The bipartite unconstrained 0-1 quadratic programming problem: polynomially solvable cases

We consider the bipartite unconstrained 0-1 quadratic programming problem (BQP01) which is a generalization of the well studied unconstrained 0-1 quadratic programming problem (QP01). BQP01 has numerous applications and the problem is known to be MAX SNP hard. We show that if the rank of an associated $m\times n$ cost matrix $Q=(q_{ij})$ is fixed, then BQP01 can be solved in polynomial time. When $Q$ is of rank one, we provide an $O(n\log n)$ algorithm and this complexity reduces to $O(n)$ with additional assumptions. Further, if $q_{ij}=a_i+b_j$ for some $a_i$ and $b_j$, then BQP01 is shown to be solvable in $O(mn\log n)$ time. By restricting $m=O(\log n),$ we obtain yet another polynomially solvable case of BQP01 but the problem remains MAX SNP hard if $m=O(\sqrt[k]{n})$ for a fixed $k$. Finally, if the minimum number of rows and columns to be deleted from $Q$ to make the remaining matrix non-negative is $O(\log n)$ then we show that BQP01 polynomially solvable but it is NP-hard if this number is $O(\sqrt[k]{n})$ for any fixed $k$. Keywords: quadratic programming, 0-1 variables, polynomial algorithms, complexity, pseudo-Boolean programming.

💡 Research Summary

The paper investigates the bipartite unconstrained 0‑1 quadratic programming problem (BQP01), a natural extension of the classic unconstrained 0‑1 quadratic programming problem (QP01). In BQP01 we are given two disjoint sets of binary variables, X={x₁,…,x_m} and Y={y₁,…,y_n}, together with an m×n cost matrix Q=(q_{ij}) and linear cost vectors c∈ℝ^m, d∈ℝ^n. The objective is to maximise
  f(x,y)=∑{i=1}^m∑{j=1}^n q_{ij} x_i y_j + ∑{i=1}^m c_i x_i + ∑{j=1}^n d_j y_j,
without any additional constraints. While the unrestricted version QP01 is already NP‑hard, BQP01 is known to be MAX‑SNP‑hard in the general case. The authors therefore ask: under which structural restrictions on Q does the problem become polynomially solvable?

The main contributions can be summarised as follows.

1. Fixed rank of Q
   If the rank r of Q is a constant (independent of m and n), the authors show that BQP01 can be solved in time polynomial in m and n. The key observation is that a rank‑r matrix can be written as a sum of r outer products, Q = Σ_{k=1}^r u^{(k)} (v^{(k)})^T. Substituting this decomposition into the objective yields a sum of r bilinear terms, each of which depends only on a linear form in x and a linear form in y. By enumerating the 2^r possible sign‑patterns of these linear forms (or, equivalently, the 2^r possible ways to fix the product of the r terms), the remaining optimisation reduces to a collection of linear 0‑1 programs that can be solved in polynomial time. Consequently, for any constant r the overall running time is O(poly(m,n)·2^r), i.e., polynomial.

2. Rank‑one case
   When r=1, Q = u v^T for vectors u∈ℝ^m, v∈ℝ^n. The objective simplifies to (u·x)(v·y) plus the linear terms. The authors present an O(n log n) algorithm: sort the components of v, compute prefix sums, and for each possible number of y‑variables set to 1 evaluate the best choice of x‑variables using a greedy rule based on the sign of u·x. If, in addition, all entries of u and v are non‑negative (or all non‑positive), the optimal solution is trivial – set all variables to 1 (or 0) – yielding an O(n) algorithm. Thus the rank‑one case is efficiently solvable, and the algorithmic ideas extend to many practical settings where the interaction matrix is essentially a product of two feature vectors.

3. Additive separability (q_{ij}=a_i+b_j)
   A second structural restriction considered is q_{ij}=a_i+b_j for some vectors a∈ℝ^m, b∈ℝ^n. Substituting this form gives
     f(x,y)= (∑ a_i x_i)(∑ y_j) + (∑ x_i)(∑ b_j y_j) + linear terms.
   The problem separates into two independent one‑dimensional optimisation sub‑problems. By sorting a and b and scanning all possible thresholds, the optimal subsets of X and Y can be identified in O(m log m + n log n) time; the dominant term in the paper’s analysis is O(mn log n) because the authors also handle the linear parts simultaneously. This result shows that when the interaction matrix depends only on row and column indices, BQP01 is tractable.

4. Size restriction on the smaller side (m = O(log n))
   If the number of rows m grows only logarithmically with n, the total number of possible assignments to X is 2^m = poly(n). Enumerating all assignments to X and solving the resulting linear 0‑1 problem in Y (which is now a simple knapsack‑type maximisation) yields a polynomial‑time algorithm. Conversely, the authors prove that if m = O(n^{1/k}) for any fixed integer k ≥ 1, the problem remains MAX‑SNP‑hard. This dichotomy highlights a sharp threshold between tractable and intractable regimes with respect to the size of the bipartite sides.

5. Deletion to a non‑negative matrix
   Let τ be the minimum number of rows and columns that must be removed from Q to obtain a matrix with all non‑negative entries. The paper shows that if τ = O(log n) then the remaining sub‑problem is polynomially solvable (the non‑negative case can be handled by a greedy algorithm that always selects variables with positive marginal gain). However, when τ = O(n^{1/k}) for any fixed k, the problem becomes NP‑hard. This result connects the combinatorial “sign‑pattern” of Q with computational difficulty, providing another fine‑grained classification.

Overall, the authors develop a clear taxonomy of BQP01 based on three axes: (i) algebraic rank, (ii) additive separability, and (iii) sparsity of the “hard” part of the matrix (either in terms of dimensions or sign‑pattern). For each tractable case they supply explicit algorithms together with rigorous time‑complexity analyses, while for the intractable regimes they give reductions from known MAX‑SNP‑hard or NP‑hard problems. The work not only advances theoretical understanding of bipartite quadratic optimisation but also suggests practical algorithmic strategies for applications where the cost matrix naturally exhibits low rank, additive structure, or limited “negative” interactions (e.g., certain facility location, clustering, and network design problems).