A note on two source location problems

We consider Source Location ($\mathsf{SL}$) problems: given a capacitated network $G=(V,E)$, cost $c(v)$ and a demand $d(v)$ for every $v\in V$, choose a min-cost $S\subseteq V$ so that $\lambda (v,S)⩾d(v)$ holds for every $v\in V$, where $\lambda (v,S)$ is the maximum flow value from v to S. In the directed variant, we have demands $d^{\mathit{in}}(v)$ and $d^{\mathit{out}}(v)$ and we require $\lambda (S,v)⩾d^{\mathit{in}}(v)$ and $\lambda (v,S)⩾d^{\mathit{out}}(v)$. Undirected $\mathsf{SL}$ is (weakly) NP-hard on stars with $r(v)=0$ for all v except the center. But, it is known to be polynomially solvable for uniform costs and uniform demands. For general instances, both directed an undirected $\mathsf{SL}$ admit a $(\ln D+1)$-approximation algorithms, where D is the sum of the demands; up to constant this is tight, unless P = NP. We give a pseudopolynomial algorithm for undirected $\mathsf{SL}$ on trees with running time $O(|V|{\Delta }^3)$, where $\Delta ={\max }_{v\in V}d(v)$. This algorithm is used to derive a linear time algorithm for undirected $\mathsf{SL}$ with $\Delta ⩽3$. We also consider the Single Assignment Source Location ($\mathsf{SASL}$) where every $v\in V$ should be assigned to a single node $s(v)\in S$. While the undirected $\mathsf{SASL}$ is in P, we give a $(\ln |V|+1)$-approximation algorithm for the directed case, and show that this is tight, unless P = NP.     

Hamilton differential Harnack inequality and W-entropy for Witten Laplacian on Riemannian manifolds

In this paper, we prove the Hamilton differential Harnack inequality for positive solutions to the heat equation of the Witten Laplacian on complete Riemannian manifolds with the $CD(?K,m)$-condition, where $m\in [n,\infty )$ and $K\ge 0$ are two constants. Moreover, we introduce the W-entropy and prove the W-entropy formula for the fundamental solution of the Witten Laplacian on complete Riemannian manifolds with the $CD(?K,m)$-condition and on compact manifolds equipped with $(?K,m)$-super Ricci flows.