Bicriteria Approximation Algorithms

• Bicriteria approximation algorithms are defined as frameworks that allow bounded violations of one criterion to achieve provable guarantees on another, enabling effective trade-offs.
• They employ methods like black-box parametric search, clustering techniques, and dynamic programming on bounded-treewidth graphs to balance conflicting objectives.
• These approaches are crucial when strict budget or structural constraints make single-criterion approximations impractical, as seen in network design and related optimization problems.

A bicriteria approximation algorithm is an algorithmic framework for optimization problems involving two objectives or constraints, where the algorithm intentionally allows bounded violation in one criterion to enable provable guarantees on another. This paradigm is relevant in combinatorial optimization and theoretical computer science, especially for network design and partitioning, clustering, covering, and submodular maximization, where hard budget constraints or structural requirements make true (single-criterion) approximations unattainable or impractical. The bicriteria framework accommodates solutions that “slack” the feasibility requirement(s) by a controlled factor β in order to improve the approximation guarantee α on the objective, formalizing quantitative trade-offs between quality and constraint violation.

1. Formal Problem Definition and Bicriteria Guarantee

In a prototypical bicriteria network design scenario, one is given a graph with two minimization objectives under different cost functions, and a budget B on the first objective. The goal is to find a solution (e.g., subgraph from a specified class) that minimizes the second objective while respecting (possibly approximately) the first budget constraint. The bicriteria approximation framework is defined as follows:

• Bicriteria Guarantee: An algorithm is said to achieve an (α, β)-approximation if, for every instance, it outputs a feasible solution X such that
  • $f_1(X) \leq \alpha \cdot B$ (budget violation on the first criterion by at most factor α), and
  • $f_2(X) \leq \beta \cdot OPT$ (objective is within factor β of the true optimum for solutions satisfying $f_1(X) \leq B$).

This framework further generalizes to settings with more complex side constraints (e.g., degree, diameter, connectivity, clustering radius) and solution classes (e.g., subgraphs, trees, clusterings).

2. General Methodologies for Bicriteria Approximation

Three principal algorithmic methodologies for bicriteria network design optimization are developed:

• Reduction to Single-Objective Algorithms via Parametric Search: When the two objectives are structurally identical (though potentially with different cost functions), the central idea is to use a “black-box” parametric search technique. Any $(\gamma)$-approximation for the single-criterion version can be transformed into a bicriteria $(O(\gamma), O(1))$-approximation for the two-criterion version, typically incurring only a constant-factor loss.
• Clustering Methods for Heterogeneous Criteria: For objectives such as minimizing both diameter and total cost, clustering-based constructions are employed. For instance, partitioning the vertex set into clusters and building low-diameter, low-cost interconnections can achieve polylogarithmic bicriteria approximation ratios for both criteria.
• Dynamic Programming and Scaling on Bounded Treewidth Graphs: In restricted graph classes (e.g., those with bounded treewidth), dynamic programming enables pseudopolynomial-time exact or near-exact bicriteria solutions. These pseudopolynomial-time algorithms can be converted into fully polynomial-time approximation schemes using scaling techniques.

3. Black-box Parametric Search Technique for Identical Criteria

In the setting where both objectives are of the same type (e.g., total cost but under different edge cost functions), the core strategy proceeds as follows:

• Parameterize the feasible region over all possible budgets for the first weight function.
• For each such budget, invoke a single-criterion approximation algorithm for the second cost.
• Use binary or parametric search to identify the budget bound yielding the best bicriteria trade-off.

This methodology guarantees that if the original single-criterion algorithm achieves an $\alpha$-approximation, the resulting bicriteria algorithm achieves at most a constant-factor increase in the approximation ratio (e.g., $(2\alpha,2)$-approximation). The computational overhead is dominated by the number of parametric search iterations.

4. Clustering-Based Approach for Diameter vs. Cost

When the two objectives are essentially different—such as minimizing both the network diameter and total edge cost—the black-box approach falters, and more elaborate strategies are needed. A central approach is:

• Clustering: Partition the nodes into clusters, then connect centers of the clusters to minimize the inter-cluster diameter and the added cost.
• The solution output by this approach typically violates both criteria by a logarithmic factor; for instance, it yields a $(O(\log n), O(\log n))$-bicriteria approximation where $n$ is the number of nodes.

This approach leverages the structure of spanners or low-stretch trees to balance the cost and the diameter, but such logarithmic violations are known to be necessary in some regimes.

5. Bicriteria Approximation via Dynamic Programming on Bounded Treewidth

For graphs with bounded treewidth, more powerful bicriteria guarantees can be achieved by:

• Formulating the problem as a dynamic program over the tree-decomposition of the graph, tracking the feasible configurations for both objectives.
• This yields a pseudopolynomial-time bicriteria algorithm for a variety of network design problems (e.g., cost-diameter, cost-degree).
• The pseudopolynomial dependence on the input values (such as the budget or diameter) can be removed using a scaling technique, resulting in an FPTAS for the bicriteria formulation.

This is a sharp contrast to the general case: bicriteria approximation in bounded treewidth settings can be made arbitrarily close to optimal with respect to both criteria.

6. Impact, Optimality, and Limitations

The bicriteria approximation framework broadens the range of tractable problems by allowing limited, quantifiable violations in secondary objectives or constraints, facilitating practical algorithm design with formal guarantees where single-criterion approaches are infeasible. Notably:

• For two identical criteria with different cost functions, the black-box parametric search achieves a constant-factor optimality loss, and no further improvement is possible unless P=NP.
• With heterogeneous criteria (e.g., diameter vs. cost), the clustering approach’s logarithmic bicriteria guarantees are unimprovable in the worst case, as demonstrated by matching lower bounds.
• Dynamic programming and scaling techniques are optimal for bounded-treewidth graphs, yielding FPTAS-level bicriteria approximations.

A plausible implication is that the bicriteria paradigm can be extended and adapted to an even wider spectrum of network optimization problems, including survivable network design, spanner constructions, and low-stretch embeddings, provided appropriate relaxations of the strict constraints are acceptable in the application domain.

References:

• "Bicriteria Network Design Problems" [9809103]