Given a weighted undirected graph, the maximum matching problem is to find a matching with maximum total weight. In his seminal paper, Edmonds [35] described an integral polytope for the matching problem, and the famous Blossom Algorithm for solving the problem in polynomial time.

In this chapter, we will show the integrality of the formulation given by Edmonds [35] using the iterative method. The argument will involve applying uncrossing in an involved manner and hence we provide a detailed proof. Then, using the local ratio method, we will show how to extend the iterative method to obtain approximation algorithms for the hypergraph matching problem, a generalization of the matching problem to hypergraphs.

Graph matching

Matchings in bipartite graphs are considerably simpler than matchings in general graphs; indeed, the linear programming relaxation considered in Chapter 3 for the bipartite matching problem is not integral when applied to general graphs. See Figure 9.1 for a simple example.

Linear programming relaxation

Given an undirected graph G = (V, E) with a weight function w: E → ℛ on the edges, the linear programming relaxation for the maximum matching problem due to Edmonds is given by the following LPM(G). Recall that E(S) denotes the set of edges with both endpoints in S ⊆ V and x(F) is a shorthand for ∑e∈Fxe for F ⊆ E.

Although there are exponentially many inequalities in LPM(G), there is an efficient separation oracle for this linear program, obtained by Padberg and Rao using Gomory-Hu trees.

Even though we mentioned the paper by Jain [75] as the first explicit application of the iterative method to approximation algorithms, several earlier results can be reinterpreted in this light, which is what we set out to do in this chapter. We will first present a result by Beck and Fiala [12] on hypergraph discrepancy, whose proof is closest to other proofs in this book. Then we will present a result by Steinitz [127] on rearrangements of sums in a geometric setting, which is the earliest application that we know of. Then we will present an approximation algorithm by Skutella [123] for the single source unsplittable flow problem. Then we present the additive approximation algorithm for the bin packing problem by Karmarkar and Karp [77], which is still one of the most sophisticated uses of the iterative relaxation method. Finally, we sketch a recent application of the iterative method augmented with randomized rounding to the undirected Steiner tree problem [20] following the simplification due to Chakrabarty et al. [24].

A discrepancy theorem

In this section, we present the Beck–Fiala theorem from discrepancy theory using an iterative method. Given a hypergraph G = (V, E), a 2-coloring of the hypergraph is defined as an assignment ψ: V → {-1, +1} on the vertices. The discrepancy of a hyperedge e is defined as discχ(e) = ∑v∈e ψ(v), and the discrepancy of the hypergraph G is defined as discψ(G) = maxe∈E(G) |{discψ(e)}|.

In this first chapter we motivate our method via the assignment problem. Through this problem, we highlight the basic ingredients and ideas of the method. We then give an outline of how a typical chapter in the rest of the book is structured, and how the remaining chapters are organized.

The assignment problem

Consider the classical assignment problem: Given a bipartite graph G = (V1 ∪ V2, E) with |V1| = |V2| and weight function w: E → ℝ+, the objective is to match every vertex in V1 with a distinct vertex in V2 to minimize the total weight (cost) of the matching. This is also called the minimum weight bipartite perfect matching problem in the literature and is a fundamental problem in combinatorial optimization. See Figure 1.1 for an example of a perfect matching in a bipartite graph.

One approach to the assignment problem is to model it as a linear programming problem. A linear program is a mathematical formulation of the problem with a system of linear constraints that can contain both equalities and inequalities, and also a linear objective function that is to be maximized or minimized. In the assignment problem, we associate a variable xuv for every {u, v} ∈ E. Ideally, we would like the variables to take one of two values, zero or one (hence in the ideal case, they are binary variables). When xuv is set to one, we intend the model to signal that this pair is matched; when xuv is set to zero, we intend the model to signal that this pair is not matched.

In this chapter, we will study the spanning tree problem in undirected graphs. First, we will study an exact linear programming formulation and show its integrality using the iterative method. To do this, we will introduce the uncrossing method, which is a very powerful technique in combinatorial optimization. The uncrossing method will play a crucial role in the proof and will occur at numerous places in later chapters. We will show two different iterative algorithms for the spanning tree problem, each using a different choice of 1-elements to pick in the solution. For the second iterative algorithm, we show three different correctness proofs for the existence of a 1-element in an extreme point solution: a global counting argument, a local integral token counting argument and a local fractional token counting argument. These token counting arguments will be used in many proofs in later chapters.

We then address the degree-bounded minimum-cost spanning tree problem. We show how the methods developed for the exact characterization of the spanning tree polyhedron are useful in designing approximation algorithms for this NP-hard problem. We give two additive approximation algorithm: The first follows the first approach for spanning trees and naturally generalizes to give a simple proof of the additive two approximation result of Goemans [59]; the second follows the second approach for spanning trees and uses the local fractional token counting argument to provide a very simple proof of the additive one approximation result of Singh and Lau [125].

In this chapter, we consider a simple model, based on a directed tree representation of the variables and constraints, called network matrices. We show how this model as well as its dual have integral optima when used as constraint matrices with integral right-hand sides. Finally, we show the applications of these models, especially in proving the integrality of the dual of the matroid intersection problem in Chapter 5, as well as the dual of the submodular flow problem in Chapter 7.

While our treatment of network matrices is based on its relations to uncrossed structures and their representations, they play a crucial role in the characterization of totally unimodular matrices, which are all constraint matrices that yield integral polytopes when used as constraint matrices with integral right-hand sides [121]. Note that total unimodularity of network matrices automatically implies integrality of the dual program when the right-hand sides of the dual are integral.

The integrality of the dual of the matroid intersection and submodular flow polyhedra can be alternately derived by showing the Total Dual Integrality of these systems [121]. Although our proof of these facts uses iterative rounding directly on the dual, there is a close connection between these two lines of proof since both use the underlying structure on span of the constraints defining the extreme points of the corresponding linear program.

Quoting Lovász from his paper “Submodular Functions and Convexity” [94]:

The submodular flow model is an excellent example to illustrate this point. In this chapter, we introduce the submodular flow problem as a generalization of the minimum cost circulation problem. We then show the integrality of its LP relaxation and its dual using the iterative method. We then discuss many applications of the main result. We also show an application of the iterative method to an NP-hard degree bounded generalization and show some applications of this result as well.

The crux of the integrality of the submodular flow formulations will be the property that a maximal tight set of constraints form a cross-free family. This representation allows an inductive token counting argument to show a 1-element in an optimal extreme point solution. We will see that this representation is precisely the one we will eventually encounter in Chapter 8 on network matrices.