[Paper Review] Quasi-concave functions on antimatroids

Vadim E. Levit, Yulia Kempner|arXiv (Cornell University)|Aug 26, 2004

Rough Sets and Fuzzy Logic3 references3 citations

TL;DR

This paper establishes a duality between quasi-concave set functions and monotone linkage functions on k-truncated antimatroids. It proves that every quasi-concave function on such a structure can be represented as the minimum of a monotone linkage function, and conversely, that the minimum of any monotone linkage function yields a quasi-concave function—provided the underlying set system satisfies the k-truncated interval property without upper bounds, which characterizes k-truncated antimatroids.

ABSTRACT

In this paper we consider quasi-concave set functions defined on antimatroids. There are many equivalent axiomatizations of antimatroids, that may be separated into two categories: antimatroids defined as set systems and antimatroids defined as languages. An algorthmic characterization of antimatroids, that considers them as set systems, was given in (Kempner, Levit 2003). This characterization is based on the idea of optimization using set functions defined as minimum values of linkages between a set and the elements from the set complement. Such set functions are quasi-concave. Their behavior on antimatroids was studied in (Kempner, Muchnik 2003), where they were applied to constraint clustering. In this work we investigate a duality between quasi-concave set functions and linkage functions. Our main finding is that quasi-concave set functions on an antimatroid may be represented as minimum values of some monotone linkage functions.

Motivation & Objective

• To establish a duality between quasi-concave set functions and monotone linkage functions on truncated antimatroids.
• To characterize the class of set systems for which every minimum of a monotone linkage function yields a quasi-concave function.
• To show that the set of monotone linkage functions defining a given quasi-concave function forms a semilattice under pointwise minimum.
• To prove that the k-truncated interval property without upper bounds is necessary and sufficient for the duality to hold.

Proposed method

• Define quasi-concave set functions on k-truncated antimatroids using the meet operation via basis intersection: F(X ∧ Y) ≥ min{F(X), F(Y)}.
• Introduce monotone linkage functions π satisfying X ⊆ Y ⇒ π(x,X) ≥ π(x,Y) for all x ∈ E.
• Represent a quasi-concave function F as F(X) = min_{x ∈ Γ(X)} π(x,X), where Γ(X) is the set of feasible continuations of X.
• Prove that any such minimum function is quasi-concave using the k-truncated interval property and basis properties.
• Construct a canonical linkage function π_F from F via π_F(x,X) = max_{A ∈ [X, E−x]_F_{k−1}} F(A), showing it defines F and is minimal among such functions.
• Demonstrate that the set of all such linkage functions forms a semilattice under pointwise minimum, with π_F as the null element.

Experimental results

Research questions

• RQ1 Can every quasi-concave function on a k-truncated antimatroid be represented as the minimum of a monotone linkage function?
• RQ2 Is the k-truncated interval property without upper bounds both necessary and sufficient for the minimum of any monotone linkage function to be quasi-concave?
• RQ3 What is the algebraic structure of the set of monotone linkage functions that represent a given quasi-concave function?
• RQ4 How does the canonical linkage function π_F relate to other linkage functions defining the same F?
• RQ5 What is the role of the basis operator B(X) in characterizing feasible continuations and ensuring the duality holds?

Key findings

• Every quasi-concave function F on a k-truncated antimatroid can be expressed as F(X) = min_{x ∈ Γ(X)} π(x,X) for some monotone linkage function π.
• The canonical linkage function π_F defined by π_F(x,X) = max_{A ∈ [X, E−x]_F_{k−1}} F(A) satisfies F(X) = min_{x ∈ Γ(X)} π_F(x,X), and is the minimal such function.
• The set of all monotone linkage functions defining a given F forms a semilattice under pointwise minimum, with π_F as the least element.
• The k-truncated interval property without upper bounds is both necessary and sufficient for the minimum of any monotone linkage function to be quasi-concave.
• If the k-truncated interval property fails, there exists a monotone linkage function whose minimum is not quasi-concave, demonstrating the necessity of the condition.
• The example with E = {1,2}, F = 2^E, and π(2,∅) = 2, π(x,X) = 1 otherwise, shows that π_F ≠ π even when F is constant, illustrating that π_F is not unique but minimal.