Biobjective optimization with M-convex functions

In this paper, we deal with two ingredients that, as far as we know, have not been combined until now: multiobjective optimization and discrete convex analysis. First, we show that the entire Pareto optimal value set can be obtained in polynomial time for biobjective optimization problems with discrete convex functions, in particular, involving an M$^\natural$-convex function and a linear function with binary coefficients. We also observe that a more efficient algorithm can be obtained in the special case where the M$^\natural$-convex function is M-convex. Additionally, we present a polynomial-time method for biobjective optimization problems that combine M$^\natural$-convex function minimization with lexicographic optimization.

💡 Research Summary

The paper pioneers the integration of multi‑objective optimization with discrete convex analysis, focusing on bi‑objective problems where one objective is an M♮‑convex (or M‑convex) function and the other is a linear function with binary coefficients. Traditional multi‑objective combinatorial optimization suffers from NP‑hardness and the exponential growth of non‑supported Pareto points, making the enumeration of the full Pareto frontier infeasible in general. The authors overcome this barrier by exploiting the exchange axioms that characterize M‑convexity and M♮‑convexity.

The core contribution is a polynomial‑time algorithm that enumerates the entire Pareto optimal value set for the problem
 min (g(x), ⟨b,x⟩) subject to x ∈ dom g,
where g: ℤᴱ → ℝ∪{+∞} is M♮‑convex and b: E → {0,1}. The method proceeds by fixing the binary linear objective ⟨b,x⟩ to each possible value k (0 ≤ k ≤ rank(M)) and solving a series of single‑objective M♮‑convex minimization problems under an equality constraint ⟨b,x⟩ = k. This reduction mirrors the “color‑induced budget” formulation used for matroids, but the authors generalize it to arbitrary M♮‑convex functions by invoking Takazawa’s result that such constrained minimizations are solvable in polynomial time via evaluated matroid intersection. Consequently, each k yields the optimal set Tₖ, and the union of all Tₖ constitutes the full Pareto value set.

When the first objective is an M‑convex function—a stricter subclass—the algorithm can be refined further. M‑convexity guarantees that the feasible domain itself is an M‑convex set, allowing stronger exchange operations and reducing the number of required oracle calls. The authors present an improved complexity bound for this special case, demonstrating that the Pareto frontier can be obtained even more efficiently.

The paper also addresses a hybrid setting where the second objective is not merely linear but a lexicographic (lex) ordering. By applying the ε‑constraint method with the lexicographic cone as the ordering structure, the authors transform the bi‑objective problem into a series of lex‑constrained M♮‑convex minimizations. They prove that solutions to these transformed problems are Pareto optimal for the original formulation. Moreover, they devise a polynomial‑time algorithm based on evaluated matroid intersection that directly enumerates the lexicographically ordered Pareto values without any post‑processing filtering.

Complexity analysis shows that each subproblem (for a fixed k) can be solved in O(poly(|E|, log U)) time, where U bounds the magnitude of g’s values. Since the number of distinct k values is at most |E|+1, the total running time remains polynomial in the input size and the numeric range of the objective functions. This is a stark contrast to the exponential blow‑up typical of non‑supported Pareto points in generic combinatorial settings.

Beyond theoretical development, the authors discuss practical implications. Many real‑world problems—such as network flow cost minimization, logistics planning, and bicycle‑sharing system design—naturally involve M‑convex cost structures. The presented algorithms enable decision makers to obtain the complete trade‑off surface between a convex cost and a binary resource usage metric, facilitating informed multi‑criteria choices.

In conclusion, the paper establishes that for bi‑objective problems combining an M♮‑convex (or M‑convex) function with a binary linear objective, the entire Pareto optimal value set is polynomially tractable. It further extends this tractability to lexicographic secondary objectives. Future work suggested includes extending the approach to more than two objectives, developing approximation schemes for larger-scale instances, and conducting extensive computational experiments to validate the methods in real applications.