Set Cover

• Ground set \(U\) and a collection of subsets \(\mathcal{S}\) 
  • \(|U|=n\), \(|\mathcal{S}|=m\)
• Cost function: \(c:\mathcal{S}\to\mathbb{R}^+\)
• Goal: Find \(S' \subseteq \mathcal{S}\) that covers \(U\) and has minimum cost 

Online Set Cover

• Ground set \(U\) and a collection of subsets \(\mathcal{S}\) 
• Cost function: \(c:\mathcal{S}\to\mathbb{R}^+\)
• Goal: Find \(S' \subseteq \mathcal{S}\) that covers \(U\) and has minimum cost 
• Online:
  • Elements \(v\in U\) arrive one by one;
  • When element \(v\) arrives, only knows which sets contains them and nothing else
  • Need to decide on the spot which set to pick 

Random Order Online Set Cover

• Ground set \(U\) and a collection of subsets \(\mathcal{S}\) 
• Cost function: \(c:\mathcal{S}\to\mathbb{R}^+\)
• Goal: Find \(S' \subseteq \mathcal{S}\) that covers \(U\) and has minimum cost 
• Online:
  • Elements \(v\in U\) arrive in random order
  • When element \(v\) arrives, only knows which sets contains them and nothing else
  • Need to decide on the spot which set to pick 

LearnOrCover (poly) 

Intuition: 

• The function either makes progress by covering elements in the universe or by learning 
  • "learning" in the sense of multiplicative weight updating 

\(\to\) Cover

\(\to\) Learn

Define potential function \[\Phi(t):=C_1\cdot KL(x^*||x^t) + C_2 \log |U^t| \]

LearnOrCover (poly) 

• \(O(k\cdot \log(mn))\) is achieved via the proving the following two bounds related to potential function: 
  • \(\Phi(0) = O(\log(mn))\) 
  • \(\mathbb{E}[\Delta \Phi] \leq - \frac{1}{k}\)
• Combining the above two will give us \(\Phi(t) = O(k\log(mn))\)

\[\Phi(t):=C_1\cdot KL(x^*||x^t) + C_2 \log |U^t| \]