EDMONDS POLYHEDRA AND A HIERARCHY OF COMBINATORIAL PROBLEMS bY

Let S be a set of linear inequalities that determine a bounded polyhedron P. The closure of S is the smallest set of inequalities that contains S and is closed under two operations: (i) taking linear combinations of inequalities, (ii) replacing an inequality c aj xj 2 ao, where al, a2, . . . , an are integers, by the inequality c a. x < a with a 1 [a,]. Obviously, if J jintegers Xl' 5’ l *a 9 ⌧n satisfy all the inequalities in S then they satisfy also all the inequalities in the closure of S. Conversely, let g c of odd components (that is, components having an odd number of vertices each) of the resulting graph G does not exceed IS;, THEOREM B (Gallai [II]). Let G be a (finite undirected) graph. Then the two following conditions are equivalent. (i> G is k-colorable, (ii) the edges of G can be directed in such a way that the resulting directed graph contains no (simple directed) path having k edges. Both of these theorems, asserting the equivalence of (i) and (ii), are characterizations? Yet there is a considerable formal difference between the twoc Theorem A gives necessary and sufficient canditions for the existence of a certain structure (perfect matching in G) in terms of the absence of another structure (a set S with ko(G-S) > jsi)n On the other hand, Theorem B gives necessary and 2 sufficient conditions for the existence of a certain structure (k-coloring of G) in terms of the existence of another structure (the directions of the edges of G). Another aspect of this difference can be illuminated as follows. It is easy to convince one's supervisor that G has a perfect matching. To do this, one only has to exhibit the matching. (The question of the difficulty of finding the matching is irrelevant for our discussion.) It is equally easy (with help of Theorem A) to convince the supervisor that G has no perfect matching -one has to exhibit a set S with kO(G-S) > ISI. On the other hand, while it is easy to convince the supervisor that G has a k-coloring, Theorem B gives no easy way of showing that G has no k-coloring. Apparently Edmonds [6] has been the first to turn attention to this feature of characterizations; he introduced the term "good characterizations" for the theorems of the first type. Hence Tutte's theorem is a good characterization while Gallai's theorem is not. Needless to say, the words "good characterization" form a nonseparable entity without any reference to the emotional cha.rge of the adjective "good". The statement "Gallai's theorem is not a good characterization" asserts nothing whatsoever about the quality and depth of the theorem. In our further considerations, the duality theorem of linear programming will play an important role, It expresses the maximum of a linear form C ci xi subject to a set of constraints (primal problem) as a minimum of another form c bi yi subject to other constraints (dual problem). Hence to show that a feasible primal solution (x1' 3’ l *a 9 Xn) is optimal, one only has to exhibit a feasible dual solution (y,, y2, . . . , Ym> with c ci xi = c bi yi. In a way, the duality theorem of linear programming is a prototype of a good characterization. Our last sentence has more into it than meets the eye. Actually, Edmonds [7] has shown how to relate Theorem A to the duality theorem and made it clear that his approach can be adopted in many different settings. It is the purpose of this paper to study various questions related to Edmonds' technique.