Problems

Here is another variant that is open. In this case we begin with a -1, +1 valued matrix with discrepancy n^3/2. Say a Hadamard matrix. Balancer picks certain +1's. Unbalancer picks certain -1's. Over the course of the game, can Balancer…

Does this set contain interior points within the manifold A ∈ R_+^m × n | rk A = k of nonnegative rank-k matrices?

If vecG is a best-balanced orientation of G := (V + s, E) and varrho_vecG(s) = δ_vecG(s) then there exist rs, st ∈ A(vecG) so that vecG_rt is a best-balanced orientation of G_rt.

For every graph G = (V + s, E) with d(s) ≥ 4 there exist rs, st ∈ E such that for every best-balanced orientation vecG_rt of G_rt, vecG := vecG_rt - rt + rs + st is a best-balanced orientation of G.

Let b: V → Z_0^+ be a symmetric crossing submodular function with b(∅) = 0 and b(X) ≡ |X ∩ T_b| mod 2. Then there exists a pairing M on T_b that satisfies (17).

An interesting open question is whether L(p, 1)-LABELING parameterized by only twin cover number is FPT or not.

Let d: V → Z_0^+ be a symmetric function that satisfies d(∅) = 0 and ∀ X, Y ⊆ V (19) and (20). Let hatR: V → Z_0^+ be an even valued, symmetric, skew-supermodular function. Suppose that hatR(X) ≤ d(X) ∀ X ⊆ V. Then there exists a pairing M…