Periodic Reporting for period 4 - CRACKNP (Finding Cracks in the Wall of NP-completeness)
Reporting period: 2024-08-01 to 2025-01-31
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 the Traveling Salesman Problem (TSP), 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.
During the project, we made considerable progress on this challenge and delivered more efficient algorithms for these classical problems such as faster approximation algorithms for Euclidean TSP, a faster exact algorithm for Bipartite TSP and a more space efficient algorithm for Knapsack. Many of these results break through a long-standing barrier and provided new algorithmic tools and structural insights on the hardness of NP-complete 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 the Traveling Salesman Problem (TSP), 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.
During the project, we made considerable progress on this challenge and delivered more efficient algorithms for these classical problems such as faster approximation algorithms for Euclidean TSP, a faster exact algorithm for Bipartite TSP and a more space efficient algorithm for Knapsack. Many of these results break through a long-standing barrier and provided new algorithmic tools and structural insights on the hardness of NP-complete problems.
We indeed managed to give improved exponential time algorithms for, among others:
- 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 graph coloring problem and the computation of the Tutte polynomial: we determined its complexity on graphs with small cutwidth in STACS’22 and ESA’23.
- the independent set problem: we determined its parameterized by rank width in STACS’23 and the complexity of approximating the optimal independent set size parameterized by treewidth in ESA’23
- the Graph Homomorphism problem: In ICALP’24 we defined a parameter of the pattern graph P that (likely) determines the complexity of the problem of detecting a homomorphism to P in a graph G with small cutwidth
- a classic scheduling problem with precedence constraints: in SODA’25 we showed that a classical scheduling problem can be solved in sub-exponential time, which is the first progress on a major open question known since the 1980’s. This was a surprising result that came out of an attempt to solve the problem in 1.9999^n time.
- 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 graph coloring problem and the computation of the Tutte polynomial: we determined its complexity on graphs with small cutwidth in STACS’22 and ESA’23.
- the independent set problem: we determined its parameterized by rank width in STACS’23 and the complexity of approximating the optimal independent set size parameterized by treewidth in ESA’23
- the Graph Homomorphism problem: In ICALP’24 we defined a parameter of the pattern graph P that (likely) determines the complexity of the problem of detecting a homomorphism to P in a graph G with small cutwidth
- a classic scheduling problem with precedence constraints: in SODA’25 we showed that a classical scheduling problem can be solved in sub-exponential time, which is the first progress on a major open question known since the 1980’s. This was a surprising result that came out of an attempt to solve the problem in 1.9999^n time.
We have made significant progress on most of the fronts discussed in the proposal. The results have 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.
The foundations for the next generation of improved exact exponential time algorithms has therefore already been made.