Max-min weight balanced connected partition

Abstract

For a connected graph $G=(V,E)$ and a positive integral vertex weight function $w$, a max-min weight balanced connected $k$-partition of $G$, denoted as $BCP_k$, is a partition of $V$ into $k$ disjoint vertex subsets $(V_1,V_2,\ldots ,V_k)$ such that each $G[V_i]$ (the subgraph of $G$ induced by $V_i$) is connected, and $\min _{1\le i\le k}\{w(V_i)\}$ is maximum. Such a problem has a lot of applications in image processing and clustering, and was proved to be NP-hard. In this paper, we study $BCP_k$ on a special class of graphs: trapezoid graphs whose maximum degree is bounded by a constant. A pseudo-polynomial time algorithm is given, based on which an FPTAS is obtained for $k=2,3,4$. A step-stone for the analysis of the FPTAS depends on a lower bound for the optimal value of $BCP_k$ in terms of the total weight of the graph. In providing such a lower bound, a byproduct of this paper is that any 4-connected trapezoid graph on at least seven vertices has a 4-contractible edge, which may have a value in its own right.

Appendix

The method to obtain an FPTAS from a pseudo-polynomial time algorithm is standard. For the convenience of readers, we present the proof for the FPTAS as follows.

Given a connected vertex-weighted graph $G=(V,E,w)$ whose maximum degree is bounded by a constant $K$. Suppose

$$\begin{aligned} \beta ^*(G,k)\ge W/q, \end{aligned}$$

where $q$ is a constant and $W=w(V(G))$ is the total weight of the graph. Let $\mathcal A$ be the pseudo-polynomial time algorithm for $BCP_k$ as described at the beginning of Sect. 5. Given a real number $\varepsilon >0$, let $\widetilde{G}$ be the graph having the same topology as $G$ but a new weight $\widetilde{w}$, where

$$\begin{aligned} \widetilde{w}(v)=\lfloor w(v)/p\rfloor \text{ for} \text{ every} v\in V \text{ and} p=\left\lfloor \frac{\varepsilon W}{nq}\right\rfloor . \end{aligned}$$

Use $\mathcal A$ on $\widetilde{G}$ to compute an optimal connected $k$-partition $\widetilde{\mathcal{P }}=(\widetilde{V}_1,\ldots ,\widetilde{V}_k)$ of $\widetilde{G}$. Then, $\widetilde{P}$ is also a connected $k$-partition of $G$. We shall show that $\widetilde{P}$ is a $(1-\varepsilon )$-approximation solution of $BCP_k$ on $G$.

Suppose $\mathcal P ^*=(V_1^*,\ldots , V_k^*)$ is an optimal solution of $BCP_k$ on $G$. Use $apx$ to denote the approximation value. Then $apx=\min _{i=1,\ldots ,k}w(\widetilde{V}_i)$. Since $\widetilde{P}$ is an optimal solution of $BCP_k$ on $\widetilde{G}$, we have

$$\begin{aligned} \min _{i=1,\ldots ,k}\widetilde{w}(V_i^*)\le \min _{i=1,\ldots ,k}\widetilde{w}(\widetilde{V}_i) \end{aligned}$$

(3)

For each vertex $v\in V$,

$$\begin{aligned} p\widetilde{w}(v)\le w(v)\le p\widetilde{w}(v)+p. \end{aligned}$$

Thus

$$\begin{aligned} \beta ^*(G,k)&=\min _{i=1,\ldots ,k}w(V_i^*)\le \min _{i=1,\ldots ,k}\{p\widetilde{w}(V_i^*)+p|V_i^*|\}\\&\le \min _{i=1,\ldots ,k}\{p\widetilde{w}(\widetilde{V}_i)\}+pn\le \min _{i=1,\ldots ,k}\{w(\widetilde{V}_i)\}+\frac{\varepsilon W}{q}\\&\le apx+\varepsilon \beta ^*(G,k). \end{aligned}$$

Hence $apx\ge (1-\varepsilon )\beta ^*(G,k)$.

By the second paragraph of Sect. 5, the time complexity is $O(n(n^{2K}\widetilde{W})^{k-1})$. Since $\widetilde{W}=O(W/p)=O(n/\varepsilon )$, the time complexity is $O(n^{((2K+1)(k-1)+1)}/\varepsilon ^{k-1})$.