Combinatorial Optimization

The front matter includes the title page, copyright page, dedication, TOC, list of figures, list of tables, preface, and Notation, Conventions, Abbreviations

A clutter C is a pair (V, E), where V is a finite set and E is a family of subsets of V none of which is included in another. The elements of V are the vertices of C and those of E are the edges. For example, a simple graph (V, E) (no multiple edges or loops) is a clutter. We refer to West [208] for definitions in graph theory. In a clutter, a matching is a set of pairwise disjoint edges. A transversal is a set of vertices that intersects all the edges. A clutter is said to pack if the maximum cardinality of a matching equals the minimum cardinality of a transversal. This terminology is due to Seymour [183]. Many min-max theorems in graph theory can be rephrased by saying that a clutter packs. We give three examples. The first is Känig's theorem.

Consider a connected graph G with nonnegative edge weights w_e for e ∊ E(G). The Chinese postman problem consists in finding a minimum weight closed walk going through each edge at least once (the edges of the graph represent streets where mail must be delivered and w_e is the length of the street). Equivalently, the postman must find a minimum weight set of edges J ⊆ E(G) such that J ∪ E(G) induces an Eulerian graph; i.e., J induces a graph the odd degree nodes of which coincide with the odd degree nodes of G. Since w ≥ 0, we can assume w.l.o.g. that J is acyclic. Such an edge set J is called a postman set.

The problem is generalized as follows. Let G be a graph and T a node set of G of even cardinality. An edge set J of G is called a T-join if it induces an acyclic graph the odd degree nodes of which coincide with T. For disjoint node sets S₁, S₂, let (S₁, S₂) denote the set of edges with one endnode in S₁ and the other in S₂. A T-cut is a minimal edge set of the form (S, V (G)-S), where S is a set of nodes with ∣T ∩ S∣ odd. Clearly every T-cut intersects every T-join.

Edmonds and Johnson [82] considered the problem of finding a minimum weight T-join. One way to solve this problem is to reduce it to the perfect matching problem in a complete graph K_p, where p = ∣T∣. Namely, compute the lengths of shortest paths in G between all pairs of nodes in T, use these values as edge weights in K_p, and find a minimum weight perfect matching in K_p. The union of the corresponding paths in G is a minimum weight T-join. There is another way to solve the minimum weight T-join problem: Edmonds and Johnson gave a direct primal-dual algorithm and, as a by-product, obtained that the clutter of T-cuts is ideal.

Chapter 1 discussed the min-max equation (1.1)=(1.2). In this chapter, we consider the max-min equation max⁡ {wx:x≥0,Mx≤1}=min{y1:y≥0,yM≥w}⁡.

A 0,1 matrix M with no column of 0's is perfect if the polytope P = {x ≥ 0 : Mx ≤ 1} is integral; i.e., all the extreme points of P are 0,1 vectors. When M is perfect, the linear program max{wx : x ≥ 0, Mx ≤ 1} has an integral optimal solution x for all w ∊ Rⁿ. Therefore, the set packing problem max{wx : Mx ≤ 1, x ∊ {0,1}ⁿ} is solvable in polynomial time. By contrast, for a general 0,1 matrix M, the set packing problem is NP-hard [94].

Edmonds and Giles [81] observed that, when a linear system Ax ≤ b, x ≥ 0 is TDI and b is integral, the polyhedron {x : Ax ≤ b, x ≥ 0} is integral. The converse is not true in general. But it is true when A is a 0,1 matrix and b = 1, as shown by Lovász.

A 0,1 clutter matrix A is ideal if the polyhedron Q(A) ≡ {x ≥ 0 : Ax ≥ 1} is integral. Ideal matrices give rise to set covering problems that can be solved as linear programs for all objective functions.

This concept was introduced by Lehman under the name width—length property. Lehman [132] showed that ideal 0,1 matrices always come in pairs (Theorem 1.17: A is ideal if and only if its blocker b(A) is ideal) and that the width—length inequality is in fact a characterization of idealness (Theorem 1.21). Another important result of Lehman about ideal 0,1 matrices is the following.

In this chapter, we consider the clutter C of odd cycles in a graph G. Seymour [183] characterized exactly the graphs for which C has the MFMC property and Guenin [110] characterized exactly when C is ideal.

For edge weights w ∊ R₊^E(G), consider the minimization problem (1.1). Recall that an integral solution to (1.1) is the incidence vector of a transversal T of C. Since T intersects all odd cycles, E(G) - T induces a bipartite graph. Therefore, a minimal transversal T of C is the complement of a cut (W, W¯ ). In particular, when C is ideal, (1.1) finds a cut of maximum weight in G; i.e., (1.1) solves the famous max cut problem.

The concepts of perfect and of ideal 0,1 matrices can be extended to 0,±1 matrices. Given a 0,±1 matrix A, denote by n(A) the column vector whose ith component is the number of -1's in the ith row of matrix A. A 0,±1 matrix A is perfect if the polytope {x : Ax ≤ 1 - n(A), 0 ≤ x ≤ 1} is integral. Similarly, a 0,±1 matrix A is ideal if the polytope {x : Ax ≥ 1 - n(A), 0 ≤ x ≤ 1} is integral. A matrix is totally unimodular if every square submatrix has determinant equal to 0,±1. In particular, all entries are 0,±1. A milestone result in the study of integral polyhedra, due to Hoffman and Kruskal [122], is that the following statements are equivalent for an integral matrix A.

• The polyhedron {x ≥ 0 : Ax ≤ b} is integral for each integral vector b,

• A is totally unimodular.

We prove this in section 6.1. It follows from this result that a totally unimodular matrix is both perfect and ideal.

A 0,±1 matrix is balanced if, in every square submatrix with exactly two nonzero entries per row and per column, the sum of the entries is a multiple of four. The class of balanced 0,±1 matrices properly includes totally unimodular 0,±1 matrices.

A 0,1 matrix is regular if its nonzero entries can be signed +1 or -1 so that the resulting matrix is totally unimodular. A result of Camion [27] shows that this signing is unique up to changing signs in rows or columns. So the recognition of totally unimodular matrices reduces to that of regular matrices. Similarly, a 0,1 matrix is balanceable if its nonzero entries can be signed +1 or -1 so that the resulting 0,±1 matrix is balanced. In this lecture, we present Tutte's characterization [202] of regular matrices and Truemper's characterization [192] of balanceable matrices.

In this chapter, we discuss Seymour's decomposition theorem for regular matrices [186], the Tseng—Truemper decomposition theorem for binary clutters with the MFMC property [198], and results of Novick and Sebö [150] and Cornuéjols and Guenin [61] on ideal binary clutters. We adopt a matroidal point of view. See Oxley's excellent textbook [153] on matroid theory for background material.

In this chapter, we present a polynomial time recognition algorithm for balanced 0,1 matrices. By contrast, no polynomial recognition algorithm is known for perfection or for idealness. The algorithm decomposes a balanced matrix into totally unimodular matrices. The decomposition result is best explained in terms of graphs. Given a 0,1 matrix A, the bipartite representation of A is the bipartite graph G = (V^r ∪ V^c, E) having a node in V^r for every row of A, a node in V^c for every column of A, and an edge ij joining nodes i ∊ V^r and j ∊ V^c if and only if the entry a_ij of A equals 1. If A is balanced (totally unimodular, respectively), the bipartite graph G is said to be balanced (totally unimodular, respectively).

A double star is a tree where at most two nodes have degree greater than one. A node set S is a double star cutset of G if G \ S has more connected components than G and S induces a double star.

Much research has been devoted to studying classes of perfect graphs that can be decomposed into “basic classes” using “perfection-preserving operations.” In this chapter, we present four basic classes of perfect graphs and several operations on graphs that preserve perfection. In the last section, we illustrate the decomposition approach on a class of perfect graphs called Meyniel graphs.

The back matter includes, Bibliography, and Index