Periodic Reporting for period 3 - CRACKNP (Finding Cracks in the Wall of NP-completeness)
Reporting period: 2023-02-01 to 2024-07-31
Assuming P does not equal NP, there are no polynomial time algorithms for any NP-complete problem. This however still leaves a huge gap between anything super-polynomial and the exponential run times of trivial exhaustive search. The study of exact (exponential time) algorithms that aims to breach this gap is as old as Theoretical Computer Science (TCS) itself: Already in the 1960's, researchers found elementary (for modern standards) algorithms that greatly improve exponential the run times. But over time, TCS seems to have hit a brick wall: Somewhat embarrassingly, as of 2018 the run times of these classic algorithms are still the best known for many classic problems.
This project aims to strike at the heart of this issue by designing the next generation of exact exponential time algorithms. To obtain these algorithms, we consider the most famous NP-complete problems such as Traveling Salesman, CNF-Sat and Knapsack, and we challenge ourselves to improve the classic currently best algorithms for them.
These problems have served as a prototypical test bed for many algorithmic techniques with extensive applications, and thus their study provides an excellent road map towards our aim.
This project aims to strike at the heart of this issue by designing the next generation of exact exponential time algorithms. To obtain these algorithms, we consider the most famous NP-complete problems such as Traveling Salesman, CNF-Sat and Knapsack, and we challenge ourselves to improve the classic currently best algorithms for them.
These problems have served as a prototypical test bed for many algorithmic techniques with extensive applications, and thus their study provides an excellent road map towards our aim.
In the first half of this project, we indeed managed to give improved exponential time algorithms already for
- the (Euclidean) Traveling Salesman Problem: Most noteworthy, in STOC'20 we managed to apply algebraic techniques to include weights and give an improved algorithm for bipartite TSP (assuming matrices can be multiplied in quadratic time). Additionally, in FOCS'21 we managed to improve the runtime of the famous approximation scheme of Arora to a runtime that cannot be further improved (under a plausible hypothesis)
- the Knapsack problem: In STOC'21, we exponentially improved the space usage of the fastest worst-case algorithm by Schroeppel and Shamir from 30 years ago.
- the Bin Packing problem: In SODA'21, we gave an improved algorithm for a classic algorithm to solve the Bin Packing problem for instances with a constant number of bins.
- the (Euclidean) Traveling Salesman Problem: Most noteworthy, in STOC'20 we managed to apply algebraic techniques to include weights and give an improved algorithm for bipartite TSP (assuming matrices can be multiplied in quadratic time). Additionally, in FOCS'21 we managed to improve the runtime of the famous approximation scheme of Arora to a runtime that cannot be further improved (under a plausible hypothesis)
- the Knapsack problem: In STOC'21, we exponentially improved the space usage of the fastest worst-case algorithm by Schroeppel and Shamir from 30 years ago.
- the Bin Packing problem: In SODA'21, we gave an improved algorithm for a classic algorithm to solve the Bin Packing problem for instances with a constant number of bins.
So far, we have made significant progress on most of the fronts discussed in the proposal. The results so far have already shifted the scientific understanding of which classic algorithms can be improved using which can kind of new tools.
The foundations for the next generation of improved exact exponential time algorithms has therefore already been made.
In the remaining half of the project, we will build on these foundations to get even more significant improvements (in terms of run time and centrality of the algorithm to be improved).
The foundations for the next generation of improved exact exponential time algorithms has therefore already been made.
In the remaining half of the project, we will build on these foundations to get even more significant improvements (in terms of run time and centrality of the algorithm to be improved).