Is Binary Encoding Appropriate for the Problem-Language Relationship?

It is proved that there exist encoding schemes which are arbitrarily as efficient as the binary encoding (in terms of compactness and arithmetic operations), with respect to which Khachiyan's algorithm for Linear Programming is exponential. This constitutes an objection to the standard translation of problems into languages via the binary encoding. When we speak about the complexity of a problem in which numbers are involved, we usually think of a formalization as a language recognition problem where the numbers are encoded in binary. Most of the people believe that the work of Khachiyan [2] has resolved the question of the complexity of linear programming. However, if we wish to be precise, Khachiyan has proven that the language of linear inequalities in binary encoding belongs to the class P. The complexity of linear programming as a problem (rather than a language) still constitutes an interesting open question. A major open question is the following: Is there an algorithm and is there a polynomial p(m, n ) such that every set of rn ~inkar inequalities in n variables can be solved by the algorithm in less than p(m, n ) arithmetic operations? We shall call such an algorithm genuinely-polynomial. Special linear programming problems for which genuinely-polynomial algorithms are known are the max-flow problem, the shortest-path problem and the assignment problem. One may argue that the distinction between polynomial and genuinely-polynomial is not essential since the amount of time required for the arithmetic operations is at least proportional to the logarithms of the numbers. More specifically, let A denote the maximal absolute value of a coefficient in a given set of rn inequalities in n variables with integral coefficients. Khachiyan's algorithm works in q(m, n, log A) time where q is a certain polynomial, whereas a genuinely-polynomial algorithm requires at least p(m, n) log A time. So, in what sense are the two notions distinct? The answer is simple. A genuinely-polynomial algorithm runs in polynomial time whenever the arithmetic operations can be carried out in polynomial time, whereas Khachiyan's 0304-3975/82/0000-0000/$02.75 @ 1982 North-Holland