Notable Resolved Problems in Computational Complexity Theory

Computational complexity theory explores the inherent difficulty of computational problems, particularly their relationship to resources such as time and space. Over the past few decades, several significant problems have been resolved, offering profound insights into the nature of computation. Let's delve into a few of these notable breakthroughs.

SL L

In the realm of computational complexity, the resolution of SL L stands out as a remarkable achievement. Introduced in 2008 by Omer Reingold, this breakthrough provides a proof that the class of problems solvable by a nondeterministic log-space Turing machine (SL) is identical to the class of problems solvable in deterministic log-space (L).

The problem in question, Undirected s-t Connectivity (USTCON), is central to this resolution. USTCON is the problem of determining whether there exists a path between two vertices in an undirected graph. This problem is known to be complete for the complexity class SL, meaning that any problem in SL can be reduced to USTCON. The resolution of SL L implies that such problems can be solved efficiently using very limited space, specifically logarithmic space, on a deterministic Turing machine. This is a highly surprising and elegant result, though it may not have immediate practical applications.

Undirected Connectivity in Log-Space

The Journal of the ACM published a seminal paper on this topic, detailing the proof that USTCON can be solved in log-space, marking a significant milestone in computational complexity. This proof not only establishes the equivalence of SL and L but also opens up new avenues for understanding the limits of computational space.

Integer Multiplication in Time O(n log n)

In 2021, David Harvey and Joris van der Hoeven achieved another milestone by demonstrating that the task of multiplying n-bit integers can be accomplished in time O(n log n). This result is notable because it represents a significant improvement over previous methods, which had been conjectured to be as efficient since 1971.

This groundbreaking work builds upon extensive research by mathematicians and computer scientists over several decades. The previous best-known algorithm for integer multiplication was O(n log n log log n), developed in 2007 by Martin Fürer. However, the new approach offers a simpler and more efficient solution, reducing the complexity to O(n log n). Although this result is theoretically profound, it is also important to note that practical implementation challenges remain.

Graph Isomorphism and the Sensitivity Conjecture

The Graph Isomorphism (GI) problem, which involves determining whether two graphs are structurally identical, has been a longstanding challenge in computational complexity. In 2015-2016, László Babai made a significant breakthrough by showing that the GI problem can be solved in quasi-polynomial time. Specifically, his algorithm runs in time exp(O((log n)^1)), marking a substantial improvement over the previous best-known algorithm from 1982.

Babai's algorithm is based on a novel approach called "group-theoretic" techniques, which have been instrumental in the development of efficient algorithms for GI. However, whether GI can be solved in polynomial time (P) still remains an open question. The quasi-polynomial time result is a significant leap forward but does not fully resolve the complexity classification of this problem.

A separate breakthrough in 2019 resolved the sensitivity conjecture, a long-standing conjecture in theoretical computer science. Formulated in 1992 by Hao Huang, the sensitivity conjecture asserts that for a Boolean function, the maximum sensitivity is bounded by the degree of the function. In 2019, Huang provided a two-page proof of this long-standing conjecture, which has profound implications for the theory of Boolean functions and complexity.

Conclusion

These resolved problems in computational complexity theory not only represent significant theoretical advances but also open up new avenues for research and practical application. From the equivalence of SL and L to the efficient multiplication of large integers and the resolution of the sensitivity conjecture, each of these breakthroughs contributes to a deeper understanding of the fundamental limits of computation.

Key Terms

Computational Complexity, SL L, Graph Isomorphism, Sensitivity Conjecture, Integer Multiplication