By Kulikov A. S.

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Definition 27 A bipolynomial reduction, denoted by OCB, from an enumeration problem E^ towards an enumeration problem E'^ is a reduction which time complexity is bounded by a polynomial ofLength[I] and |S'ocß(/)| for any instance I of E^. Fig. 3. The world of SAfV Definition 28 An enumeration problem E is SAfV-complete if and only if E G £NV, and \/E' G SNV, 3 a ^ such that E' ocß E. The class ofSMVcomplete problems is denoted by £MVC. Henceforth, the notion of completeness in class £J\fV is defined by mean of bipolynomial reductions which are no more than a straigth adaptation of polynomial reductions to enumeration problems.

Complexity of problems and algorithms Definition 19 [Fukuda, 1996] An algorithm which solves an enumeration problem E is a bipolynomial algorithm if and only if, for any instance I, it requires a number of iterations bounded by a polynomial function of Length[I] and | 5 / | . Definition 20 [Fukuda, 1996] An enumeration problem E belongs to the class SV if there exists a bipolynomial algorithm which solves it. e. there are enumeration problems E G £V for which we cannot provide the outcome in polynomial time of only Length.

With K an input of the problem. If this decision problem is A/'T^-complete, then at least so is the optimisation problem. 1 it is possible to derive straight complexity classes for optimisation problems. This is achieved by using a generalisation of polynomial reductions: the polynomial Turing reductions. 2 Complexity of problems 39 1. e. calculates for an instance / a solution of Sj if it exists. 2. A uses a procedure S which solves the problem 0\ 3. If 5 solves the problem O' in polynomial time, then A solves O in polynomial time.

### A 2E4-time algorithm for MAX-CUT by Kulikov A. S.

by Ronald

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