Applications of Discrepancy Theory in Multiobjective Approximation

We apply a multi-color extension of the Beck-Fiala theorem to show that the multiobjective maximum traveling salesman problem is randomized 1/2-approximable on directed graphs and randomized 2/3-approximable on undirected graphs. Using the same technique we show that the multiobjective maximum satisfiablilty problem is 1/2-approximable.

💡 Research Summary

The paper presents a novel framework for approximating Pareto sets in multi‑objective combinatorial optimization, focusing on three canonical problems: the k‑objective maximum asymmetric traveling salesman problem (k‑MaxATSP), the k‑objective maximum symmetric traveling salesman problem (k‑MaxSTSP), and the k‑objective maximum weighted satisfiability problem (k‑MaxWeightedSAT). Each problem is defined on a complete graph (or a Boolean formula) whose edges (or clauses) carry k‑dimensional non‑negative weights, and the goal is to find a Hamiltonian cycle (or a truth assignment) that simultaneously maximizes all k objectives.

Traditional single‑objective approximation algorithms achieve ratios such as 1/2 for Max‑ATSP, 2/3 for Max‑STSP, and 7/9 for Max‑SAT, but these do not directly extend to the multi‑objective setting because solutions optimal for one objective can be arbitrarily bad for another. Consequently, a naïve “enumerate several solutions and pick the best” approach fails to provide a meaningful approximation of the Pareto front.

The authors overcome this difficulty by leveraging discrepancy theory, specifically a multi‑color extension of the Beck‑Fiala theorem due to Doerr and Srivastava. Their algorithm consists of two main stages.

Stage 1 – Fixed‑edge cycle‑cover approximation.
For a given instance they first select a small set F of edges (size ℓ, where ℓ depends only on k) by brute force. They then solve a cycle‑cover problem that is required to contain all edges of F. This is the k‑2‑MaxDCC F problem for directed graphs and the k‑3‑MaxUCC F problem for undirected graphs. Prior work (Manth­ey & Ram, 2009) provides a fully polynomial‑time randomized approximation scheme (FPRAS) for the underlying unrestricted cycle‑cover problems; the authors show how to adapt the FPRAS to the fixed‑edge variant, preserving the approximation guarantee.

Stage 2 – Multi‑color discrepancy rounding.
From each cycle in the obtained cover they pick a constant number of candidate edges (two per cycle for directed graphs, three per cycle for undirected graphs). These candidate edges form a matrix A whose rows correspond to the k objectives and whose columns correspond to the possible “colors” (i.e., which candidate edge will be removed). Applying the multi‑color Beck‑Fiala theorem yields a coloring χ such that the discrepancy ‖A(p − χ)‖∞ is bounded by 2‖A‖₁, independent of the number of cycles. In concrete terms, the coloring guarantees that, for each objective i, the total weight of the edges removed from the cycles deviates from the ideal half‑(or one‑third‑) share by at most a factor that translates into a loss of at most 1/2 of the optimal weight in the directed case and at most 1/3 in the undirected case.

After coloring, exactly one edge is removed from each cycle (the edge whose color is selected), and the remaining paths are arbitrarily linked to form a Hamiltonian cycle. Because the discrepancy bound holds simultaneously for all k objectives, the resulting Hamiltonian cycle is a randomized 1/2‑approximation of the Pareto set for k‑MaxATSP and a randomized 2/3‑approximation for k‑MaxSTSP. The success probability is at least 1/2, matching the definition of a randomized α‑approximation algorithm used in the paper.

For k‑MaxWeightedSAT, the authors apply an analogous idea. For each clause they consider the two complementary truth assignments (setting the clause to true or false) and treat the choice of assignment as a “color”. Using the same discrepancy rounding, they guarantee that the total weight of satisfied clauses in the chosen assignment is at least half of the optimum, yielding a 1/2‑approximation for the multi‑objective weighted SAT problem.

The paper improves upon the previously best known results for multi‑objective TSP, which were randomized (1/2 − ε) for the asymmetric case and (2/3 − ε) for the symmetric case (Manth­ey, 2009). By eliminating the ε‑loss, the authors achieve the optimal constants known for the single‑objective versions, now extended to the multi‑objective setting via a clean, general technique.

Complexity analysis shows that the brute‑force enumeration of the edge set F requires O(n^ℓ) time, where ℓ = O(k) is a constant for fixed k, and the subsequent FPRAS and discrepancy rounding run in polynomial time in the input size. Hence the overall algorithm runs in polynomial time for any fixed number of objectives.

Finally, the authors argue that their approach—combining fixed‑edge cycle‑cover approximations with multi‑color discrepancy rounding—is a versatile template that could be applied to other multi‑objective combinatorial problems where a set of “candidate modifications” can be represented as a matrix with bounded column sums. This opens a promising research direction for leveraging discrepancy theory in the design of multi‑objective approximation algorithms.