For a matrix $A$ over a field of characteristic two, let $\Phi(A)$ be the sum of its permutation monomials indexed by odd permutations. Although determinant and permanent coincide in this characteristic, this parity sub-sum retains information that neither gives separately. We show that $\Phi(A)$ and all its first part...
We give a deterministic algorithm that computes the parity of the number of Hamiltonian cycles in an $n$-vertex directed graph in $O(n^4(3/2)^n)$ time and $O(n^2)$ bits of working space, improving the $O^*(\varphi^n)$ bound of Bj\"orklund and Husfeldt. Their local-degree formula reduces the problem to a weighted sum ov...
We study the weighted feedback vertex set problem in tournaments. For every fixed integer $k\geq 2$, we give a deterministic $(2+1/k)$-approximation algorithm with running time $n^{2^{O(k)}}$, apart from polynomial dependence on the encoding length of the weights. Consequently, for every fixed $\varepsilon>0$, weighted...
We give a deterministic algorithm for the parity of the number of directed Hamiltonian cycles in an $n$-vertex digraph. For $n\geq 2$, the running time is $O^*((3/2)^n)$ and the space usage is $2^{O(n/\log n)}$. The construction starts from the local-degree identity of Bj\"orklund and Husfeldt. Their surviving terms ar...
Han-Qing Li· 0 citations
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