Determinant Sums for Undirected Hamiltonicity (1008.0541v1)

Abstract: We present a Monte Carlo algorithm for Hamiltonicity detection in an $n$-vertex undirected graph running in $O^*(1.657^{n})$ time. To the best of our knowledge, this is the first superpolynomial improvement on the worst case runtime for the problem since the $O^*(2^n)$ bound established for TSP almost fifty years ago (BeLLMan 1962, Held and Karp 1962). It answers in part the first open problem in Woeginger's 2003 survey on exact algorithms for NP-hard problems. For bipartite graphs, we improve the bound to $O^*(1.414^{n})$ time. Both the bipartite and the general algorithm can be implemented to use space polynomial in $n$. We combine several recently resurrected ideas to get the results. Our main technical contribution is a new reduction inspired by the algebraic sieving method for $k$-Path (Koutis ICALP 2008, Williams IPL 2009). We introduce the Labeled Cycle Cover Sum in which we are set to count weighted arc labeled cycle covers over a finite field of characteristic two. We reduce Hamiltonicity to Labeled Cycle Cover Sum and apply the determinant summation technique for Exact Set Covers (Bj\"orklund STACS 2010) to evaluate it.

• The paper introduces a Monte Carlo algorithm that detects Hamiltonicity in undirected graphs using determinant sums, markedly improving previous exponential bounds.
• It employs algebraic sieving and labeled cycle cover sums to achieve O(1.657^n) time on general graphs and O(1.414^n) on bipartite ones.
• The approach offers practical benefits for network routing and TSP adaptations, advancing computational methods for NP-hard problems.

Determinant Sums for Undirected Hamiltonicity

The paper "Determinant Sums for Undirected Hamiltonicity" by Andreas Björklund presents a novel and efficient Monte Carlo algorithm for detecting Hamiltonicity in an undirected graph with n vertices, exhibiting a significant reduction in computational complexity compared to previous methods. Traditional approaches for solving Hamiltonicity, a well-recognized NP-hard problem, such as those by BeLLMan, Held, and Karp, operate in O(n^2) time. This paper, however, introduces an algorithm that achieves time complexity of O(1.657^n) for general graphs and O(1.414^n) for bipartite graphs—significant improvements over the longstanding exponential base of approximately two.

Algorithm and Techniques

The author innovatively applies several concepts to develop this performance gain, including the algebraic sieving method used in the k-Path problem and determinant summation techniques from Exact Set Covers. A crucial aspect of the approach is reducing the problem of Hamiltonicity to the Labeled Cycle Cover Sum, which is then evaluated using determinants, effectively accounting for exact set covers.

The Monte Carlo algorithm ensures no false positives, and the probability of false negatives is exponentially small in n. For graphs with specific characteristics, such as bipartite graphs, the algorithm's efficiency is further improved to O(2^n-i) time where i represents an independent set size.

Implications and Impact

The implications of this work are notable both theoretically and practically. Theoretically, the research answers an open question from Woeginger's 2003 survey on exact algorithms for NP-hard problems, refining the understanding of possible time complexity bounds for Hamiltonian detection. This contributes to ongoing discussions and investigations into the Exponential Time Hypothesis and feasible improvements of algorithms for NP-hard problems.

Practically, the development supports more efficient solutions to problems related to network routing, computational biology, and circuit design, where Hamiltonicity needs to be examined as a subroutine. The potential for practical applications is expanded by the paper's provision that the Hamiltonicity algorithm can be adapted for the Traveling Salesman Problem (TSP) with integer weights through self-reducibility.

Future Directions

Considering the improvements achieved, this work propounds further research into approximate algorithms and potential modifications to address broader TSP instances or apply the inclusion–exclusion principles in different combinatorial contexts. Further exploration into deterministic solutions that utilize insights from this approach could enhance computational strategies, ultimately achieving similar efficiencies without reliance on probabilistic methods.

In summary, Björklund's paper represents a significant advancement in the field of algorithmic graph theory, offering both a detailed technical contribution and an effective tool for practical computational problems in combinatorics and related domains. It advances the discourse on exact solutions to NP-hard problems and opens avenues for further research on the optimization of such methods.