Improved Approximation Algorithms for the Maximum Happy Vertices and Edges Problems

Abstract

The Maximum Happy Vertices (MHV) problem and the Maximum Happy Edges (MHE) problem are two fundamental problems arising in the study of the homophyly phenomenon in large scale networks. Both of these two problems are NP-hard. Interestingly, the MHE problem is a natural generalization of Multiway Uncut, the complement of the classic Multiway Cut problem. In this paper, we present new approximation algorithms for MHV and MHE based on randomized LP-rounding technique and non-uniform approach. Specifically, we show that MHV can be approximated within $\frac{1}{\varDelta +1/g(\varDelta )}$, where $\varDelta$ is the maximum vertex degree and $g(\varDelta ) = (\sqrt{\varDelta } + \sqrt{\varDelta +1})^2 \varDelta$, and MHE can be approximated within $\frac{1}{2} + \frac{\sqrt{2}}{4}f(k) \ge 0.8535$, where $f(k) \ge 1$ is a function of the color number k. These results improve over the previous approximation ratios for MHV, MHE as well as Multiway Uncut in the literature.

Appendix: An Example of MHV

In the MHV instance in Fig. 3, there are k vertices $s_1$, $s_2$, $\ldots$, $s_k$ with the pre-specified colors $1, 2, \ldots , k$, respectively. The remaining vertices $t_1$, $t_2$, $\ldots$, $t_k$, and u are uncolored. Every vertex has unit weight. Since each $t_i$ ($1\le i \le k$) is adjacent to $s_1$, $s_2$, $\ldots$, $s_k$, an optimal solution is to color $t_1$, $t_2$, $\ldots$, $t_k$, and u in color 1 and the optimum is 2.

Then consider the fractional optimal value $\text{ OPT }_f = \sum _v\sum _{i=1}^k x_v^i$ of linear program (LP-V) on the instance in Fig. 3. For a vertex v, define $x_v = \sum _{i=1}^k x_v^i$; this is the fraction to which vertex v is happy in a fractional solution. Denote by $(y_v^1, y_v^2, \ldots , y_v^k)$ the color vector assigned to vertex v. Then the color vectors of $s_1$, $s_2$, $\ldots$, $s_k$ are $(1, 0, \ldots , 0)$, $(0, 1, \ldots , 0)$, and $(0, \ldots , 0, 1)$, respectively. Since each $t_i$ ($1\le i \le k$) is adjacent to $s_1$, $s_2$, $\ldots$, $s_k$, no matter what color vector is assigned to $t_i$, $t_i$ cannot be happy to any fraction (that is, $x_{t_i} = 0$).

For each $s_i$ ($1 \le i \le k$), and for all colors $j \ne i$, $x_{s_i}^j$ is always zero; only $x_{s_i}^i$ can be greater than zero. Since each $s_i$ is adjacent to $t_1$, $t_2$, $\ldots$, $t_k$, and since $x_{s_i}^i = \min _{u\in B(s_i)} \{y_u^i\}$, this means the minimum of $y_{t_1}^i$, $y_{t_2}^i$, and $y_{t_k}^i$ is greater than zero. Since the optimal solution would make $x_{s_i}^i$ as large as possible, and since the structures of all $s_i$’s in the instance are symmetric, the optimal solution must assign color vector $(\frac{1}{k}, \frac{1}{k}, \ldots , \frac{1}{k})$ to each $t_i$. As a result, the same color vector will be assigned to vertex u.

In this optimal solution, $x_{s_i} = \frac{1}{k}$ for $1\le i \le k$, $x_u = \sum _i x_u^i = \sum _i \frac{1}{k} = 1$. Therefore, the fractional optimum $\text{ OPT }_f = 2$.

Since $\sum _{i=1}^k y_v^i = 1$, it is natural to color vertex v in color i using $\{y_v^i\}$ as a distribution. However, this randomized coloring scheme only gives poor expected approximation ratio on the instance in Fig. 3. The analysis is given below.

For each $1 \le i \le k$,

$$\begin{aligned}&\Pr [u {\text { is happy by color }}i] \\&\quad = \Pr [u{\text { is colored in }}i] \cdot \Pr [t_1{\text { is colored in }}i] \cdots \Pr [t_k{\text { is colored in }}i] \\&\quad = \left( \frac{1}{k}\right) ^{k+1}. \end{aligned}$$

So,

$$\begin{aligned} \Pr [u{\text { is happy}}] = k \cdot \left( \frac{1}{k}\right) ^{k+1} = \left( \frac{1}{k}\right) ^k. \end{aligned}$$

For each $1 \le i \le k$,

$$\begin{aligned}&\Pr [s_i{\text { is happy}}] \\&\quad = \Pr [s_i{\text { is happy by color }}i] \\&\quad = \Pr [t_1{\text { is colored in }}i] \cdots \Pr [t_k {\text { is colored in }}i] \\&\quad = \left( \frac{1}{k}\right) ^k. \end{aligned}$$

Therefore, the expected solution value is $\left( \frac{1}{k}\right) ^k + k \cdot \left( \frac{1}{k}\right) ^k = \left( 1+\frac{1}{k}\right) \frac{1}{k^{k-1}}$. Consequently, the expected approximation ratio (using $\text{ OPT }_f$ as an upper bound of $\text{ OPT }$) is $\varOmega \left( \frac{1}{k^{k-1}}\right)$, which is $\varOmega \left( \frac{1}{(\varDelta -1)^{\varDelta -2}}\right)$, as the maximum degree is $k+1$ in the instance.