A greedy algorithm for interval greedoids

Hua Mao

doi:10.1515/math-2018-0026

Artikel Open Access

A greedy algorithm for interval greedoids

Hua Mao

Veröffentlicht/Copyright: 31. März 2018

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$Open Mathematics$

Aus der Zeitschrift Open Mathematics Band 16 Heft 1

Abstract

We show that the greedy algorithm provided in this paper works for interval greedoids with positive weights under some conditions, and also characterize an exchangeable system to be an interval greedoid with the assistance of the greedy algorithm.

Keywords: Interval greedoid; Exchangeable system; Greedy algorithm; Positive weight

MSC 2010: 90C27; 05B35

1 Introduction and Preliminaries

Let 𝓕 (called feasible sets) be a set system on E (i.e. a non-empty family 𝓕 ⊆ 2^E where 2^E is the set of all subsets of a finite set E). We can suppose ∪𝓕 = E in this paper, since E ∖ ∪𝓕 ≠ ∅ will bring x ∈ E ∖ ∪𝓕 to own nothing in 𝓕 which is not interesting for studying. Actually, such supposition is also done in [1].

Let ω : E → ℝ⁺ be a weighting on E. Abbreviating ω(X)=∑x∈Xω(x), especially ω(∅) = 0, we want to find an A ∈ 𝓕 satisfying ω(A) = max{ω(X)∣ X ∈𝓕}. We call this problem (𝓕, ω). An element of 𝓕 is optimal if it has the maximal weight. The greedy algorithm for (E, 𝓕) attempts to solve the above problem. In fact, Helman et al. [2] point that obtaining an exact characterization of the class of problems for which the greedy algorithm returns an optimal solution has been an open problem. The process of greedy algorithm (cf.[3, p.14]) is as follows.

Set X = ∅.
Set T = {x ∈ E ∖ X ∣ X ∪ x ∈ 𝓕},
If T = ∅, stop;
If T ≠ ∅, choose x ∈ T such that ω(x) ≥ ω(y) for all y ∈ T.
Set X = X ∪ x and go to (2).

Björner et al. indicate [1] that greedoids were invented around 1980 by Korte and Lovász. The relative definitions to greedoids are reviewed as follows.

Definition 1.1

([1, 3]). Let 𝓕 be a set system on E.

A greedoid is a pair (E, 𝓕), where 𝓕 satisfies the following conditions:
1. For every non-empty X ∈ 𝓕, there is an x ∈ X such that X ∖ {x} ∈ 𝓕. (accessible)
2. For X, Y ∈𝓕 such that |X| > |Y|, there is an x ∈ X ∖ Y such that Y ∪ {x} ∈ 𝓕. (exchangeable)
A greedoid (E, 𝓕) has the interval property (or to be an interval greedoid) if A ⊆ B ⊆ C, A, B, C ∈ 𝓕 and x ∈ E ∖ C, then A ∪ {x} ∈𝓕 and C ∪ {x} ∈ 𝓕 imply B ∪ {x} ∈ 𝓕.
A maximal element in (𝓕, ⊆) is called a basis.
A loop in (E, 𝓕) is an element x ∈ E that is contained in no basis.
A language 𝓛 over E is a non-empty set 𝓛 ⊆ E^*(the free monoid of all words over the alphabet E) of words over the alphabet E; it is called simple if every word in 𝓛 is simple (i.e. it does not contain any letter more than once).

A greedoid language over E is a pair (E, 𝓛), where 𝓛 is a simple language 𝓛 ⊆ Es∗(the set of simple words in E^*) satisfying the following conditions:

If α = β γ and α ∈ 𝓛, then β ∈ 𝓛. (hereditary)
If α, β ∈ 𝓛 and |α| > |β|, then α contains a letter x such that β x ∈𝓛. (exchange).

Björner et al. indicate [1] that greedoids were originally developed to give a unified approach to the optimality of various greedy algorithms known in combinatorial optimization. Such algorithms can be loosely characterized as having locally optimal strategy and no backtracking. Nowadays, researchers provide different greedy algorithms to characterize the different kinds of greedoids (cf. [1, 2, 3, 4]). Helman et al. [2] characterize greedy structures. [2, Theorem 1] is the best and main result in [2]. That is, let (E, 𝓕) be an accessible set system. Then (E, 𝓕) has an optimal greedy basis for every positive weighted linear function if and only if (E, 𝓕) is a matroid embedding (cf. [2, Definition 2], i.e. A matroid embedding is an accessible set system which is extensible, closure-congruent, and the hereditary closure of which is a matroid). Below [2, Definition 2], Helman et al. say that (S, 𝓒₁) in [2, Example 1] is a matroid embedding, yet not a greedoid. Combining [2, Example 2] and the definition of a matroid embedding, (S, 𝓒₂) in [2, Example 2] is a greedoid, yet not a matroid embedding. In other words, [2, Theorem 1] does not characterize a greedoid structure with greedy algorithm. Korte et al. say [4, p.358, Theorem 14.7]: Let (E, 𝓕) be a greedoid. The greedy algorithm finds a set F ∈ 𝓕 of maximum weight for each modular weight function if and only if (E, 𝓕) has the so-called strong exchange axiom (see [2], [3, p.160], [4, p.358], or say: For all A ∈ 𝓕, B is a maximal in 𝓕 and A ⊆ B. If x ∈ E ∖ B with A ∪ {x} ∈ 𝓕, then there exists a y ∈ B ∖ A such that A ∪ {y} ∈ 𝓕 and (B ∖ {y}) ∪ {x} ∈ 𝓕). Actually, Korte et al. [3, p.160, Theorem 2.2] is the same result as [4, p.358, Theorem 14.7]. However, from [3], it is easily seen that a greedoid can not be ensured to satisfy the strong exchange axiom. Hence, we may be asserted that [3, p.160, Theorem 2.2] or [4, p.358, Theorem 14.7] is not a characterization for all of greedoids with greedy algorithm, but only a characterization for a part of class of greedoids. In addition, among the known characterizations relative to greedoids with greedy algorithm, we think [1, Theorem 8.5.2] (the same as [3, p.157, Theorem 1.4]) to be better, that is: suppose (E, 𝓛) is a simple hereditary language, then (E, 𝓛) is a greedoid if and only if greedy algorithm gives an optimal solution for every compatible objective function on 𝓛. In the characterizations using Definition 1.1 and the greedy algorithms for a greedoid (E, 𝓕) proved in [1, Theorem 8.5.2] and [3, p.157, Theorem 1.4], 𝓕 must be hereditary (i.e. X ⊆ Y, Y ∈ 𝓕 ⇒ X ∈ 𝓕).

Now returning to our question: under what conditions on a greedoid, can every linear function be optimized by the greedy algorithm? Up to now, we do not find an answer for all of greedoids. Though we do not find out the solution to the open problem for all of greedoids, using the research methods in [1, 2, 3, 4] for reference, we can pay our attention to some special class of greedoids to look for the answer. By [1, 3], an interval greedoid (E, 𝓕₀) does not ask 𝓕₀ to be hereditary or satisfy strong exchange axiom. The authors describe [1] that the ‘interval property’ characterizes a very large class of greedoids and interval greedoids behave better than general greedoids in many respects. In some types of study, the interval property has to be assumed to obtain meaningful results [1, 3, 5, 6]. Hence, this paper will focus on interval greedoids in hope to find the answer for the open problem.

We may find from Definition 1.1 that for a greedoid (E, 𝓕), if 𝓕 is hereditary, then (E, 𝓕) is interval. In addition, it is necessary to generalize the results in [1, Theorem 8.5.2] and [3, p.157, Theorem 1.4] for interval greedoids. This is done in this paper.

Lemma 1.2

Let 𝓕 ⊆ 2^Eand x ∈ E be a loop. Then 𝓕 is a set system on E ∖ {x}.

Proof

Suppose that a loop x is contained in a X ∈ 𝓕. Then, there is a basis B_X satisfying X ⊆ B_X according to Definition 1.1(3). This follows x ∈ B_X, a contrary to the loop of x. Therefore, we demonstrate that 𝓕 is defined on E ∖ {x}.

By Lemma 1.2, this paper only considers the set systems with no loops.

Lemma 1.3

([3, p.47]). For a given set system 𝓕 on E, the property (G2) holds if and only if for any A ⊆ E, all bases of A have the same cardinality.

According to Lemma 1.3, we can state that in a set system such that (G2), then X ⊆ E is a basis of A ⊆ E if and only if X ∈ 𝓕, X ⊆ A satisfies |X|=maxY∈F,Y⊆A|Y|.

2 Main results

We give some notions for a set system (E, 𝓕):

𝓕^(k) = {X ∈ 𝓕 ∣ |X| = k};
𝓕|_A = {X ∣ X ⊆ A, X ∈ 𝓕} for any A ∈ 𝓕;
n = maxX∈F|X|;
Let ω : E → ℝ⁺ be a positive weight function (i.e. ω(x) > 0 for any x ∈ E). For X ⊆ E, define ω_X: X → ℝ⁺ as ω_X(x) = ω(x) for any x ∈ X.

We know that, generally, the solution of the greedy algorithm in Section 1 is not optimal. The already existing greedy algorithms for greedoids (see [1, 3, 4]) are satisfied (or say, characterized) by some different classes of greedoids. In order to search out a characterization of a type of greedy algorithms for some class of interval greedoids, we provide a type of greedy algorithm (i.e. Algorithm 1) as follows. After that, we will demonstrate under what conditions for a set system, Algorithm 1 has an optimal solution. We also find under what conditions Algorithm 1 characterizes an interval greedoid.

Algorithm 1

(Interval Greedy Algorithm). Input: 𝓕, a set system on E; ω: E → ℝ⁺, a positive weight function; n, maxX∈F |X|.

Output: S, the greedy solution.

SetS = ∅, j = 0.
If j < n − 1, then go to 3.
If j = n − 1, then go to 4.
If j ≥ n, thenS: = S, stop.
Set D_j+1 = {e ∣ there exist S_j+1 ∈ 𝓕^(j+1)andS ⊂ S_j+1such that e ∈ E ∖ S_j+1and S_j+1 ∪ {e} ∈ 𝓕}, and G_j = {e ∈ E ∖ S ∣ S ∪ {e} ∈ 𝓕}.
1. If D_j+1 ∩ G_j ≠ ∅, then choose e_j+1 ∈ D_j+1 ∩ G_j such that ω(e_j+1) = maxe∈Dj+1∩Gjω(e), and setS: = S ∪ {e_j+1}, j: = j+1, go to {2}.
2. If D_j+1 ∩ G_j = ∅, thenS: = S, and j: = j+1, go to 2.
Set G_j = {e ∈ E ∖ S ∣ S ∪ {e} ∈ 𝓕}.

If G_j ≠ ∅, then choose e_j+1 ∈ G_j such that ω(e_j+1) = maxe∈Gjω(e), and setS := S ∪ {e_j+1}, j := j+1, go to 2
If G_j = ∅, thenS := Sand j: = j+1, go to 2.

We say the greedy algorithm works if ω(S) ≥ ω(A) for ∀ A ∈ 𝓕. In the process of Algorithm 1, we can use S_t+1 to stand for the solution when the cyclic variable j is t ≤ n − 1.

Example 2.1

Let E₁ = {a₁, a₂, a₃, a₄} and 𝓕₁ = {∅, {a₁}, {a₄}, {a₁, a₂}, {a₁, a₄}, {a₁, a₂, a₃}, {a₁, a₃, a₄}, {a₁, a₂, a₃, a₄}}. We can easily check that 𝓕₁satisfies (G1) and (G2) in Definition 1.1 (1).

Let A = ∅, B = {a₁, a₂} and C = {a₁, a₂, a₃}. We easily find A ⊂ B ⊂ C. For a₄ ∈ E₁ ∖ C, we obtain A ∪ {a₄} = {a₄} ∈ 𝓕₁, C ∪ {a₄} = {a₁, a₂, a₃, a₄} ∈ 𝓕₁ and B ∪ {a₄} = {a₁, a₂, a₄} ∉ 𝓕₁. Using Definition 1.1 (2), (E₁, 𝓕₁) is not an interval greedoid.

Define ω₁ : E₁ → ℝ⁺ as ω₁(a₁) = 5, ω₁(a₂) = 4, ω₁(a₃) = 3, ω₁(a₄) = 2. Then, we can demonstrate that {a₁, a₂, a₃, a₄} is an optimal set. Applying Algorithm 1 on (𝓕₁, ω₁), we look for the solution S of Algorithm 1 as follows: There is n = 4.

When j = 0, there are S₀ = ∅, D₁ = {a₁, a₂, a₄}, G₀ = {a₁, a₄}, and so S₁ = {a₁}.

When j = 1, there are D₂ = {a₃}, G₁ = {a₂, a₄}, and so D₂ ∩ G₁ = ∅. Thus, we attain S₂ = S₁ = {a₁}.

When j = 2, there are D₃ = {a₂, a₄}, G₂ = {a₂, a₄}, and so S₃ = {a₁, a₂}.

When j = 3, there is j = n − 1. We find G₃ = {a₃}. Hence, there is S₄ = {a₁, a₂, a₃}.

When j = 4, there is j ≥ n. So, we obtain S = S₄.

Actually, ω₁({a₁, a₂, a₃}) = 12 < 14 = ω₁({a₁, a₂, a₃, a₄}) indicates that S is not optimal.

Remark 2.2

After analyzing Example 2.1, we attain some properties as follows.

If we hope Algorithm 1 to work for (E, 𝓕), then (E, 𝓕) should be an interval greedoid.
There are {a₁}, {a₄} ∈ 𝓕₁ and {a₂}, {a₃} ∉ 𝓕₁holds. Hence, we can ask {x} ∈ 𝓕 for any x ∈ E if we hope Algorithm 1 to work for a set system 𝓕 with any positive weight function ω.

Example 2.3

Let E₂ = {a₁, a₂, a₃, a₄} and 𝓕₂ = {∅, {a₁}, {a₄}, {a₁, a₂}, {a₁, a₄}, {a₁, a₂, a₃}, {a₁, a₃, a₄}}. We easily check up 𝓕₂to satisfy (G1) and (G2) and the interval property. Hence, (E₂, 𝓕₂) is an interval greedoid.

Define ω₂ : E₂ → ℝ⁺ as ω₂(a₁) = 5, ω₂(a₂) = 4, ω₂(a₃) = 3, ω₂(a₄) = 2. Then, we can demonstrate that {a₁, a₂, a₃} is an optimal set. Applying Algorithm 1 on (𝓕₂, ω₂), we look for the solution S of Algorithm 1 as follows: There is n = 3.

When j = 0, there are S₀ = ∅, D₁ = {a₁, a₂, a₄}, G₀ = {a₁, a₄}, and so S₁ = {a₁}.

When j = 1, there are D₂ = {a₃}, G₁ = {a₂, a₄}, and so D₂∩ G₁ = ∅. Thus, we attain S₂ = S₁ = {a₁}.

When j = 2, there is j = n − 1. We find G₂ = {a₂, a₄}. Hence, there is S₃ = {a₁, a₂}.

When j = 3, there is j ≥ n. So, we obtain S = S₃.

Actually, ω₂({a₁, a₂, a₃}) = 12 > 9 = ω₂({a₁, a₂}) indicates that S is not optimal.

Remark 2.4

InExample 2.3, B₁ = {a₁, a₂, a₃} andB₂ = {a₁, a₃, a₄} are two bases of 𝓕₂such thatB₁ ∖ {a₃} = {a₁, a₂},B₂ ∖ {a₃} = {a₁, a₄} ∈ 𝓕₂. Butω₂(a₁) > ω₂(a₂) > ω₂(a₃) > ω₂(a₄) and 𝓕⁽²⁾ = {{a₁, a₂}, {a₁, a₄}} imply that (E₂,𝓕₂) does not satisfy the following condition:

(G3) LetX, Ybe bases in 𝓕 andS = Y ∖ {y₀} ∈ 𝓕 for somey₀ ∈ Y. Then there isx₀ ∈ X ∖ Ssuch thatS ∪ {x₀} ∈ 𝓕 andω(x₀) = maxx∈Xω(x).

Hence, we should ask if 𝓕 with ω satisfy (G3) if we hope Algorithm 1 to work for (𝓕,ω) though (E, 𝓕) is an interval greedoid.

Lemma 2.5

Let 𝓕 ⊆ 2^Ewith ∅ ∈ 𝓕 andω : E → ℝ⁺be a positive weight function.

If 𝓕 satisfies (G2), then there is 𝓕^(k) ≠ ∅ for anyk = 0, 1, …, n.
An optimal set in (𝓕,ω) is a basis.

Proof

Using Lemma 1.3 and Definition 1.1 (3), all of bases in 𝓕 have the cardinality n. Let B ∈ 𝓕 be a basis. From Björner et al. [1, 8.2.A], we know that ∅ ∈ 𝓕 and (G2) together define greedoids as well as (G1) and (G2). Considering (G1) on B, we may easily obtain 𝓕^(k) ≠ ∅, (k = 0, 1, …, n).
Since an optimal set S satisfies ω(S) ≥ ω(B) for any basis B of 𝓕. If S is not a basis, then S ⊂ B_S holds for some basis B_S according to Definition 1.1(3). Thus, there is ω(S) < ω(B_S) since ω is positive, a contradiction with ω(S) ≥ ω(B). Hence S is a basis.

□

Theorem 2.6

Let 𝓕 ⊆ 2^Esatisfy ∅ ∈ 𝓕 and {x} ∈ 𝓕 for anyx ∈ E. Letω : E → ℝ⁺be a positive weight function. If (E, 𝓕) is an interval greedoid satisfying the condition (G3), thenAlgorithm 1works for (𝓕,ω).

Proof

Let 𝓕_k = {X ∣ X ∈ 𝓕, |X| ≤ k} (k ≥ 1). We prove the following statements.
(st1) ∅ ∈ 𝓕_k,{x} ∈ 𝓕_k for any x ∈ E.
(st2) (E, 𝓕_k} is an interval greedoid.
{x} ∈ 𝓕 and |{x}| = 1 ≤ k for any x ∈ E imply {x} ∈ 𝓕_k for any x ∈ E. ∅ ∈ 𝓕 and | ∅ | = 0 ≤ k together means ∅ ∈ 𝓕_k. Hence, we can say that 𝓕_k is a set system with no loops. According to 𝓕_k ⊆ 𝓕 and 𝓕 satisfying both of (G1) and (G2), we easily obtain that 𝓕_k satisfies both of (G1) and (G2).
Let A ⊆ B ⊆ C, A, B, C ∈ 𝓕_k and a ∈ E_k ∖ C satisfy A ∪ {a} ∈ 𝓕_k and C ∪ {a} ∈ 𝓕_k. Using 𝓕_k ⊆ 𝓕 and the interval property of 𝓕, we obtain B ∪ {a} ∈ 𝓕. C ∪ {a} ∈ 𝓕_k follows | C ∪ {a}| ≤ k. Combining with B ∪ {a} ⊆ C ∪ {a}, we decide | B ∪ {a}| ≤ k. Hence, there is B ∪ {a} ∈ 𝓕_k.
Therefore, (E, 𝓕_k) is an interval greedoid.
We will prove that Algorithm 1 works for (𝓕,ω) by induction on n.
If n = 0. This means 𝓕 = {∅. Hence, the needed result follows.
If n = 1. By Lemma 2.5 (1) and Definition 1.1 (2), there are 𝓕⁽⁰⁾ ≠ ∅ and 𝓕⁽¹⁾ ≠ ∅.
Then, in the process of Algorithm 1, when j = 0 = 1 − 1 = n − 1, according to S₀ = ∅, there is G₀ = {e ∈ E ∖ ∅ ∣ ∅ ∪ {e} ∈ 𝓕} = E since {x} ∈ 𝓕 for any x ∈ E. Hence, G₀ ≠ ∅ holds. Choose ω(e₀₊₁) = ω(e₁) = maxe∈G0ω(e), and put S₁ = ∅ ∪ {e₁} = {e₁} and j : = 0 + 1 = 1. When j = 1, then j ≥ 1 = n follows the process of Algorithm 1 to stop. Therefore, the solution of Algorithm 1 is optimal. That is to say, Algorithm 1 works for (𝓕,ω).
Suppose that if n ≤ m − 1, then the needed result is correct. Now, let n = m.
Since (E, 𝓕) is an interval greedoid, 𝓕 satisfies (G1) and (G2). Combining Lemma 2.5, there is m = |B| for any basis B of 𝓕. Utilizing Lemma 2.5 (1), we obtain 𝓕^(k) ≠ ∅ where k = 0, 1, …, m.
Let S be the solution of Algorithm 1 for (𝓕,ω). During the process of Algorithm 1, when j < m − 1, according to the interval property, ∅ ∪ {e₀} ∈ 𝓕 for any e₀ ∈ E, ∅ ⊆ S_j ⊆ S_j+1, S_j+1 ∪ {e} ∈ 𝓕 for any e ∈ D_j+1, and the definitions of D_j+1 and G_j, there is S_j ∪ {e} ∈ 𝓕 for any e ∈ D_j+1, and so D_j+1 ∩ G_j ≠ ∅. Considering Lemma 2.5 (1), we arrive at 𝓕^(m) ≠ ∅. So, there is G_m−1 = {e ∈ E ∖ S_m−1 ∣ S_m−1 ∪ {e} ∈ 𝓕} ≠ ∅. Therefore, we can demonstrate that S is a basis. Hence, |S| = m holds.
Since S is accessible according to the process of Algorithm 1 for the interval greedoid (E, 𝓕) and ω, there is S = S_m. Using Step 1 and the inductive supposition, Algorithm 1 works for (E, 𝓕_m−1). That is to say, the solution S^m−1 of Algorithm 1 for (E, 𝓕_m−1) satisfies ω(S^m−1) ≥ ω(X) for any X ∈ 𝓕_m−1. Combining the process of Algorithm 1, we confirm S^m−1 = S_m−1.
Considering G_m−1 = {e ∈ E ∖ S_m−1 ∣ S_m−1 ∪ {e} ∈ 𝓕} and 𝓕^(m) ≠ ∅, we obtain S_m = S_m−1 ∪ {e_m} where ω(e_m) = maxe∈Gm−1ω(e). Moreover, S_m is the solution of Algorithm 1 for (𝓕,ω).
Let B be a basis of 𝓕. We easily find |B| = |S_m−1| + 1. Thus, S_m−1 ∪ {b} ∈ 𝓕 holds for some b ∈ B ∖ S_m−1 according to (G2) satisfied by 𝓕. Thus, S_m−1 ∪ {b} is a basis of 𝓕. Using (G3), there is S_m−1 ∪ {b₀} ∈ 𝓕 where ω(b₀) = maxx∈B∖S¯m−1ω(x).
On the other hand, for any x ∈ B, if B ∖ {x} ∈ 𝓕, then ω(B ∖ {x}) ≤ ω(S^m−1) = ω(S_m−1) in view of the inductive supposition. Since (B ∖ {x}) ∪ {x} is a basis, there is ω((B ∖ {x}) ∪ {x}) ≤ ω(S_m−1 ∪ {b₀}) ≤ ω(S_m−1 ∪ {e_m}) = ω(S_m) in virtue of (G3) and the process of searching S_m.
Therefore, ω(S_m) ≥ ω(B) holds for any basis B in 𝓕. Furthermore, ω(S_m) ≥ ω(X) is correct for any X ∈ 𝓕 since X must be contained in a basis and ω is positive. Thus, S_m is optimal.
Summing up, Algorithm 1 works for (𝓕,ω).

□

Remark 2.7

Example 2.1shows that (E₁, 𝓕₁) is not an interval greedoid. In addition, for {a₁} ⊂ {a₁, a₂} anda₃ ∈ E₁ ∖ {a₁, a₂}, there are {a₁, a₂} ∪ {a₃} = {a₁, a₂, a₃} ∈ 𝓕₁and {a₁} ∪ {a₃} = {a₁, a₃} ∉ 𝓕₁. That is to say, 𝓕 does not satisfy thesemi-interval property (that is, ifY ⊆ Z, Y, Z ∈ 𝓕, a ∈ E ∖ Z, thenZ ∪ {a} ∈ 𝓕 ⇒ Y ∪ {a} ∈ 𝓕)}.

It is more interesting that the converse of Theorem 2.6 is also true under some pre-conditions.

Theorem 2.8

Let 𝓕 be a set system onEwith ∅ ∈ 𝓕. If for any positive weight functionω : E → ℝ⁺, there are the following statements:

Algorithm 1works for (𝓕|_A, ω_A) for anyA ∈ 𝓕.
𝓕 has semi-interval property.
𝓕 satisfies (G2).

Then, (E, 𝓕) is an interval greedoid.

Proof

To prove : 𝓕 is accessible.
Let A ∈ 𝓕. Since A is the basis of 𝓕|_A and ω is positive, there is ω(A) = maxX∈F|Aω(X). Let S_A be the solution of Algorithm 1 for (𝓕|_A, ω_A). Consider the process of Algorithm 1 and A ∈ 𝓕, we can decide S_A ⊆ A. Hence, we obtain ω_A(S_A) ≤ ω_A(A) since ω_A is positive. Furthermore, since ω_A is positive, we find S_A ⊂ A ⇔ ω_A(S_A) < ω_A(A). By (s1), we confirm ω_A(S_A) = maxX∈F|Aω_A(X). Summing up the above results, we may follow S_A = A.
From ∅ ∈ 𝓕 and the process of Algorithm 1 for (𝓕|_A, ω_A), we may assert that S_A is accessible. We will prove this assertion as follows.
Let m = |S_A|, that is, m = |A| since S_A = A.
If m = 0. Then S_A = ∅. So, S_A is accessible.
If m = 1. Then A = {a} and S₀ = ∅.
When j = 0 = m − 1 = 1 − 1. There is G_j = G₀ = {a}. Hence, G_j ≠ ∅ and ω(e₀₊₁) = ω(a) and S₁ = ∅ ∪ {a} = {a}.
When j = 0 + 1 = 1 = m, then stop, and S_A = {a}. We easily obtain S_A ∖ {a} = ∅ ∈ 𝓕. Hence, S_A is accessible.
Suppose that if m ≤ k − 1, then S_A is accessible. Now, let m = k > 1.
If for every j < k − 1, there is D_j+1 ∩ G_j ≠ ∅ in the process of Algorithm 1, then according to the definitions of D_j+1 and G_j, there are S₀ = ∅ ⊂ S₁ ⊂ S₂ ⊂ … ⊂ S_j+1 and S_i+1 = S_i ∪ {e_i+1} for some e_i+1 ∈ A ∖ S_i, (i = 0, …, j + 1), we can state that S_j is accessible (j = 0, …, k − 1).
If for some j₀ < k − 1, there is D_j₀+1 ∩ G_j₀ = ∅ in the process of Algorithm 1, then S_j₀+1 = S_j₀. This follows |S_j₀+1| < j₀+1. Furthermore, we obtain |S_k| < k according to the process of Algorithm 1. Thus, we attain S_A (that is S_k) satisfying |S_A| < k = |A| = |S_A|, a contradiction. Hence, for every j < k − 1, there is D_j+1 ∩ G_j ≠ ∅.
So, S_j is accessible (j = 0, 1, …, k − 1).
Let j = k − 1. If G_k−1 ≠ ∅, then we obtain S_k = S_k−1 ∪ {e_k}. Thus, we may easily find S_k to be accessible since S_k ∖ e_k−1 = S_k−1 ∈ 𝓕 and the above discussion.
If G_k−1 = ∅, then S_k = S_k−1. Thus, there is |S_k| = |S_k−1| = k − 1. On the other hand, S_k = S_m = S_A = A hints |S_k| = |A| = m = k. This is a contradiction to |S_k| = k − 1. Therefore, we confirm G_k ≠ ∅.
Adding up the above discussion with the induction, we can state that S_A, that is A, is accessible.
According to the arbitrariness of A, we attain that 𝓕 is accessible.
To prove : 𝓕 satisfies the interval property.
Let X, Y, Z ∈ 𝓕 satisfy X ⊆ Y ⊆ Z. Let a ∈ E ∖ Z satisfy X ∪ {a}, Z ∪ {a} ∈ 𝓕. Using the statement (s2), there is Y ∪ {a} ∈ 𝓕. Therefore, 𝓕 satisfies the interval property.
𝓕 is exchangeable according to the statement (s3).
Combining Steps 1, 2 and 3 with Definition 1.1, (E, 𝓕) is an interval greedoid.

□

Example 2.9

LetE₃ = {a₁, a₂, a₃, a₄} and 𝓕₃ = {∅, {a₁}, {a₄}, {a₁, a₂}, {a₃, a₄}}. There is |{a₃, a₄}| = 2 = |{a₁}| + 1, but no elementa ∈ {a₃, a₄} ∖ {a₁} satisfies {a₁} ∪ {a} ∈ 𝓕₃. Defineω₃ : E₃ → ℝ⁺asω₃(a₁) = 5, ω₃(a₂) = 1, ω₃(a₃) = ω₃(a₄) = 4. The solution ofAlgorithm 1for (𝓕₃,ω₃) is {a₁}. But, ω₃({a₁}) = 5 < ω₃({a₃, a₄}) = 8 impliesAlgorithm 1not to work for (𝓕₃,ω₃).

Example 2.9 shows that if we want Algorithm 1 to work for (𝓕, ω), then 𝓕 should satisfy (G2). Combining Theorems 2.6 and 2.8, we give the following characterization for interval greedoids.

Theorem 2.10

Let 𝓕 be an exchangable set system onEwith ∅ ∈ 𝓕 and {x} ∈ 𝓕 for anyx ∈ E. 𝓕_Asatisfies (G3) for anyA ∈ 𝓕. Then 𝓕 is the set of feasible sets of an interval greedoid onEif and only if for all positive weight functionsω : E → ℝ⁺, 𝓕 satisfies the statements (s1) and (s2).

Proof

We easily prove (A, 𝓕|_A) to be an interval greedoid for any A ∈ 𝓕. Combining Theorem 2.1 and 𝓕|_A satisfying (G3), we obtain the correctness of (s1). ∅, {x} ∈ 𝓕 and the interval of 𝓕 follow the correctness of (s2).
Using Theorem 2.8, all of needed results are straightforward.

□

Next, we will compare our results with some known results for greedoids.

To compare our results with [1, Theorem 8.5.2] (or say [3, p.157, Theorem 1.4]).

In [1, Theorem 8.5.2] and [3, p.157, Theorem 1.4], the authors give a kind of greedy algorithm to characterize a greedoid (E, 𝓕₁₃), where 𝓕₁₃ is asked to be hereditary. In other words, if a greedoid (E, 𝓕) does not satisfy the hereditary property for 𝓕, then the characterizations with greedy algorithms in [1, Theorem 8.5.2] and [3, p.157, Theorem 1.4] will not be successful.
Let 𝓕 be a set system on E satisfying the hereditary. Then, we easily find that 𝓕 has the following properties:

∅, {x} ∈ 𝓕 holds for any x ∈ E; 𝓕 satisfies the condition (G3).
Let Y ⊆ Z, Y, Z ∈ 𝓕 and a ∈ E ∖ Z. If Z ∪ {a} ∈ 𝓕 is correct, then Y ∪ {a} ∈ 𝓕 holds according to Y ∪ {a} ⊆ Z ∪ {a} and the hereditary property of 𝓕. This implies that every hereditary sets system satisfies the semi-interval property.
Considered items (1) and (2), we know that the characterization for greedoids with the greedy algorithm provided in [1, Theorem 8.5.2] and [3, p.157, Theorem 1.4] are really effective only for some of interval greedoids and not for the other kinds of greedoids.
Evidently, the given conditions in Theorems 2.6, 2.8 and 2.10 do not ask 𝓕 to be hereditary. Combining the above three items, we can say that for a hereditary set system 𝓕, Theorem 2.6 and Theorem 2.8 are satisfied by much more greedoids than that in [1, Theorem 8.5.2] (or say, [3, p.157, Theorem 1.4]) respectively.
Moreover, the characterization (i.e. Theorem 2.3) proposed in this paper for interval greedoids generalize the results in [1, Theorem 8.5.2] and [3, p.157, Theorem 1.4] respectively.
Therefore, Algorithm 1 generalizes the greedy algorithm for [1, Theorem 8.5.2] and [3, p.157, Theorem 1.4].
To compare our results with [4, p.358, Theorem 14.7].
It is well known that a greedoid is perhaps not satisfying the strong exchange axiom. In other words, not every greedoid has strong exchange axiom, though any greedoid is exchangeable. We also know that an interval greedoid can not be ensured to satisfy strong exchange axiom. Thus, we can state that Theorem 2.10 is a characterization of a greedy algorithm for some class of interval greedoids. [4, p.358, Theorem 14.7] can not substitute for the results in this paper. Therefore, Algorithm 1 is a new algorithm and not covered by the algorithm for [4, p.358, Theorem 14.7].
More generalized characterization for greedoids with greedy algorithms will be studied in the future. We also hope to give more answers to the open problem stated in Section 1.

Acknowledgement

This research is supported by NNSF of China (61572011) and NSF of Hebei Province (A2018201117).

References

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Received: 2016-12-31

Accepted: 2018-02-08

Published Online: 2018-03-31

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https://doi.org/10.1515/math-2018-0026

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Interval greedoid; Exchangeable system; Greedy algorithm; Positive weight

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