles and warehouse graphs when the depots are both exogenously and endogenously determined. Specifically, we develop linear time algorithms for the VRPD problem on tree graphs and polynomial algorithms on cycle and warehouse graphs. (C) 2002 Elsevier Science B.V. All rights reserved.
In this paper, a generalized formulation of a classical single machine scheduling problem is considered. A set of n jobs characterized by their release dates, deadlines and a start time-dependent processing time funct...
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We study remoteness function R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlengt...
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We study remoteness function R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}$$\end{document} of impartial games introduced by Smith in 1966. The player who moves from a position x can win if and only if R(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}(x)$$\end{document} is odd. The odd values of R(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}(x)$$\end{document} show how soon the winner can win, while even values show how long the loser can resist, provided both players play optimally. This function can be applied to the conjunctive compounds of impartial games, in the same way as the Sprague-Grundy function is applicable to their disjunctive compounds. We provide polynomial algorithms computing R(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}(x)$$\end{document} for games Euclid and generalized Wythoff. For Moore's NIM we give a simple explicit formula for R(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {R}}(x)$$\end{document} if it is even and show that computing it becomes an NP-hard probl
Given a set H of binary vectors of length n, is there a cyclic listing of H so that every two successive vectors differ in a single coordinate? The problem of the existence of such a listing, which is called a cyclic ...
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Given a set H of binary vectors of length n, is there a cyclic listing of H so that every two successive vectors differ in a single coordinate? The problem of the existence of such a listing, which is called a cyclic Gray code of H, is known to be NP-complete in general. The goal of this paper is therefore to specify boundaries between its intractability and polynomial decidability. For that purpose, we consider a restriction when the vectors of H are of a bounded weight. A weight of a vector u is an element of {0, 1}(n) is the number of l's in u. We show that if every vertex of H has weight k or k + 1, our problem is decidable in polynomial time for k <= 1 and NP-complete for k >= 2. Furthermore, if k = 2 and for every i is an element of [n] there are at most m vectors of H of weight two having one in the i-th coordinate, then the problem becomes decidable in polynomial time for m <= 3 and NP-complete for m >= 13. The following complementary problem is also known to be NP-hard: given an F subset of (0, 1)(n), which now plays the role of a set of faults to be avoided, is there a cyclic Gray code of {0, 1}(n) \ F? We show that if every vertex of F has weight at most k, the problem is decidable in polynomial time for k <= 2 and NP-hard for k >= 5. It follows that there is a function f(n) = Theta(n(4)) such that the existence of a cyclic Gray code of {0, 1}(n) \ F for a given set F subset of {0, 1}(n) of size at most f(n) is NP-hard. In addition, we study the cases when the Gray code does not have to be cyclic, and moreover, when the first and the last vectors of the code are prescribed. For these two modifications, all NP-hardness and NP-completeness results hold as well. (C) 2011 Elsevier B.V. All rights reserved.
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