Approximation Algorithms for NP-Hard Problems

Analysis

Why This Paper Matters

This paper, published in 1997, provides a comprehensive survey of approximation algorithms for NP-hard problems, a topic of central importance in theoretical computer science and operations research. The impossibility of solving many optimization problems exactly in polynomial time has driven the development of approximation algorithms that trade optimality for efficiency. Hochba's survey systematically organizes the design techniques and inapproximability results that had emerged over the preceding three decades, making it an essential reference for both newcomers and experts.

The paper's significance is underscored by its 3217 citations, reflecting its role as a standard resource in the field. It addresses a fundamental question: how close can we get to the optimal solution in polynomial time? By covering both upper bounds (approximation algorithms) and lower bounds (inapproximability), the survey provides a balanced view of what is achievable and what is provably hard.

Technical Contributions

The survey's key contributions include:

• Classification of design techniques: It reviews methods such as greedy algorithms, local search, dynamic programming, linear programming rounding, and primal-dual approaches.
• Inapproximability results: The paper discusses how to derive lower bounds on approximation ratios, often based on the PCP theorem and other complexity-theoretic tools.
• Structural constraints: It covers algorithms that must operate online (without full knowledge of the input) or under strict memory limits, broadening the scope beyond traditional offline approximation.
• Historical perspective: The survey traces the evolution of the field from its inception to the mid-1990s, highlighting key milestones and open problems.

Results

As a survey, the paper does not present new experimental results. Instead, it summarizes known approximation guarantees for classic NP-hard problems such as the traveling salesman problem, set cover, vertex cover, and bin packing. For example, it notes that the greedy algorithm for set cover achieves an approximation ratio of O(log n), and that the vertex cover problem admits a 2-approximation via maximal matching. Inapproximability results are also cataloged, such as the fact that the traveling salesman problem cannot be approximated within any constant factor unless P = NP.

Significance

The broader impact of this survey lies in its role as a foundational text for researchers and practitioners. By organizing a vast body of knowledge, it has enabled subsequent advances in approximation algorithms and influenced the design of practical heuristics. The paper also highlights the interplay between algorithmic design and complexity theory, showing how lower bounds guide the search for better approximations. Even decades after publication, it remains a frequently cited reference, testament to its enduring value in the AI and optimization communities.