Online MinCut: Competitive and Regret Analysis

In this paper we study the mincut problem in the online setting. We consider two distinct models: A) competitive analysis and B) regret analysis. In the competitive setting we consider the vertex arrival model; whenever a new vertex arrives it's neighborhood with respect to the set of known vertices is revealed. An online algorithm must make an irrevocable decision to determine the side of the cut that the vertex must belong to in order to minimize the size of the final cut. Various models are considered. 1) For classical and advice models we give tight bounds on the competitive ratio of deterministic algorithms. 2) Next we consider few semi-adversarial inputs: random order of arrival with adversarially generated and sparse graphs. 3) Lastly we derive some structural properties of \mc-type problems with respect to greedy strategies. Finally we consider a non-stationary regret setting with a variational budget $V_T$ and give tights bounds on the regret function. Specifically, we show that if $V_T$ is sublinear in $T$ (number of rounds) then there is a deterministic algorithm achieving a sublinear regret bound ($O(V_T)$). Further, this is optimal, even if randomization is allowed.