System of unbiased representatives for a collection of bicolorings

Abstract

Let B denote a set of bicolorings of [n], where each bicoloring is a mapping of the points in [n] to {−1, +1}. For each B ∈ B, let YB = (B(1), . . . , B(n)). For each A ⊆ [n], let XA ∈ {0, 1} n denote the incidence vector of A. A non-empty set A is said to be an 'unbiased representative' for a bicoloring B ∈ B if XA, YB = 0. Given a set B of bicolorings, we study the minimum cardinality of a family A consisting of subsets of [n] such that every bicoloring in B has an unbiased representative in A.

The Balanced Connected Subgraph Problem

“…In [2], Aichholzer et al considered the balanced island problem and devised polynomial algorithms for points considered on plane. From combinatorial side, Balanchandran et al [4] studied the problem of unbiased representatives in a set of bicolorings. In this paper, they have mentioned the usefulness of the unbiased representatives in drug testing.…”