A Generalization of the Matroid Polytope Theorem to Local Forest Greedoids

We prove a theorem that generalizes an equivalent formulation of Edmonds' classic matroid polytope theorem to local forest greedoids -- a class of greedoids that contains matroids as well as branching greedoids. On our way, we prove new results on the optimality of the greedy algorithm on greedoids and correct some mistakes that have been present in the literature for almost three decades. We also describe an application of the obtained results in the field of measuring the reliability of networks by game-theoretical tools.


Introduction
Greedoids were introduced by Korte and Lovász at the beginning of the 1980s as a generalization of matroids. The motivation behind the concept was the observation that in the proofs of various results on matroids subclusiveness (that is, the property that all subsets of independent sets are also independent) is not needed. Besides matroids, the class of greedoids includes some further very important combinatorial objects such as the edge sets of subtrees of a graph rooted at a given node.
Although the research of greedoids was very active until the mid-1990s, the topic seems to have faded away since then. Most of the known results on This work is connected to the scientific program of the "Development of qualityoriented and harmonized R+D+I strategy and functional model at BME" project greedoids are already included in the comprehensive book of Korte, Lovász and Schrader [7] published in 1991. The fact that greedoids have not gained as much importance within combinatorial optimization as matroids is probably due to the fact that the class of greedoids is much more diverse than that of matroids and classic concepts and results on matroids do not seem to generalize easily to greedoids.
The motivations behind the results of this paper are twofold. Firstly, we aim at generalizing some results obtained in [10]. There we considered some attacker-defender games played on graphs with the aim of defining new security metrics of graphs and better understanding others that had been known in the literature. For this purpose, we defined a general framework involving matroids: the Matroid Base Game is a two-player, zero sum game in which the Attacker chooses an element of the ground set of a given matroid and the Defender chooses a base of the same matroid; then the payoff depends on both of their choices in such a way that it is favorable for the Attacker if his chosen element belongs to the base chosen by the Defender. The results of [10] on the Matroid Base Game served as a common generalization of some results that had been known in literature on measuring the security of networks via game-theoretical means. In particular, the Nash-equilibrium payoff of the Matroid Base Game was determined and it was proved that it is a common generalization of some known graph reliability metrics. However, there are other known metrics of a very similar nature which did not fit into the framework provided by the Matroid Base Game. In this paper we further generalize the definition of the Matroid Base Game by replacing matroids with local forest greedoids and we prove that some of the results of [10] generalize to this case too. We also show that this more general framework is capable of handling and generalizing some further graph reliability metrics known from the literature beyond the ones already contained in the framework provided by the matroid base game.
The other motivation behind the results of this paper is to identify a class of greedoids, local forest greedoids, that includes both matroids and branching greedoids and that admits a generalization of a fundamental polyhedral result on matroids: Edmonds' classic theorem on the polytope spanned by incidence vectors of independent sets of a matroid. In particular, we prove a generalization of an eqivalent formulation of this theorem to local forest greedoids. To the best of our knowledge no generalization of (any form of) the matroid polytope theorem to greedoids has been known. We do this in the (perhaps vain) hope that further fundamental results on matroids will turn out to be generalizable to this class of greedoids.
The name greedoid comes from "a synthetic blending of the words greedy and matroid" [7] which indicates that one of the basic motivations of the notion was to extend the theoretical background behind greedy algorithms beyond the well-known results on matroids. Accordingly, Korte and Lovász proved some fundamental results on the optimality of the greedy algorithm on greedoids in [5] and [6] which were also presented and further extended in [7]. Most surprisingly however, they seem to have overlooked a detail which led them to some false claims. Since the optimality of the greedy algorithm will turn out to be fundamental for proving our main result, these mistakes will be pointed out and corrections will be proposed and proved.
This paper is structured as follows. In Section 2 all the necessary background on greedoids is given. In Section 3 the main result of the paper is claimed. Section 4 is dedicated to some results on the optimality of the greedy algorithm on greedoids. These will enable us to prove the main result of the paper in Section 5. In Section 6 we briefly outline the above mentioned application concerning the measurement of the reliability of networks.

Preliminaries on Greedoids
All the definitions and claims in this section are taken from [7].
A greedoid G = (S, F) is a pair consisting of a finite ground set S and a collection of its subsets F ⊆ 2 S such that the following properties are fulfilled: (2.1) ∅ ∈ F (2.2) If X, Y ∈ F and |X| < |Y | then there exists a y ∈ Y − X such that X + y ∈ F.
Members of F are called feasible sets. Obviously, the definition of greedoids is obtained from that of matroids by relaxing subclusiveness, that is, subsets of feasible sets are not required to be feasible any more. On the other hand, (2.2) immediately implies that every X ∈ F has a feasible ordering: (x 1 , x 2 , . . . , x k ) is a feasible ordering of X if X = {x 1 , x 2 , . . . , x k } and {x 1 , x 2 , . . . , x i } ∈ F holds for every 1 ≤ i ≤ k. The existence of a feasible ordering, in turn, implies the accessible property of greedoids: for every ∅ = X ∈ F there exists an x ∈ X such that X − x ∈ F.
In this paper, the following notations will be (and have been) used: for a subset X ⊆ S and an element x ∈ S we will write X + x and X − x instead of X ∪ {x} and X − {x}, respectively. Furthermore, given any function c : S → R and a subset X ⊆ S, c(X) will stand for {c(x) : x ∈ X}.
Some of the well-known terminology on matroids can be applied to greedoids without any modification. In particular, the rank r(A) of a set A ⊆ S is r(A) = max{|X| : X ⊆ A, X ∈ F}. Given a subset A ⊆ S, a base of A is a subset X ⊆ A, X ∈ F of maximum size. This, by property (2.2) is equivalent to saying that X + y / ∈ F for every y ∈ A − X. A base of S is called a base of the greedoid G = (S, F). The set of bases of G will be denoted by B.
Minors of greedoids can also be defined almost identically to those of matroids. If G = (S, F) is a greedoid and X ⊆ S is an arbitrary subset then the deletion of X yields the greedoid Then a minor of G is obtained by applying these two operations on G. It is straightforward to check that minors are indeed greedoids. (Note, however, that G/X was only defined here in the X ∈ F case. The definition could be extended to a wider class of subsets, but unless some further structural properties are imposed on the greedoid, not to arbitrary ones. See [7,Chapter V.] for the details.) In this paper the following terminology will also be used: X ⊆ S will be called subfeasible if there exists a Y ∈ F such that X ⊆ Y . The set of subfeasible sets will be denoted by F ∨ .
There are many known examples of greedoids beyond matroids and they arise in diverse areas of mathematics, see [7] for an extensive list. For the purposes of this paper, branching greedoids will be of importance. Let H = (V, E u , E d ) be a mixed graph (that is, it can contain both directed and undirected edges) with V , E u and E d being its set of nodes, undirected edges and directed edges, respectively. Furthermore, let r ∈ V be a given root node. The ground set of the branching greedoid on H is E u ∪ E d and F consists of all subsets A ⊆ E u ∪ E d such that disregarding the directions of the arcs in A ∩ E d , A is the edge set of a tree containing r and for every path P in A starting in r all edges of P ∩ E d are directed away from r. It is straightforward to check that G = (E u ∪ E d , F) is indeed a greedoid. G is called an undirected branching greedoid or a directed branching greedoid if H is an undirected graph (that is, E d = ∅) or a directed graph (that is, E u = ∅), respectively.
Most of the known results on greedoids are about special classes of greedoids, that is, further structural properties are assumed. Among these, the following will be of relevance in this paper: Obviously, all matroids are local forest greedoids and it is easy to check that so are branching greedoids. However, there are further examples that do not belong to either of these classes: for example, the direct sum of the uniform matroid U 3,2 and a branching greedoid that is not a matroid is also a local forest greedoid but it is neither a matroid nor a branching greedoid (since all these classes are closed under taking minors and U 3,2 is clearly not a branching greedoid). Another type of example can be obtained from any local forest greedoid (even a matroid): let G = (S, F) be a local forest greedoid, X ∈ F a feasible set and (x 1 , x 2 , . . . , x k ) an arbitrary (not necessarily feasible) ordering of X; then is also a local forest greedoid on S.
Observe that in interval greedoids (2.3) implies that every X ∈ F ∨ has a unique base; indeed, it is the union of all feasible sets in X. In this paper, this unique base will be denoted by ∆(X). Analogously, (2.4) implies that if X ⊆ A and A ∈ F then there is a unique minimum size feasible set containing X in A. This gives rise to the definition of paths: if G = (S, F) is a local poset greedoid, A ∈ F and x ∈ A then the x-path in A, denoted by P A x (or simply P x if this is unambiguous) is the unique feasible set in A containing x such that no proper feasible subset of P A x contains x. Clearly, in case of branching greedoids this notion translates to paths starting in the root node r that are directed in the sense that all directed edges in the path are directed away from r. The following theorem was proved in [8]; we also give a simple proof here for the sake of self-containedness. (i) G is a local forest greedoid (that is, it fulfills (2.5)); (ii) every path in G has a unique feasible ordering; (iii) if (a 1 , a 2 , . . . , a k ) is the feasible ordering of a path then {a 1 , a 2 , . . . , a i } is also a path for every 1 ≤ i ≤ k.
Proof. Assume by way of contradiction that (i) is fulfilled but (ii) is not and choose a path P z of minimum size that has two different feasible orderings: (a 1 , . . . , a k , x, z) and (b 1 , . . . , b k , y, z). We first show that x = y can be assumed without loss of generality. Indeed, if x = y then {a 1 , . . . , a k , x} is not a path by the minimality of |P z |, but since it is feasible, it contains an x-path P x as a proper subset. Then augmenting a feasible ordering of P x from P z by (2.2) we get a feasible ordering of P z the second to last element of which is not y. So assume x = y and let A = P z − {x, y, z}. Then clearly A + x ∈ F and A + y ∈ F hold by the feasibility of the two orderings and A ∪ {x, y, z} = P z ∈ F. Hence by (2.3) and (2.4) we have A ∪ {x, y} ∈ F and A ∈ F too. Therefore (2.5) implies A ∪ {x, z} ∈ F or A ∪ {y, z} ∈ F. In both cases we get a smaller feasible set containing z than P z contradicting the definition of P z . Proving (iii) from (ii) is almost immediate: if {a 1 , a 2 , . . . , a i } were not a path then it would contain an a i -path by definition which could be augmented by (2.2) from P = {a 1 , . . . , a k } to obtain a different feasible ordering of P .
Finally, we show (i) from (iii). Let A and x, y, z be given according to (2.5) and let B = A ∪ {x, y, z}. Clearly, if A ∩ {x, y, z} = ∅ or |{x, y, z}| < 3 then (2.5) is automatically fulfilled, so we can assume that neither of these is the case. Since A + x ∈ F, we have P B x ⊆ A + x and hence y / ∈ P B x . Similarly, x / ∈ P B y . This, by (iii), implies that P B z contains at most one of x and y; indeed, if it contained both and, for example, x preceded y in a feasible ordering of P B z then since the prefix of this ordering up to y would be a path by (iii), we would get x ∈ P B y . So assume y / ∈ P B z without loss of generality. Then applying (2.3) on A + x and P B z , both of which are subsets of B ∈ F, we get A ∪ {x, z} ∈ F as claimed.

Main Result
The classic matroid polytope theorem of Edmonds [4] is the following.
Theorem 2. Let M = (S, F) be a matroid with rank function r and let P ind (M ) denote the polytope spanned by the incidence vectors of all independent sets of M . Then The theorem has some equivalent formulations, the one of relevance for the purposes of this paper is the following. The up-hull of a polyhedron P ⊆ R n , denoted by P ↑ , is defined as P ↑ = {x ∈ R n : ∃z ∈ P, z ≤ x}; in other words, P ↑ is the Minkowski-sum of P and the non-negative orthant of R n (and as such, it is also a polyhedron).
As claimed above, this theorem is just a reformulation of Theorem 2. Indeed, by applying Theorem 2 to the dual of a matroid one gets a description of the polytope spanned by the incidence vectors of all spanning sets (that is, sets containing a base of M ); then it is easy to check that this polytope is nothing but the intersection of P ↑ base (M ) and the hypercube [0, 1] S . The details are given in [9,Chapter 40.2].
The shadow polytope P shadow (G) of G is defined as the polytope spanned by the shadow vectors of all bases of G.
M is a matroid then its shadow polytope is nothing but its base polytope P base (M ) (since sh A (x) = 1 for every x ∈ A). Therefore the following theorem is indeed a direct generalization of Theorem 3.
Theorem 5. Let G = (S, F) be a local forest greedoid with rank function r. Then The proof will be given in Section 5.

Optimality of the Greedy Algorithm in Greedoids
Let G = (S, F) be an arbitrary greedoid and w : F → R an objective function. Assume that we are interested in finding a base B ∈ B that maximizes w(B) across all bases of G.
Then the greedy algorithm for the above problem can be described as follows [5,7]: Step 1. Set A = ∅.
Step 2. If Γ(A) = ∅ then stop and output A.
Step 3. Choose an x ∈ Γ(A) such that w(A + x) ≥ w(A + y) for every y ∈ Γ(A).
Step 4. Replace A by A + x and continue at Step 2.
Obviously, if one is interested in minimizing w(B) across all bases then, since this is equivalent to maximizing −w(B), the only modification needed in the algorithm is to require w(A + x) ≤ w(A + y) for every y ∈ Γ(A) in Step 3.
Many of the well-known, elementary algorithms in graph theory fall under this framework as shown by the following examples.
Example 6. If M is a matroid and w is linear (meaning that w(A) = c(A) for some weight function c : S → R) then the above greedy algorithm is nothing but the well-known greedy algorithm on matroids. In particular, we get Kruskal's algorithm for finding a maximum weight spanning tree in case of the cycle matroid.
Example 7. Let G be the branching greedoid of the undirected graph H and w a linear objective function. Then the greedy algorithm translates to Prim's well-known algorithm for finding a maximum weight spanning tree. (Note that this algorithm cannot be interpreted in a matroid-theoretical context.) Example 8. Let G be the branching greedoid of the mixed graph H = (V, E u , E d ) with root node r and let c : E u ∪ E d → R + be a non-negative valued weight function. Then let w(A) = {c(P A e ) : e ∈ A} for every A ∈ F. Korte and Lovász observed [5] that in this case the greedy algorithm for minimizing w(B) translates to Dijkstra's well-known shortest path algorithm. Indeed, Dijkstra's algorithm constructs a spanning tree on the set of nodes reachable from r such that the unique path from r to every other node in this tree is a shortest path and hence it clearly minimizes w.
Although the greedy algorithm finds an optimum base in the above examples, it is obviously not to be expected that this is true in general. The first sufficient condition for the optimality of the greedy algorithm was given by Korte and Lovász in [5]. There they introduced an even broader framework: they considered objective functions defined on all feasible orderings of feasible sets. Given a greedoid G = (S, F), let L(F) denote the set of all feasible orderings of all feasible sets. Extending the greedy algorithm to the case of an objective function w : L(F) → R is obvious: instead of augmenting a feasible set A ∈ F, it keeps maintaining and updating a feasible ordering of A that is always augmented by the best possible choice x. . Let G = (S, F) be an arbitrary greedoid and w : L(F) → R an objective function. Assume that whenever (a 1 , . . . , a i , x) is a feasible ordering of a set A + x ∈ F (where i = 0 is possible) such that w ((a 1 , . . . , a i , x)) ≥ w ((a 1 , . . . , a i , y)) for every y ∈ Γ(A) then the following conditions hold: if both of these strings are in L(F) (and j = 0 or k = 0 is possible).
if both of these strings are in L(F) (and j = 0 or k = 0 is possible).
Then the greedy algorithm finds a maximum base with respect to w.
Since in most applications the objective function only depends on the feasible sets themselves and not on their orderings, one would want to formulate the corresponding corollary of Theorem 9. Obviously, (4.2) is automatically fulfilled in these cases, however, it is not at all straightforward to specialize (4.1) to such objective functions. Both in [5] and [7, Chapter XI] it is claimed that for objective functions w : F → R (4.1) is equivalent to the following: Unfortunately, as innocuous as the above mistake might look, it led the authors of [7] to the following false claim (see [7, page 156] x ∈ A} for a c : S → R + analogously to Example 8, then the greedy algorithm finds a minimum base with respect to w. To disprove this, let S = {x, y, z, u}, Then it is easy to check that (S, F) is a local poset greedoid, but since the greedy algorithm starts with choosing y, it terminates with {x, y, z} which is not minimum as w({x, y, z}) = 10 and w({x, z, u}) = 9.
Moreover, it is worth noting that while the optimality of Dijkstra's algorithm does follow from Theorem 9 for directed graphs, it does not follow in the undirected case as shown by the example of Figure 2: although x is the best continuation of the empty set, 11 = w((x, b, a)) > w((z, b, a)) = 10, hence (−w) violates (4.1). On the other hand, it was shown in [1] that the greedy algorithm does indeed minimize the objective function w(A) = {c(P A x ) : x ∈ A} for all non-negative valued weight functions c : S → R + in local forest greedoids. We will prove a generalization of this result below, see Theorem 13.
We start with the following theorem which seems to be new, but its proof is just an adaptation of a result of Korte and Lovász [6], [7, Theorem XI.2.2] on the optimality of the greedy algorithm in case of linear objective functions.
Theorem 10. Let G = (S, F) be an arbitrary greedoid and w : F → R an objective function that fulfills the following property: Then the greedy algorithm gives a maximum base with respect to w.
Proof. Assume by way of contradiction that the greedy algorithm gives the base B g = {a 1 , a 2 , . . . , a r } choosing the elements in this order, but B g is not maximum with respect to w. Choose a maximum base B m with respect to w such that max{i : a 1 , . . . , a i ∈ B m } is maximum possible, let this maximum be k and A = {a 1 , . . . , a k }. Then A ∈ F, A ⊆ B m , a k+1 / ∈ B m and w(A + a k+1 ) ≥ w(A + u) for every u ∈ Γ(A) by the operation of the greedy algorithm. Therefore, by (4.4), there exists a y ∈ B m − A such that B m −y +a k+1 ∈ B and w(B m −y +a k+1 ) ≥ w(B m ). Therefore B m −y +a k+1 is also a maximum base with respect to w, but {a 1 , . . . , a k , a k+1 } ⊆ B m −y+a k+1 contradicts the choice of B m .
It is worth noting that, in spite of its simplicity, the above theorem implies the optimality of the greedy algorithm in all three examples listed at the beginning of this section. This is easy to check in case of Examples 6 and 7 and in case of Example 8 it will follow from the results below. Furthermore, it is not too hard to show that Theorem 10 also implies Theorem 9 in case of objective functions w : F → R that are independent of the ordering. (This could be proved by an argument similar to that of Theorem 12 below, we omit the details here.) Moreover, Theorem 10 is in a sense best possible as shown by the following theorem. To claim the theorem, we need to extend the formerly given definition of minors of greedoids to incorporate modifying the objective function w G : F → R in an obvious way: in case of a deletion G \ X w G is simply restricted to S −X, while in case of a contraction G/X the modified objective function becomes w G/X (A) = w G (A ∪ X).
The following theorem will be weaker than Theorem 10 -not only because it applies to interval greedoids only, but also because it will not cover Example 7 (or the case of linear objective functions in general). However, it can also be regarded as a corrected version of (4.3) and it will be easier to work with later on.
Recall that ∆(X) denotes the unique base of a subfeasible set X ∈ F ∨ in interval greedoids.
Theorem 12. Let G = (S, F) be an interval greedoid and w : F → R an objective function that fulfills the following property: B+z ∈ B such that ∆(B)∪{x, z} / ∈ F it holds that w(A+x) ≥ w(A+u) for every u ∈ Γ(A) then w(B + x) ≥ w(B + z).
Then the greedy algorithm gives a maximum base with respect to w.
Proof. We will show that (4.5) implies (4.4) which will obviously settle the proof by Theorem 10. So let A, B and x be given such that A ⊆ B, A, A+x ∈ F, B ∈ B and w(A + x) ≥ w(A + u) for every u ∈ Γ(A). We need to show the existence of a y ∈ B − A according to (4.4).
Let (b 1 , . . . , b k ) be a feasible ordering of A and, using (2.2), augment this to get a feasible ordering (b 1 , . . . , b k , b k+1 , . . . , b r ) of B. Denote B 0 = ∅ and B i = {b 1 , . . . , b i } for every 1 ≤ i ≤ r. Let t ∈ {1, . . . , r} be the largest index such that B t−1 + x ∈ F. Obviously, t exists and t ≥ k + 1 since B k + x = A + x ∈ F. Now set y = b t ; we claim that this is a suitable choice for (4.4).
Trivially, y ∈ B−A by t ≥ k+1. To show B−y+x ∈ F, augment B t−1 +x from B t+1 ; then augment the obtained feasible set from B t+2 and continue like this until a base is obtained. Then b t can never occur as an augmenting element during this process by the choice of t which implies B − y + x ∈ F as claimed.
Let C = B − y. We claim that ∆(C) ∪ {x, y} / ∈ F, so assume the opposite towards a contradiction. Since B t−1 ∈ F and B t−1 ⊆ C, we have B t−1 ⊆ ∆(C). Furthermore, B t−1 +x ∈ F by the choice of t and B t−1 +y = B t ∈ F is also true. Since B t−1 +x, B t−1 +y ⊆ ∆(C)∪{x, y}, B t−1 ∪{x, y} = B t +x ∈ F follows by the local union property (2.3). This either contradicts the choice of t if t < r or the fact that B is a base if t = r.
Consequently, since we have C + y = B ∈ B, C + x = B − y + x ∈ B and w(A + x) ≥ w(A + u) for every u ∈ Γ(A), we get w(C + x) ≥ w(C + y) from (4.5), which concludes the proof by C + x = B − y + x and C + y = B.
The next theorem will serve as a theoretical background for the optimality of Dijkstra's algorithm according to Example 8.
Theorem 13. Let G = (S, F) be a local forest greedoid, P its set of paths and f : P → R a function that satisfies the following monotonicity constraints: Finally, let w(A) = {f (P x ) : x ∈ A} for every A ∈ F. Then the greedy algorithm gives a minimum base with respect to w.
We will need the following lemma for proving the above theorem. Now we are ready for proving Theorem 13. The proof follows the argument of [7, page 156] where they showed that property (4.3) is fulfilled by a similarly defined objective function w in local poset greedoids. As mentioned above, that was insufficient for guaranteeing the optimality of the greedy algorithm, however, a similar argument will work well with Theorem 12.
Proof of Theorem 13. We will show that (4.5) is fulfilled by (−w). So let A, B, x and z given such that Since ∆(B) ∈ F, we have Proof. Immediate from Theorem 13 as f (P ) = c(P ) obviously fulfills the monotonicity constraints (i) and (ii).
Another application of Theorem 13 is to set f (P ) = max{c(x) : x ∈ P } for a weight function c : S → R, which again obviously fulfills conditions (i) and (ii). Theorem 13 implies the fact, which was also proved in [1], that in local forest greedoids the greedy algorithm finds a minimum base with respect to w in this case. If applied to the branching greedoid (and for maximizing (−w)), this implies the well-known fact that the corresponding modification of Dijkstra's algorithm solves the widest path problem (also known as the bottleneck shortest path problem) in graphs.

Proof of the Main Result
To prepare the proof of Theorem 5, we need the following lemmas.
Lemma 16 (Local Supermodularity Property). If G = (S, F) is a local poset greedoid then r(A) Proof. Let X = ∆(A) and Y = ∆(B). Then X ∩ Y ∈ F by the local intersection property. Furthermore, since for every feasible set Z ⊆ A ∩ B, Z ∪ X ∈ F and Z ∪ Y ∈ F by the local union property, Z ⊆ X ∩ Y must hold by the definition of ∆(A) and ∆(B). Therefore X ∩ Y = ∆ (A ∩ B). Finally, since X ∪ Y ∈ F is also true by the local union property, we have Note that the above local supermodularity property also characterizes local poset greedoids among all greedoids since it implies both the local intersection and the local union properties if applied to feasible sets.  Therefore all vertices of P shadow (G) are in Q which implies P shadow (G) ⊆ Q. Consequently, P ↑ shadow (G) ⊆ Q ↑ = Q. It can happen that P ↑ shadow (G) is a proper subset of Q in the above proposition as shown by the example already seen in Section 4: let S = {a, b, c, d} and F = {∅, {a}, {b}, {a, b}, {a, d}, {a, b, c}, {a, c, d}}. Then G = (S, F) is a local poset greedoid, the shadow vectors of its two bases are (2, 2, 1, 0) and (3, 0, 2, 2) (if the elements are arranged in alphabetical order), both of which fulfill x a + x b + x c ≥ 5, hence this inequality is fulfilled by every member of P ↑ shadow (G). However, (2, 1, 1, 1) ∈ Q is easy to check which shows that Q − P ↑ shadow (G) = ∅. As already clamied above, the main result of the paper will be that ⊆ can be replaced by = in Proposition 18 in case of local forest greedoids. The proof will follow the argument of Edmonds' original proof of Theorem 2: the greedy algorithm will be used to construct an optimum dual solution. However, it should be noted that the construction we give below is not an extension of that of Edmonds: even if applied to matroids it gives a different optimum dual solution. In particular, Edmonds' construction (even if adapted to prove Theorem 3, which can easily be done) yields a chain of subsets of the ground set which is not true for the construction given below.
Theorem 19. Let G = (S, F) be a local forest greedoid, |S| = n, c : S → R + a non-negative valued weight function, w(A) = {c(P A x ) : x ∈ A} for every A ∈ F and B m a minimum base with respect to w. Then there exist the subsets U 1 , U 2 , . . . , U n ⊆ S and corresponding values y(U 1 ), y(U 2 ), . . . , y(U n ) such that y(U i ) ≥ 0 for all 1 ≤ i ≤ n, {y(U i ) : x ∈ U i } = c(x) holds for every x ∈ S and n i=1 (r(S) − r(S − U i )) · y(U i ) = w(B m ). Proof. Assume that a running of the greedy algorithm gives the base B = {s 1 , s 2 , . . . , s r } choosing the elements in this order and let B 1 = ∅ and B i = {s 1 , s 2 , . . . , s i−1 } for every 2 ≤ i ≤ r. Let S − B = {s r+1 , . . . , s n } with the elements ordered arbitrarily. Finally, denote P 0 = ∅ and P i = P B s i for every 1 ≤ i ≤ r. Then let We prove that the above choice of U i and y(U i ) fulfills all requirements of the theorem through a series of claims. Claim 1. Let x ∈ r i=1 U i and j = min{i : x ∈ U i }. Then P B j +x x = P j−1 +x.
Proof. If j = 1 then P B j +x x = {x} and thus the claim is obvious, so assume j ≥ 2 and hence |P ∈ F follows from the local union property. This implies s j−1 ∈ P follows from the definition of a path. The second to last element in the unique ordering of P B j +x x is obviously s j−1 otherwise s j−1 ∈ P t ⊆ B t+1 would follow from Theorem 1 for some t < j − 1, a contradiction. Therefore P Proof. From x ∈ U j ∩ U k we have B j + x ∈ F and B k + x ∈ F which, by the local union property, imply B i + x ∈ F and therefore x ∈ U i for every j ≤ i ≤ k as claimed. Consequently, Proof. If s i ∈ U i−1 for some 2 ≤ i ≤ r then c(P i−1 ) ≤ c(P i ) is implied by the fact that the greedy algorithm could have chosen s i instead of s i−1 . If, on the other hand, s i / ∈ U i−1 then P i = P i−1 + s i follows from Claims 1 and 2. Hence c(P i ) = c(P i−1 ) + c(s i ) which proves the claim. ♦ Proof. Let first x = s t for some 1 ≤ t ≤ r. Then y(U t ) ≥ 0 is immediate from Claim 3. Define j and k as in Claim 3. Then k = t is obvious and Claim 1 gives P t = P j−1 + s t . Therefore from Claim 2 we have {y(U i ) : on the other hand, s t ∈ r i=1 U i then again define j and k as in Claim 3. Since the greedy algorithm could have chosen x instead of s k by x = s t ∈ U k , we have c(P B k +x x ) ≥ c(P k ). This, together with Claims 1 and 2 implies hence we have the claim by the definitions of U i and y(U i ). ♦ Claim 5.
where the last equation follows from Corollary 15.
As a final piece of preparation for proving our main result we need the following lemma.
Lemma 20. Let G = (S, F) be a local poset greedoid, A ∈ F and c : S → R a weight function. Then x∈A c(x) · sh A (x) = x∈A c(P A x ).
Proof. We claim that y ∈ P A x if and only if x ∈ A − ∆(A − y) for every x . The converse follows from the local union property: since P A x , ∆(A − y) ∈ F, P A x , ∆(A − y) ⊆ A, P A x ∪ ∆(A − y) ∈ F holds and thus y ∈ P A x by the definition of ∆(A − y). Then the lemma follows by

Now we are ready for the
Proof of Theorem 5. Let P = P ↑ shadow (G) for short. By Proposition 18 we To show equality it suffices to prove that min{cx : x ∈ P } = min{cx : x ∈ Q} holds for every c ∈ R S , c ≥ 0. (Indeed, since P ↑ = P holds, P can be written in the form P = {x : Ax ≥ b} for some matrix A ≥ 0. If a z ∈ Q − P existed then z would violate a constraint cx ≥ δ of Ax ≥ b and hence min{cx : x ∈ P } > min{cx : x ∈ Q} would follow.) So let a c ∈ R S , c ≥ 0 be fixed, let w(A) = {c(P A x ) : x ∈ A} for every A ∈ F and B m a minimum base with respect to w. Using Lemma 20 and since min{cx : x ∈ P shadow (G)} is attained on a vertex of P shadow (G) and P shadow (G) ⊆ P ⊆ Q, we get Theorem 19 implies that this maximum is at least w(B m ), which in turn implies that every inequality in (5.1) is fulfilled with equation and hence concludes the proof. Proof. It follows from the proof Theorem 5 that the minimum of the primal problem is attained on the shadow vector of a base of G which is obviously integer. Furthermore, the construction of the proof of Theorem 19 yields an integer optimum solution of the dual problem if c is integer.
Corollary 22. If G = (S, F) is a local forest greedoid then the system is totally dual integral.
Proof. Immediately from Corollary 21 after observing that the minimum of the primal program obviously does not exist if c contains a negative component.
We remark that no similar description of P shadow (G) is to be hoped for, not even for branching greedoids. Indeed, it follows from Lemma 20 that maximizing a linear objective function over P shadow (G) translates to maximizing e∈E(H) c(P e ) which is, as it was pointed out in [7,Chapter XI.], NP-hard as it contains the Hamilton path problem. Therefore the existence of such a description of P shadow (G) would imply that, for example, the Hamilton path problem is in co-NP, which is highly unlikely.

An Application: Reliability of Networks via Game Theory
The problem of measuring the robustness or reliability of a graph arises in many applications. The most widely applied reliability metrics are obviously the connectivity based ones, however, these are unsuitable in many casesfor example because in many applications the network is almost completely functional if removing some nodes or links results in the loss of only a small number of nodes that are in some sense insignificant or peripheral. Applying game-theoretical tools for measuring the reliability of a graph has become very common. The basic idea is very natural: define a game between two virtual players, the Attacker and the Defender, such that the rules of the game capture the circumstances under which reliability is to be measured. Then analyzing the game might give rise to an appropriate security metric: the better the Attacker can do in the game, the lower the level of reliability is. This kind of analysis can give rise to new graph reliability metrics and in some cases it can shed a new light on some well-known ones.
To illustrate this, consider the following Spanning Tree Game: a connected, undirected graph G, a positive valued damage function d : E(G) → R + and a cost function c : E(G) → R are given. For each edge, d(e) represents the "damage" caused by the loss of e (or in other words, the "importance" of e) and c(e) represents the cost of attacking e. The Attacker chooses (or "attacks and destroys") an edge e of G and the Defender (without knowing the Attacker's choice) chooses a spanning tree T of G (that she intends to use as some kind of "communication infrastructure"). Regardless of the Defender's choice, the Attacker has to pay the cost of attack c(e) to the Defender. There is no further payoff if e / ∈ T . If, on the other hand, e ∈ T then the Defender pays the Attacker the damage value d(e). Since this game is a two-player, zero-sum game, it has a unique Nash-equilibrium payoff (or, in simpler terms, game value) V by Neumann's classic Minimax Theorem. Since V is the highest expected gain the Attacker can guarantee himself by an appropriately chosen mixed strategy (that is, probability distribution on the set of edges), it makes sense to say that 1 V is a valid reliability metric. After some preliminary results on some special cases in the literature (see [11] for the details), the Spanning Tree Game was solved in the above defined general form in [10]. In fact, it was considered there in a more general, matroidal setting: the Matroid Base Game was defined analogously to the Spanning Tree Game with the only difference being that the Attacker chooses an element of the ground set of a matroid M = (S, F) and the Defender chooses a base B of M . Then the following result was proved.
Theorem 23 ( [10]). For every input of the Matroid Base Game the game value is where p(s) = 1 d(s) and q(s) = c(s) d(s) for all s ∈ S. Furthermore, if M is given by an independence testing oracle then there exists a strongly polynomial algorithm that computes the game value of the Matroid Base Game and an optimum mixed strategy for both players.
If specialized to the Spanning Tree Game and to the c ≡ 0 case, the above theorem implies that the game value is the reciprocal of a well-known graph reliability metric: the strength of a graph is defined as σ p (G) = min p(U ) comp(G−U )−1 : U ⊆ E(G), comp(G − U ) > 1 , where comp(G − U ) is the number of components of the graph obtained from G by deleting U and p : E(G) → R + is a weight function. This notion was defined in the weighted case and its computability in strongly polynomial time was proved in [3].
While the Matroid Base Game has further relevant applications beyond the Spanning Tree Game (see [10]), there are other types of games of a similar nature which do not fit into this framework. The following Rooted Spanning Tree Game was considered in [2]: a (mixed) graph H with a "headquarters" node r is given such that every node is reachable from r. (The role of r can be that all other nodes need to communicate with r only, for example to transmit some collected data to r.) Furthermore assume that a cost function c : E(H) → R is also given. Again, the Attacker chooses an edge e, the Defender chooses a spanning tree T and the cost of attack c(e) is payed by the Attacker to the Defender in all cases and there is no further payoff if e / ∈ T . However, if e ∈ T then the payoff from the Defender to the Attacker is the number of nodes that become unreachable from r in T after removing e. Since this number is nothing but the shadow sh T (e) in case of the branching greedoid, the definition of the Local Forest Greedoid Base Game presents itself: given a local forest greedoid G = (S, F) and weight functions d, c ∈ R S with d > 0, the Attacker chooses an element s ∈ S, the Defender chooses a base B of G and then the payoff from the Defender to the Attacker is d(s) · sh B (s) − c(s). Clearly, this game is a direct generalization of the Matroid Base Game mentioned above. Then, using Theorem 5 and following the proof of [10, Theorem 5] we can prove the following. for all s ∈ S. Then the vector x ∈ R S is nothing but an element of P shadow (G) by definition (since the values δ(B) form the set of coefficients of a convex combination). Since, by definition, the Defender's objective is to minimize the maximum expected loss she has to suffer, her task amounts to the following by (6.1): min µ : ∃x ∈ P shadow (G), d(s) · x(s) − c(s) ≤ µ for all s ∈ S . (6.2) In other words, the minimum in (6.2) is equal to V by Neumann's Minimax Theorem. Rearranging (6.2): V = min µ : ∃x ∈ P shadow (G), x ≤ µ · p + q}.
Using the definition of P ↑ shadow (G) this is further equivalent to the following: V = min µ : µ · p + q ∈ P ↑ shadow (G)}. holds for all U ⊆ S. Then simple rearranging (and observing that this inequality is trivial for U = ∅) immediately gives that µ · p + q ∈ P ↑ shadow (G) is true if and only if Hence V , the minimum of all such µ's is exactly this maximum.
If specialized to the branching greedoid and to the c ≡ 0 case it follows that the value of the Rooted Spanning Tree Game is the reciprocal of another known graph reliability metric, also defined in [3]. Interested readers are referred to [11] for the details. Furthermore, the above theorem also generalizes the first statement of Theorem 23. However, generalizing the algorithmic statement of Theorem 23 to the Local Forest Greedoid Base Game is left as an open problem.