Irreducible Width 2 Posets of Linear Discrepancy 3

Abstract

The linear discrepancy of a poset P is the least k such that there is a linear extension L of P such that if x and y are incomparable in P,Tannenbaum, Trenk, and Fishburn characterized the posets of linear discrepancy 1 as the semiorders of width 2 and posed the problem for characterizing the posets of linear discrepancy 2. Howard et al. (Order 24:139-153, 2007) showed that this problem is equivalent to finding all posets of linear discrepancy 3 such that the removal of any point reduces the linear discrepancy. In this paper we determine all of these minimal posets of linear discrepancy 3 that have width 2. We do so by showing that, when removing a specific maximal point in a minimal linear discrepancy 3 poset, there is a unique linear extension that witnesses linear discrepancy 2.

When linear and weak discrepancy are equal

“…However, we can exploit the structure of elements of W d to provide a more natural description of the class as interval orders. This characterization of W d as a collection of interval orders joins with results such as the forbidden subposet characterization of posets with linear discrepancy at most 2 [7,8], the NP-completeness of linear discrepancy [4], and the behavior of online algorithms for linear discrepancy [9] in emphasizing the centrality of interval orders in the study of linear and weak discrepancy.…”

“…In light of Theorem 2, rather than attempting to explicitly characterize all posets for which linear and weak discrepancy are the same, we follow the work in [1,7,8] and determine essential characteristics of posets with equal linear and weak discrepancy. To that end, we recall that a poset P is d-linear-discrepancy-irreducible if ld(P) = d and for any x ∈ P we have ld(P − x) < d. We define being d-weak-discrepancy-irreducible analogously.…”

“…In this paper we answer a question of Fishburn et al [11] and characterize the tight examples. More precisely, we expand upon the idea of irreducibility with respect to linear discrepancy, introduced in [1] and expanded upon in [7,8], to define and characterize the class of irreducible posets with equal linear and weak discrepancy.…”

Degree bounds for linear discrepancy of interval orders and disconnected posets

“…A poset P is k-discrepancy irreducible if ld (P) = k and ld (P − {x}) < k for any x ∈ P. This concept has been used in [10,11] to provide, together with the work of Tanenbaum et al in [15], a complete forbidden subposet characterization of posets with linear discrepancy at most two. However, without Theorem 6, it is not immediate that linear discrepancy irreducibility is analogous to dimension irreducibility.…”