By Satoru Iwata, Satoko Moriguchi, Kazuo Murota

This paper provides a swifter set of rules for the M-convex submodular How challenge, that is a generalization of the minimum-cost How challenge with an M-convex fee functionality for the How-boundary, the place an M-convex functionality is a nonlinear nonseparable cliserete convex functionality on integer issues. The set of rules extends the potential sealing strategy lor the submodular How challenge by way of Fleischer. Iwata and MeCormiek (2002) by means of a singular means of altering the aptitude through fixing greatest submodular How difficulties.

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35. 0 Let M be a reduction of a decision problem PI to a decision problem P 2' If M runs in time f(n), we say that M is an f(n) time reduction of PI to P 2 and that PI reduces in time 1(n) to P 2 (or is f(n) time reducible to P 2)' Anf(n) time reduction is a polynomial time reduction if f(n) is O(nk) for some constant k. For example, the reduction of P h • 1t to Paccept given at the end of the previous section is obviously a polynomial time reduction. Polynomial time reductions play an important role in the classification of solvable decision problems.

The size of a rewriting system G = (V, P), denoted by IG I, is defined as the sum of the lengths of the rules in P, or the size of V, whichever is larger. In other words, The norm of G, denoted by IIGII, is defined by IIGII = IG I log IV I . 45 Any rewriting system G=(V, P) can be encoded uniquely as a binary string of length 0 (II GI ). Proof Let # be a symbol not found in V. V can then be represented as the string # rx #, where rx contains exactly one occurrence of each symbol in V. Any rule W 1 -+W2 in P can be represented as the string #Wl #W2.

44, Pen) is true for all n. Note that in this case it was not necessary to assume in the induction hypothesis that all of the statements P(O), ... , P(n-l) are true: assuming only the truth of Pen -1) would have been sufficient. 44 replaced by (1') For all n>O, P(n-l) implies Pen) . However, in many cases it is harder or even impossible to formulate the claim to be proved in such a way that this form of induction can be used. 44. We now proceed to prove, again by induction, the inclusion In this case the statement Pen) takes the form Pen): "For all strings ,)" S=>" }' implies }' =O"Sl" or ,),=0"-11"-1".