The isoperimetric number of random regular graphs
European Journal of Combinatorics
The traveling salesman problem with distances one and two
Mathematics of Operations Research
Proof verification and the hardness of approximation problems
Journal of the ACM (JACM)
A threshold of ln n for approximating set cover
Journal of the ACM (JACM)
On bounded occurrence constraint satisfaction
Information Processing Letters - Special issue analytical theory of fuzzy control with applications
Non-approximability results for optimization problems on bounded degree instances
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Some optimal inapproximability results
Journal of the ACM (JACM)
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
On Some Tighter Inapproximability Results (Extended Abstract)
ICAL '99 Proceedings of the 26th International Colloquium on Automata, Languages and Programming
STACS '00 Proceedings of the 17th Annual Symposium on Theoretical Aspects of Computer Science
On the Edge-Expansion of Graphs
Combinatorics, Probability and Computing
8/7-approximation algorithm for (1,2)-TSP
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Journal of Computer and System Sciences
On the complexity of approximating k-set packing
Computational Complexity
On The Approximability Of The Traveling Salesman Problem
Combinatorica
Optimal Inapproximability Results for MAX-CUT and Other 2-Variable CSPs?
SIAM Journal on Computing
Vertex cover might be hard to approximate to within 2-ε
Journal of Computer and System Sciences
The Steiner tree problem on graphs: Inapproximability results
Theoretical Computer Science
Approximation hardness of dominating set problems in bounded degree graphs
Information and Computation
Approximation hardness for small occurrence instances of NP-hard problems
CIAC'03 Proceedings of the 5th Italian conference on Algorithms and complexity
An O(log n/ log log n)-approximation algorithm for the asymmetric traveling salesman problem
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Beating the Random Ordering Is Hard: Every Ordering CSP Is Approximation Resistant
SIAM Journal on Computing
A Randomized Rounding Approach to the Traveling Salesman Problem
FOCS '11 Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
On the approximability of Dodgson and Young elections
Artificial Intelligence
IEEE Transactions on Information Theory
Steiner Tree Approximation via Iterative Randomized Rounding
Journal of the ACM (JACM)
Improved inapproximability results for the shortest superstring and related problems
CATS '13 Proceedings of the Nineteenth Computing: The Australasian Theory Symposium - Volume 141
Improved Approximation for 3-Dimensional Matching via Bounded Pathwidth Local Search
FOCS '13 Proceedings of the 2013 IEEE 54th Annual Symposium on Foundations of Computer Science
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After two decades of progress in hardness of approximation we finally completely understand the extent to which many optimization problems can be approximated in polynomial time. Unfortunately, however, and despite significant efforts, many important problems continue to resist such an understanding. One example is the famous Traveling Salesman Problem, for which the best currently known hardness of approximation bounds are strongly believed to be quite far from the truth. In this article, we describe the main tools and techniques used in the currently best known approximation lower bounds for this problem. Among them are expander-like graph constructions called amplifiers and bounded-occurrence instances of standard constraint satisfaction problems. We also discuss how these ideas could be (modestly) improved, how (and whether) they may prove useful in eventually resolving the problem and what ingredients are still missing.