A polynomial relaxation-type algorithm for linear programming

Transcription

1 A polynomial relaxation-type algorithm for linear programming Sergei Chubanov Institute of Information Systems at the University of Siegen, Germany 6th February 20 Abstract The paper proposes a polynomial algorithm for solving systems of linear inequalities. The algorithm uses a polynomial relaxation-type procedure which either finds a solution for Ax = b, 0 x, or decides that the system has no integer solutions. Polynomial algorithm The polynomial algorithm described in [2] either solves a linear system Ax = b, 0 x, or decides that it has no 0,-solutions. Here, 0 and denote all-zero and all-one vectors, respectively. The core of the algorithm is a divide-and-conquer procedure which may be considered as a generalization of the relaxation method (see [] and [5]). This short paper describes a way of converting the mentioned polynomial algorithm, which is further called the LP algorithm, into a polynomial algorithm for solving linear systems. This approach substantially differs from well known polynomial algorithms like the ellipsoid method by Khachiyan [4] and the projective algorithm by Karmarkar [3].

2 Consider a linear system Ax = b, 0 x u, (.) where all coefficients are integer. Without loss of generality we assume that u j > 0 for all j =,..., n. Let us introduce additional variables ξ and x = ( x,..., x n ) T and consider the system Ax bξ = b, x + x uξ = u, 0 x ( + T n )u, 0 x ( + T n )u, 0 ξ T n, (.2) where T is n! times absolute value of the product of all components of A, b, and u. The size of T is polynomially bounded. Lemma. The system (.2) has an integer solution if and only if (.) is feasible. Proof. Indeed, if (x, x, ξ ) is an integer solution of (.2), then +ξ x is feasible for (.). Now let us assume that the system (.) is feasible. Then (.) defines a bounded polyhedron. Consider a vertex x # of this polyhedron. Since all coefficients of the system are integer, Cramer s theorem implies that x # is representable as x # = ( x,..., x n n ) T where x j and j are integers bounded by T. Let ξ = n l= x l. The ξ is a nonnegative integer bounded by T n. Let us set and x := ( + ξ )x # x := ( + ξ )u x. We have Ax = ( + ξ )b because Ax # = b. The vector x is nonnegative because ξ 0 and x # 0. Moreover, it is integer because ( + ξ ) x # j = n l= l x j 2 j = n l j l x j.

3 We also have x (+T n )u because x # u and ξ T n. The vector x is integer because x and u are integer vectors and ξ is integer. The vector x satisfies x = ( + ξ )(u x # ) 0, because x # u, and x ( + T n )u because x 0. It follows that (x, x, ξ ) is an integer solution of (.2). Let (.2) be written in the form A x = b, 0 x u. (.3) Recalling that s 2 s = 2 l +, and using the binary representation of u j, we derive u j = σ js 2 s = ( s ) σ js 2 l +. Here, σ js {0, } are the digits in the binary representation of u j. (We assume that equals zero.) Let us denote the elements of A by a ij and consider another system whose size is polynomial in the size of (.3): n j= a ij ( log u j σ js ( s 2l z jsl + z jss ) ) = b i, i =,..., m, 0 z. (.4) Lemma.2 The system (.4) has an integer solution if and only if (.3) has an integer solution. Proof. If (.4) has an integer solution z, we may set ( s ) σ js 2 l zjsl + zjss (.5) for every j =,..., n. Using the fact that the components of z are nonnegative and not greater than one, we get 0 x j ( s ) σ js 2 l + = u j. 3

4 At the same time, A x = b due to (.4). It follows that the obtained solution x is feasible for (.3). Let x be an integer solution of (.3). Since x j u j, we may represent x j in the form σ js2 s where σ js {0, } are the digits in the binary representation of x j. If x j = u j, we put z jsl = for all s and l. Otherwise, since x j < u j, there exists s j such that σ js j = 0 and σ js j =, and the digits σ js with s > s j in the binary representation of x j are equal to the corresponding digits σ js in the binary representation of u j. Then s=s j + σ js2 s + = s=s j + σ js s j ( s s j σjl2 l = σ js 2 s + (2 l σjl + 0) s=s j + ) s j 2 l + + σ js j (2 l σjl + 0). According to the first sum in the latter expression, we set z jss := and z jsl := for s > s j and l = 0,..., s. According to the second sum we set z js j l := σ jl for all l = 0,..., s j. For the remaining components we set zjsl := 0. The {0, }-solution z obtained in the above way satisfies (.5) for all j. Since A x = b, it follows that z is feasible for (.4). Thus, we get the following procedure: Given the system (.), transform it to the corresponding system (.2). Write this system as (.3). Then transform (.3) to the corresponding system (.4). Let (.4) be written as A = b, 0. (.6) Apply the LP algorithm to (.6). If the algorithm decides that (.6) has no integer solution, then (.3) also has no integer solution and we decide that the original system (.) is infeasible. Otherwise, let be a solution of (.6). Of course, it must not necessarily be integer. Then we translate to a solution x of (.). 4

5 Theorem. The above procedure gives a polynomial algorithm for solving systems of inequalities of the form Ax = b, 0 x u, where all coefficients are integer. Proof. By Lemma.2, the system (.6) has an integer solution if and only if (.3) has an integer solution. By Lemma., the system (.3) has an integer solution if and only if (.) is feasible. If the LP algorithm delivers a solution for (.6), then it can be translated to a solution x of (.3) in polynomial time, by the procedure which obtains x from z by the formula (.5) in the proof of Lemma.2, because the sizes of the systems are polynomially bounded in the size of the original system (.). It is proved in [2] that the LP algorithm runs in polynomial time. The solution x can in turn be translated to a solution of (.) in polynomial time. To see this, it is sufficient to note that if x = (x, x, ξ ), where (x, x, ξ ) satisfies (.2), then +ξ x is feasible for (.). If the LP algorithm decides that (.6) has no integer solution, then (.) is infeasible by Lemmas.2 and.. References [] Agmon, Sh The relaxation method for linear inequalities. Canad. J. Math. 6, [2] Chubanov, S A strongly polynomial algorithm for linear systems having a binary solution. Working paper. [3] Karmarkar, N A new polynomial-time algorithm for linear programming. Combinatorica 4, [4] Khachiyan, L.G A polynomial algorithm in linear programming. Dokl. Akad. Nauk SSSR 244 (English translation: Soviet Math. Dokl. 20, 9-94). [5] Motzkin, Th., and Schoenberg, I. J The relaxation method for linear inequalities. Canad. J. Math