Accessible Set Systems

We conclude this chapter with a brief report on further generalizations of the greedy algorithm from independence systems to even more general systems of sets. As the methods used are rather similar to the methods we have been using (although more involved), we shall skip all proofs and refer the reader to the original literature instead.

Aset systemis simply a pairM= (E,S), whereE is a finite set andSis a nonempty subset of the power set ofE. The elements ofSare calledfeasible sets of M; maximal feasible sets will again be called bases. As the greedy algorithm always chooses single elements and adds them one by one to the feasible set under construction, it would not make sense to consider entirely arbitrary set systems. At the very least, we have to ensure that every feasible set can be obtained by successively adding single elements to the empty set.

Formally, we require the followingaccessibility axiom:

(A) For each nonempty feasible set X∈S, there exists an element x∈X such thatX\ {x} ∈S.

In particular, the empty set is contained in S, as S=∅. A set system M satisfying axiom (A) is called an accessible set system. Any independence system is an accessible set system, but axiom (A) is a much weaker condition than the requirement of being closed under inclusion. Given an accessible

158 5 The Greedy Algorithm

set system M and a weight function w:E→R, we consider the following optimization problem:

(BMAX) determine max

w(B) :B is a basis ofM .

This generalizes the corresponding problem for independence systems. We also need to modify the greedy algorithm5.1.1so that it applies to accessible set systems.⁷This can be done as follows.

Algorithm 5.6.1 LetM= (E,S) be an accessible set system with a weight functionw:E→R.

Procedure GREEDY(E,S, w;T) (1) T← ∅;X←E;

(2) while there existsx∈X withT∪ {x} ∈S do (3) choose somex∈X with T∪ {x} ∈Sand

w(x)≥w(y) for ally∈X withT∪ {y} ∈S;

(4) T←T∪ {x}; X←X\ {x} (5) od

Of course, we want to characterize those accessible set systems for which Algorithm5.6.1always finds an optimal solution for (BMAX). Before describ-ing this result, we consider a special class of accessible set systems introduced by Korte and Lov´asz [KorLo81].

An accessible set system M satisfying the exchange axiom (2) of Theo-rem5.2.1is called agreedoid. Greedoids have been studied intensively because many interesting objects in combinatorics and optimization are greedoids. In particular, the so-calledantimatroids are greedoids. Antimatroids constitute a combinatorial abstraction of the notion of convexity; they play an impor-tant role in convexity, partially ordered sets, and graph theory. Greedoids occur as well in the context of matchings and of Gauß elimination. We will not go into detail here, but recommend that the reader consult the exten-sive survey [BjoZi92] or the monograph [KorLS91]. Unfortunately, the greedy algorithm does not find an optimal solution of (BMAX) for all greedoids.⁸ However, Korte and Lov´asz were able to characterize those greedoids for

7Note that it does not make sense to apply the original version of the greedy algorithm if Sis not closed under inclusion: in this case, it might happen that an elementxcannot be added to the feasible setT constructed so far, becauseT∪ {x}is not feasible; nevertheless, it might be permissible to addxat some later point to the setT=T∪A. Ifw(x)> w(y) for somey∈A, the original greedy algorithm5.1.1would fail in this situation, as the element xwould already have been dismissed. To avoid this, we simply keep the strategy of always selecting the largest available element; all that is required is a different formulation.

8Characterizing greedoids in terms of the greedy algorithm requires the use of certain non-linear objective functions; see [KorLS91].

which the greedy algorithm works. There is a simpler characterization due to Bryant and Brooksbank [BruBr92], which uses the followingstrong exchange axiom. We note that this condition holds for every matroid, but not for all greedoids.

(SE) ForJ, K∈Swith|J|=|K|+ 1, there always exists somea∈J\Ksuch that K∪ {a}and J\ {a} are inS.

Result 5.6.2 LetM= (E,S)be a greedoid.Then the greedy algorithm5.6.1 finds an optimal solution of (BMAX) for all weight functions w:E→R if and only ifM satisfies axiom (SE).

We need some further preparations to be able to formulate the charac-terization of those accessible set systems M= (E,S) for which the greedy algorithm computes an optimal solution. Given a feasible setA, we write

ext(A) :=

x∈E\A:A∪ {x} ∈S .

Now there are some situations where the greedy algorithm does not even construct a basis, but stops with some feasible set which is not maximal.

This happens if there exists a basisB with a proper feasible subset A⊂B such that ext(A) =∅. In this case, we may define a weight functionw by

w(x) :=

⎧⎨

⎩

2 forx∈A, 1 forx∈B\A, 0 otherwise;

then the greedy algorithm constructsA, but cannot extendAto the optimal basisB. The accessibility axiom (A) is too weak to prevent such situations:

it merely ensures that a basisB can be obtained somehow by adding single elements successively to the empty set, but not necessarily by adding ele-ments to agiven feasible subset ofB. To avoid this, we require the following extensibility axiom:

(E) For every basis B and every feasible subset A⊂B with A=B, there exists somex∈B\AwithA∪ {x} ∈S.

Note that this axiom is satisfied for all greedoids. We need one more definition.

For any set systemM= (E,S), define

S:={X⊆E: there isA∈SwithX⊆A},

and callM := (E,S) thehereditary closureofM. Now we require the follow-ingclosure congruence axiom:

(CC) For every feasible setA, for allx, y∈ext(A), and for each subsetX⊆ E\(A∪ext(A)),A∪X∪ {x} ∈SimpliesA∪X∪ {y} ∈S.

160 5 The Greedy Algorithm

Exercise 5.6.3 Show that an accessible set system M = (E,S) for which the greedy algorithm works correctly has to satisfy axiom (CC).

One may show that axiom (CC) is independent of the exchange axiom, even if we only consider accessible set systems satisfying the extensibility axiom. In fact, there are greedoids not satisfying (CC); on the other hand, independence systems always satisfy (CC), because the only choice forX is X=∅. We need one final axiom:

(ME) The hereditary closureM ofM is a matroid.

(ME) is called thematroid embedding axiom. Now we can state the following characterization due to Helman, Mont and Shapiro [HelMS93]:

Result 5.6.4 LetM= (E,S)be an accessible set system.Then the following statements are equivalent:

(1) M satisfies axioms (E), (CC) and(ME).

(2) For every weight function w:E→R, the optimal solutions of (BMAX) are precisely those bases of M which are found by the greedy algo-rithm5.6.1(given an appropriate order of the elements of equal weight).

(3) For every weight functionw:E→R,the greedy algorithm5.6.1yields an optimal solution of(BMAX).

The reader might try to fill in the missing parts of the proof; this is a more demanding exercise, but can be done using the methods we have presented.

Alternatively, we recommend a look at the original paper [HelMS93], which contains some further interesting results. In particular, the authors consider bottleneck problems, that is, problems of the form

(BNP) Maximize min

w(x) :x∈B over all basesB ofM for a given weight function w:E→R. The greedy algorithm constructs an optimal solution for (BNP) in the situation of Result 5.6.4. In fact, this holds even under considerably weaker conditions. We need one further axiom, namely thestrong extensibility axiom:

(SE) For every basisB and each feasible set A with |A|<|B|, there exists x∈B\A withA∪ {x} ∈S.

Then the following characterization holds [HelMS93]:

Result 5.6.5 Let M = (E,S) be an accessible set system. The greedy algo-rithm5.6.1constructs an optimal solution for(BNP)for all weight functions w:E→R if and only ifM satisfies axiom(SE).

For partially ordered set systems, the greedy algorithm was studied by Faigle [Fai79] who obtained characterizations analogous to Results 5.6.4

and5.6.5. Further characterizations of related structures by the greedy algo-rithm (or appropriately modified versions) can be found in [Fai85], [Goe88], and [BoyFa90], where ordered languages, greedoids of Gauß elimination, and antimatroids are studied, respectively. There are further important general-ization of the notion of a matroid such asoriented matroids. We will not con-sider these structures here, but refer the reader to the monographs [BacKe92]

and [BjoLSW92].

Chapter 6

Flows

What need you flow so fast?

Anonymous

In this chapter, we investigate flows in networks: How much can be trans-ported in a network from a source s to a sink t if the capacities of the connections are given? Such a network might model a system of pipelines, a water supply system, or a system of roads. With its many applications, the theory of flows is one of the most important parts of combinatorial optimiza-tion. In Chap. 7 we will encounter several applications of the theory of flows within combinatorics, and flows and related notions will appear again and again throughout the book. The once standard reference,Flows in Networks by Ford and Fulkerson [ForFu62], is still worth reading; an extensive, more recent treatment is provided in [AhuMO93].