Maximizing the Number of Spanning Trees in a Connected Graph

We study the problem of maximizing the number of spanning trees in a connected graph with <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> vertices and <inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> edges, by adding at most <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> edges from a given set of <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> candidate edges, a problem that has applications in many domains. We give both algorithmic and hardness results for this problem: 1) We give a greedy algorithm that obtains an approximation ratio of <inline-formula> <tex-math notation="LaTeX">$(1 - 1/e - \epsilon)$ </tex-math></inline-formula> in the exponent of the number of spanning trees for any <inline-formula> <tex-math notation="LaTeX">$\epsilon > 0$ </tex-math></inline-formula> in time <inline-formula> <tex-math notation="LaTeX">$\widetilde {O}(m \epsilon ^{-1} + (n + q) \epsilon ^{-3})$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$\widetilde {O}(\cdot)$ </tex-math></inline-formula> hides <inline-formula> <tex-math notation="LaTeX">${\mathrm{ poly}}\log (n)$ </tex-math></inline-formula> factors. Our running time is optimal with respect to the input size, up to logarithmic factors, and improves on the <inline-formula> <tex-math notation="LaTeX">$O(n^{3})$ </tex-math></inline-formula> running time of the previous proposed greedy algorithm with an approximation ratio <inline-formula> <tex-math notation="LaTeX">$(1 - 1/e)$ </tex-math></inline-formula> in the exponent. Notably, the independence of our running time of <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> is novel, compared to conventional top-<inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> selections on graphs that usually run in <inline-formula> <tex-math notation="LaTeX">$\Omega (mk)$ </tex-math></inline-formula> time. 2) We show the exponential inapproximability of this problem by proving that there exists a constant <inline-formula> <tex-math notation="LaTeX">$c > 0$ </tex-math></inline-formula> such that it is NP-hard to approximate the optimum number of spanning trees in the exponent within <inline-formula> <tex-math notation="LaTeX">$(1 - c)$ </tex-math></inline-formula>.

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