Edmonds polytopes and a hierarchy of combinatorial problems

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 aj xj ≤ a0, where a1, a2, . . . , an are integers, by the inequality aj xj ≤ a with a ≥ [a0]. Obviously, if integers x1, x2, . . . , xn satisfy all the inequalities in S, then they satisfy also all inequalities in the closure of S. Conversely, let cj xj ≤ c0 hold for all choices of integers x1, x2, . . . , xn, that satisfy all the inequalities in S. Then we prove that cj xj ≤ c0 belongs to the closure of S. To each integer linear programming problem, we assign a nonnegative integer, called its rank. (The rank is the minimum number of iterations of the operation (ii) that are required in order to eliminate the integrality constraint.) We prove that there is no upper bound on the rank of problems arising from the search for largest independent sets in graphs. © 1973 Published by Elsevier B.V. 1. Characterizations and good characterizations Let us examine the formal structure of the following theorems. Theorem 1.1 (Tutte [17]). Let G be a (finite undirected) graph. Then the two following conditions are equivalent. (i) G has a perfect matching (that is, a set of pairwise disjoint edges that cover all the vertices of G), (ii) if an arbitrary set S of vertices is deleted fromG, then the number k0(G\S) of odd components (that is, components having an odd number of vertices each) of the resulting graph G\S does not exceed |S|. Theorem 1.2 (Gallai [11]). 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 two. Theorem 1.1 gives necessary and sufficient conditions for the existence of a certain structure (perfect matching inG) in terms of the absence of another structure (a set Swith k0(G\S)> |S|). On the other DOI of original article: 10.1016/0012-365X(73)90167-2 The original article was published in Discrete Mathematics 4 (1973) 305–337 0012-365X/$ see front matter © 1973 Published by Elsevier B.V. doi:10.1016/j.disc.2006.03.009 V. Chvátal /Discrete Mathematics 306 (2006) 886–904 887 hand, Theorem 1.2 gives necessary and 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 the help of Theorem 1.1) to convince the supervisor that G has no perfect matching— one has to exhibit a set Swith k0(G\S)> |S|. On the other hand, while it is easy to convice the supervisor thatG has a k-coloring, Theorem 1.2 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 charge 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 cixi subject to a set of constraints (primal problem) as a minimum of another form biyi subject to other constraints (dual problem). Hence to show that a feasible primal solution (x1, x2, . . . , xn) is optimal, one only has to exhibit a feasible dual solution (y1, y2, . . . , ym) with cixi = biyi . 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 1.1 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. 2. Edmonds polytopes Let G be a graph with vertices v1, v2, . . . , vm and edges e1, e2, . . . , en; for each j = 1, 2, . . . , m we set S(j)= {i : vj is an endpoint of ei}. The problem of finding a perfect matching in G can be formulated as the following integer linear programming problem: Maximize