Feasible combinatorial matrix theory Polytime proofs for König’s Min-Max and related theorems

We show that the well-known König’s Min-Max Theorem (KMM), a fundamental result in combinatorial matrix theory, can be proven in the first order theory LA with induction restricted to Σ1 formulas. This is an improvement over the standard textbook proof of KMM which requires Π2 induction, and hence does not yield feasible proofs — while our new approach does. LA is a weak theory that essentially captures the ring properties of matrices; however, equipped with Σ1 induction LA is capable of proving KMM, and a host of other combinatorial properties such as Menger’s, Hall’s and Dilworth’s Theorems. Therefore, our result formalizes Min-Max type of reasoning within a feasible framework.