# Determinant Sums for Undirected Hamiltonicity

@article{Bjrklund2014DeterminantSF,
title={Determinant Sums for Undirected Hamiltonicity},
author={Andreas Bj{\"o}rklund},
journal={SIAM J. Comput.},
year={2014},
volume={43},
pages={280-299}
}
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 the traveling salesman problem (TSP) over 50 years ago [R. Bellman, J. Assoc. Comput. Mach., 9 (1962), pp. 61--63], [M. Held and R. M. Karp, J. Soc. Indust. Appl. Math., 10 (1962), pp. 196--210]. ($O^*(f(n… Bipartite TSP in o(1.9999ⁿ) time, assuming quadratic time matrix multiplication A fast algorithm for MinHamPair is given based on a new sparse cut-based factorization of the ‘matchings connectivity matrix’, introduced by Cygan et al. An Asymptotically Fast Polynomial Space Algorithm for Hamiltonicity Detection in Sparse Directed Graphs A polynomial space Monte Carlo algorithm that given a directed graph on$n$vertices and average outdegree$\delta$, detects if the graph has a Hamiltonian cycle in$2^{n-\Omega(\frac{n}{2^\delta})}$time, which matches the fastest known exponential space algorithm by Bjorklund and Williams ICALP 2019. Quantum Speedups for Exponential-Time Dynamic Programming Algorithms • Computer Science, Mathematics SODA • 2019 This paper gives a quantum algorithm that solves path in the hypercube in time and combines Grover's search with computing a partial dynamic programming table, and uses this approach to solve a variety of vertex ordering problems on graphs in the same time. Hamiltonicity below Dirac's condition • Mathematics WG • 2019 The results extend the range of tractability of the Hamiltonian cycle problem, showing that it is fixed-parameter tractable when parameterized below a natural bound and for the first parameterization it is shown that a kernel with$O(k)$vertices can be found in polynomial time. A Simple Algorithm for Hamiltonicity • Computer Science, Mathematics ArXiv • 2014 A new algebraic technique is developed that solves the problem of deciding whether a black box that contains an arithmetic circuit, expressed as an equivalent multivariate polynomial, contains a multilinear monomial of degree d. Computing Permanents and Counting Hamiltonian Cycles by Listing Dissimilar Vectors • Computer Science, Mathematics ICALP • 2019 We show that the permanent of an n x n matrix over any finite ring of r <= n elements can be computed with a deterministic 2^{n-Omega(n/r)} time algorithm. This improves on a Las Vegas algorithm Spotting Trees with Few Leaves • Mathematics ICALP • 2015 It is demonstrated that the iterated random bipartition employed by the algorithm can be improved whenever the host graph admits a vertex coloring with few colors; it can be an ordinary proper vertex coloring, a fractional graph coloring, or a vector coloring. Approximating Asymmetric TSP in exponential Time A very simple algorithm is proposed that, for any 0 < e < 1, finds (1+e)-approximation to asymmetric TSP in 2ne−1 time and e−1 · poly(n, log M) space. Finding the Minimum-Weight k-Path • Mathematics, Computer Science WADS • 2013 Given a weighted n-vertex graph G with integer edge-weights taken from a range [−M,M], we show that the minimum-weight simple path visiting k vertices can be found in time$\tilde{O}(2^k