Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: This study strengthens the links beten Mean Payoff Games (\MPG{s}) and Energy Games (EG{s}). Firstly,  offer a faster $O(|V|^2|E|W)$ pseudo-polynomial time and $\Theta(|V|+|E|)$ space deterministic algorithm for solving the Value Problem and Optimal Strategy Synthesis in \MPG{s}. This improves the best previously known estimates on the pseudo-polynomial time complexity to: \[ O(|E|\log |V|) + \Theta\Big(\sum_{v\in V}\texttt{deg}_{\Gamma}(v)\cdot\ell_{\Gamma}(v)\Big) = O(|V|^2|E|W), \] where $\ell_{\Gamma}(v)$ counts the number of times that a certain energy-lifting operator $\delta(\cdot, v)$ is applied to any $v\in V$, along a certain sequence of Value-Iterations on reighted \EG{s}; and $\texttt{deg}_{\Gamma}(v)$ is the degree of $v$. This improves significantly over a previously known pseudo-polynomial time estimate, i.e. $\Theta\big(|V|^2|E|W + \sum_{v\in V}\texttt{deg}_{\Gamma}(v)\cdot\ell_{\Gamma}(v)\big)$ \citep{CR15, CR16}, as the pseudo-polynomiality is now confined to depend solely on $\ell_\Gamma$. Secondly,  further explore on the relationship beten Optimal Positional Strategies (OPSs) in \MPG{s} and Small Energy-Progress Measures (SEPMs) in reighted \EG{s}. It is observed that the space of all OPSs, $\texttt{opt}_{\Gamma}\Sigma^M_0$, admits a unique complete decomposition in terms of extremal-SEPM{s} in reighted EG{s}. This points out what  called the "Energy-Lattice $\mathcal{X}^*_{\Gamma}$ associated to $\texttt{opt}_{\Gamma}\Sigma^M_0$". Finally, it is offered a pseudo-polynomial total-time recursive procedure for enumerating (w/o repetitions) all the elements of $\mathcal{X}^*_{\Gamma}$, and for computing the corresponding partitioning of $\texttt{opt}_{\Gamma}\Sigma^M_0$.
['Carlo Comin', 'Romeo Rizzi']
I am writing a research paper on a idea Faster O(|V|^2|E|W)-Time Energy Algorithms for Optimal Strategy Synthesis in Mean Payoff Games
[{"from": "human", "value": "I am writing a research paper on a idea Faster O(|V|^2|E|W)-Time Energy Algorithms for Optimal Strategy Synthesis in Mean Payoff Games"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: This study strengthens the links beten Mean Payoff Games (\\MPG{s}) and Energy Games (EG{s}). Firstly,  offer a faster $O(|V|^2|E|W)$ pseudo-polynomial time and $\\Theta(|V|+|E|)$ space deterministic algorithm for solving the Value Problem and Optimal Strategy Synthesis in \\MPG{s}. This improves the best previously known estimates on the pseudo-polynomial time complexity to: \\[ O(|E|\\log |V|) + \\Theta\\Big(\\sum_{v\\in V}\\texttt{deg}_{\\Gamma}(v)\\cdot\\ell_{\\Gamma}(v)\\Big) = O(|V|^2|E|W), \\] where $\\ell_{\\Gamma}(v)$ counts the number of times that a certain energy-lifting operator $\\delta(\\cdot, v)$ is applied to any $v\\in V$, along a certain sequence of Value-Iterations on reighted \\EG{s}; and $\\texttt{deg}_{\\Gamma}(v)$ is the degree of $v$. This improves significantly over a previously known pseudo-polynomial time estimate, i.e. $\\Theta\\big(|V|^2|E|W + \\sum_{v\\in V}\\texttt{deg}_{\\Gamma}(v)\\cdot\\ell_{\\Gamma}(v)\\big)$ \\citep{CR15, CR16}, as the pseudo-polynomiality is now confined to depend solely on $\\ell_\\Gamma$. Secondly,  further explore on the relationship beten Optimal Positional Strategies (OPSs) in \\MPG{s} and Small Energy-Progress Measures (SEPMs) in reighted \\EG{s}. It is observed that the space of all OPSs, $\\texttt{opt}_{\\Gamma}\\Sigma^M_0$, admits a unique complete decomposition in terms of extremal-SEPM{s} in reighted EG{s}. This points out what  called the \"Energy-Lattice $\\mathcal{X}^*_{\\Gamma}$ associated to $\\texttt{opt}_{\\Gamma}\\Sigma^M_0$\". Finally, it is offered a pseudo-polynomial total-time recursive procedure for enumerating (w/o repetitions) all the elements of $\\mathcal{X}^*_{\\Gamma}$, and for computing the corresponding partitioning of $\\texttt{opt}_{\\Gamma}\\Sigma^M_0$."}]