[Paper Review] Balanced Combinations of Solutions in Multi-Objective Optimization

Christian Glaßer, Christian Reitwießner+1 more|arXiv (Cornell University)|Jul 30, 2010

Complexity and Algorithms in Graphs16 references3 citations

TL;DR

This paper introduces a novel balancing technique for multi-objective optimization using topological degree theory to partition vector sequences into intervals that nearly cancel out across multiple objectives. It achieves a deterministic 1/2-approximation for multi-objective maximum weighted satisfiability and a randomized 1/2-approximation for multi-objective maximum asymmetric TSP, improving upon prior results with simpler algorithms.

ABSTRACT

For every list of integers x_1, ..., x_m there is some j such that x_1 + ... + x_j - x_{j+1} - ... - x_m \approx 0. So the list can be nearly balanced and for this we only need one alternation between addition and subtraction. But what if the x_i are k-dimensional integer vectors? Using results from topological degree theory we show that balancing is still possible, now with k alternations. This result is useful in multi-objective optimization, as it allows a polynomial-time computable balance of two alternatives with conflicting costs. The application to two multi-objective optimization problems yields the following results: - A randomized 1/2-approximation for multi-objective maximum asymmetric traveling salesman, which improves and simplifies the best known approximation for this problem. - A deterministic 1/2-approximation for multi-objective maximum weighted satisfiability.

Motivation & Objective

• To develop a method for balancing k-dimensional integer vectors using k alternations between addition and subtraction, enabling polynomial-time computation of balanced solutions.
• To apply this balancing result to multi-objective optimization problems where conflicting objectives prevent a single optimal solution.
• To design efficient approximation algorithms for multi-objective variants of MaxATSP and MaxSAT, achieving a 1/2-approximation with simpler algorithms than prior work.
• To demonstrate that the Pareto set of multi-objective problems admits a polynomial-size (1−ε)-approximation, enabling efficient approximation even when the full Pareto set is exponentially large.

Proposed method

• The core method uses the Odd Mapping Theorem from topological degree theory, discretized to handle integer vectors, to prove the existence of k disjoint intervals whose vector sums nearly cancel across all components.
• For a sequence of k-dimensional vectors, the method guarantees that the sum of vectors inside the intervals minus the sum outside differs by at most 4kz in each component, where z bounds the vector entries.
• The algorithm exhaustively searches over all possible combinations of k intervals (for fixed dimension) to find a balanced partition, enabling polynomial-time computation when k is constant.
• In MaxSAT, the method assigns truth values iteratively to variables, maintaining a partial assignment and using the balancing result on remaining unsatisfied clauses to improve the approximation.
• For MaxATSP, the algorithm extends the classical cycle cover approach by applying the balancing result to edge weights in multiple objectives, ensuring balanced trade-offs.
• The key innovation lies in reducing the multi-objective approximation problem to a discrete balancing problem, which is then solved via exhaustive search over interval combinations.

Experimental results

Research questions

• RQ1 Can a sequence of k-dimensional integer vectors be partitioned into k intervals such that the sum of vectors inside the intervals nearly cancels the sum outside in all components?
• RQ2 Is there a polynomial-time computable method to balance conflicting objectives in multi-objective optimization problems with k criteria?
• RQ3 Can the balancing result be applied to derive improved approximation algorithms for multi-objective MaxATSP and MaxSAT?
• RQ4 Does the existence of a balanced partition imply a 1/2-approximation for multi-objective MaxSAT and MaxATSP, even when the Pareto set is exponentially large?
• RQ5 Can the topological degree theory result be discretized and applied effectively to combinatorial optimization problems with multiple objectives?

Key findings

• The paper proves that for any sequence of k-dimensional integer vectors bounded by z, there exists a partition into k intervals such that the imbalance in each component is at most 4kz.
• A randomized 1/2-approximation algorithm is developed for the k-objective maximum asymmetric TSP, improving upon previous randomized approximations that were worse than 1/2.
• A deterministic 1/2-approximation is achieved for the multi-objective maximum weighted satisfiability problem, simplifying and improving upon prior approaches.
• The algorithm for MaxSAT uses a truth assignment strategy that combines iterative variable selection with interval balancing on remaining clauses, ensuring a 1/2-approximation guarantee.
• The method demonstrates that the 1/2-approximation is achievable not only for MaxATSP but also for MaxSAT, despite the complexity of handling multiple objectives simultaneously.
• The results show that the balancing framework enables efficient approximation even when the full Pareto set is intractable to compute, providing a general tool for multi-objective optimization.