  We study the Combinatorial Thompson Sampling policy (CTS) for combinatorial
multi-armed bandit problems (CMAB), within an approximation regret setting.
Although CTS has attracted a lot of interest, it has a drawback that other
usual CMAB policies do not have when considering non-exact oracles: for some
oracles, CTS has a poor approximation regret (scaling linearly with the time
horizon $T$) [Wang and Chen, 2018]. A study is then necessary to discriminate
the oracles on which CTS could learn. This study was started by Kong et al.
[2021]: they gave the first approximation regret analysis of CTS for the greedy
oracle, obtaining an upper bound of order $\mathcal{O}(\log(T)/\Delta^2)$,
where $\Delta$ is some minimal reward gap. In this paper, our objective is to
push this study further than the simple case of the greedy oracle. We provide
the first $\mathcal{O}(\log(T)/\Delta)$ approximation regret upper bound for
CTS, obtained under a specific condition on the approximation oracle, allowing
a reduction to the exact oracle analysis. We thus term this condition
REDUCE2EXACT, and observe that it is satisfied in many concrete examples.
Moreover, it can be extended to the probabilistically triggered arms setting,
thus capturing even more problems, such as online influence maximization.
