Approximation Algorithms for the Submodular Load Balancing with Submodular Penalties

Abstract

In this paper, we study the submodular load balancing problem with submodular penalties. The objective of this problem is to balance the load among sets, while some elements can be rejected by paying some penalties. Officially, given an element set V, we want to find a subset R of rejected elements, and assign other elements to one of m sets $A_1,A_2,\cdots ,A_m$. The objective is to minimize the sum of the maximum load among $A_1,A_2,\cdots ,A_m$ and the rejection penalty of R, where the load and rejection penalty are determined by different submodular functions. We study the submodular load balancing problem with submodular penalties under two settings: heterogenous setting (load functions are not identical) and homogenous setting (load functions are identical). Moreover, we design a Lovász rounding algorithm achieving a worst-case guarantee of $m+1$ under the heterogenous setting and a $min\{m,\lceil \frac{n}{m}\rceil +1\}=\mathcal{O}\left(\sqrt{n}\right)$-approximation combinatorial algorithm under the homogenous setting.

1. Introduction

The load balancing problem, also called the minimum makespan scheduling problem, is one of the most classical and active topics in the field of combinational optimization and approximation algorithms that have been widely studied for more than five decades. The load balancing problem is to find an optimal m-partition assigning n elements to m sets so as to minimize the maximum load among all sets, where $p_{ij}$ is the load for element j to be assigned to the set i. If $p_{ij}=p_{i^{\prime }j}$ for any $1\le i<i^{\prime }\le m$, we call it load balancing under the homogeneous setting. Otherwise, we call it load balancing under the heterogeneous setting.

Under the homogeneous setting, Graham [1] proved that load balancing is strong $NP$-hard, and achieved 2-approximation using the classical list scheduling (LS, for short) algorithm, which was improved by himself with a $4/3$-approximation algorithm [2]. When m is a given constant, Jansen and Porkolab [3] presented a fully polynomial-time approximation scheme(FPTAS, for short) running in time $\mathcal{O}\left(n{(m/\epsilon )}^{\mathcal{O}\left(m\right)}\right)$, which can be improved to $\mathcal{O}\left(n\right)+{(1/\epsilon )}^{\mathcal{O}\left(m\right)}$ when $1/\epsilon >m$ [4]. When m is an arbitrary input, Hochbaum and Shmoys [5] designed a polynomial-time approximation scheme (PTAS, for short), which can give an $(1+\epsilon )$-approximation for any $\epsilon >0$. PTAS can be further improved to an efficient polynomial-time approximation scheme (EPTAS, for short). Alon et al. [6], Jansen et al. [7,8] designed EPTAS for this problem and [8] provided the best result among them.

Under the heterogeneous setting, when m is a given constant, Horowitz and Sahni [9] designed an FPTAS running in time $\mathcal{O}\left(nm{(nm/\epsilon )}^{m-1}\right)$ requiring space $\mathcal{O}\left({(nm/\epsilon )}^{m-1}\right)$. The required space was reduced by Lenstra et al. [10], which is bounded by m and $log(1/\epsilon )$. This paper also proved that when m is an arbitrary input, this problem cannot be approximated within $3/2-\epsilon$ for any $\epsilon >0$ and provided a 2-approximation algorithm using the LP relaxation algorithm. Recently, Kones and Levin [11] devised a unified framework for designing EPTAS for load balancing on parallel machines. Besides, Bhaskara et al. [12] and Chen et al. [13] studied a special case when the rank of the matrix formed by the load for any element is small.

All elements have to be assigned in load balancing. However, assign all elements would be too costly or even impossible in the real world due to resource limitations or certain policies. With this motivation, Bartal et al. [14] proposed the multiprocessor scheduling with a rejection, where an element is either rejected, in which case a rejection penalty has to be paid, or accepted and assigned to the set. The objective changes slightly to minimize the maximum load among all sets plus the sum of the penalties of the rejected elements. For multiprocessor scheduling with a rejection under the homogeneous setting, Bartal et al. [14] proposed a 2-approximation algorithm, which is improved by Ou et al. [15] with a $3/2+\epsilon$-approximation algorithm. More related results can be found in [16,17,18].

In recent years, submodular load balancing, which is a generalization of load balancing, is investigated. Submodular functions have the property of decreasing marginal returns which is verified in mathematical models in various fields such as operations research, economics, engineering, computer science, management science, etc. They are extensively utilized to solve practical problems in image segmentation, parallel computing, viral marketing and data summarization [19,20,21,22]. Since the submodular property covers a much wider range of functions compared with modular property, problems with submodular settings are often harder to cope with. Even in the homogeneous setting of load balancing problem, Svitkina and Fleischer [23] showed that submodular load balancing is information-theoretically hard to approximate within $\mathit{o}\left(\sqrt{n/logn}\right)$. Using a similar method, Wei et al. [22] showed that submodular load balancing is information-theoretically hard to approximate within m if $m\ll n$. Svitkina and Fleischer [23] provided a balanced partitioning algorithm yielding a factor of $min\{m,n/m\}$ and a sampling-based algorithm achieving $\mathcal{O}(\sqrt{n/logn)}$ under the homogeneous setting, respectively. Using ellipsoid approximation techniques, Goemans et al. [24] presented an $\mathcal{O}(\sqrt{n}logn)$-approximation algorithm for submodular load balancing under the homogeneous setting. Using Lovász rounding algorithm, Wei et al. [22] presented an m-approximation algorithm for submodular load balancing under the homogeneous setting. Meanwhile, combinatorial optimization problems with submodular penalties are receiving increasing attentions [25,26,27,28,29]. Most recently, Zhang et al. [30] proposed a 3-approximation algorithm for precedence-constrained scheduling with submodular rejection on parallel machines. Liu and Li [31] proposed a 2-approximation algorithm for single machine scheduling with releasing time and submodular rejection.

In this paper, we consider the submodular load balancing with submodular penalties, which is a combination of multiprocessor scheduling with rejection and submodular load balancing, where the load and rejection penalty are all determined by submodular functions. Compared with the submodular load balancing, our problem is closer to the real situation because some elements can be rejected. Besides, in the real world, there is no linear relationship between the number of rejected elements and the rejection penalties which can be considered as the loss of the manufacturer’s prestige, but as the number of rejected elements increases, it will have fewer and fewer rejection penalties. Therefore, the submodular load balancing with submodular penalties is closer to reality than all problems introduced above.

We have dealt with this problem with two settings (heterogenous and homogenous) in this paper. As shown in

Table 1. The results in this paper.

Case	Setting	Our Result
The submodular load balancing with submodular penalties	Heterogeneous	$m+1$ (non-combinational)
	Homogenous	$min\{m,\lceil \frac{n}{m}\rceil +1\}=\mathcal{O}\left(\sqrt{n}\right)$ (combinational)

, under the heterogeneous setting, we design a non-combinational Lovász rounding algorithm achieving a worst-case guarantee of $m+1$. Under the homogeneous setting, we design an $\mathcal{O}\left(\sqrt{n}\right)$-approximation combinational algorithm by combining two approximation algorithms, one is an m-approximation combinational algorithm based on the greedy technique and another is $\lceil \frac{n}{m}\rceil +1$-approximation combinational algorithm based on the average rule.

The rest of this paper is structured as follows. In Section 2, we provide basic definitions and a formal problem statement. In Section 3, we focus on the heterogeneous setting and propose a Lovász rounding algorithm. In Section 4, we further consider the homogenous setting and provide a combinatorial m-approximation algorithm. In Section 5, we give a brief conclusion.

2. Preliminaries

Let $V=\{v_1,v_2,\dots ,$$v_n\}$ be a given ground set of n elements, and $f(·):2^V\to {\mathbb{R}}_{\ge 0}$ be a real-valued function defined on all the subsets of V. The following definitions are used frequently in this paper.

• $f(·)$ is called submodular if $f\left(S\right)+f\left(T\right)\ge f(S\cup T)+f(S\cap T),\forall S,T\subseteq V.$
• $f(·)$ is called nondecreasing if $f\left(S\right)\le f\left(T\right)$ for any $S\subseteq T\subseteq V.$
• $f(·)$ is called normalized if $f\left(\mathsf{Ø}\right)=0$.
• $f(·)$ is called polymatriod if $f(·)$ is normalized, nondecreasing and submodular.

The submodular load balancing (SLB, for short) with submodular penalties can be described as follows. Given a ground set $V=\{v_1,v_2,\dots ,v_n\}$, m ($<n$) polymatriod load functions $f_1(·),\dots ,f_m(·):2^V\to {\mathbb{R}}_{\ge 0}$, and a polymatriod rejection function $\pi (·):2^V\to {\mathbb{R}}_{\ge 0}$. The SLB with submodular penalties is to choose a partition $(A_1,\dots ,A_m,R)$ of the ground set V, where $A={\cup }_{i=1}^m{A}_i$ is the set of accepted elements, R is the set of rejected elements, and

$$\begin{array}{c}\hfill A_{i_1}\cap A_{i_2}=\mathsf{Ø},\forall 1\le i_1<i_2\le n,\mathrm{and}A\cup R=V.\end{array}$$

The objective value of partition $(A_1,\dots ,A_m,R)$ is ${max}_i{f}_i\left(A_i\right)+\pi \left(R\right)$, where $f_i\left(A_i\right)$ is the load of the ith partition. The objective is to find a partition $(A_1,\dots ,A_m,R)$ with the minimum objective value.

For each subset $S\subseteq V$ and $i\in \{1,2,\dots ,m\}$, we introduce a binary variable $x_{i,S}$, where

$$\begin{array}{c}\hfill x_{i,S}=\left\{\begin{array}{cc}& 1,\mathrm{if}{A}_i=S;\hfill \\ & 0,\mathrm{otherwise},\hfill \end{array}\right.\end{array}$$

For each subset $S\subseteq V$, we introduce a binary variable $z_S$, where

$$\begin{array}{c}\hfill z_S=\left\{\begin{array}{cc}& 1,\mathrm{if}S=R;\hfill \\ & 0,\mathrm{otherwise}.\hfill \end{array}\right.\end{array}$$

Correspondingly, the load of $A_i$ is defined as

$$f_i\left(A_i\right)=\sum _{S:S\subseteq V}{f}_i\left(S\right)x_{i,S}.$$

Thus, the integer linear program (ILP) for the SLB with submodular penalties can be formulated as

$$\begin{array}{ccc}\hfill & & min(\underset{i}{max}\sum _{S:S\subseteq V}{f}_i\left(S\right)x_{i,S}+\sum _{S:S\subseteq V}\pi \left(S\right)z_S)\hfill \\ \hfill \mathit{s}.\mathit{t}.& & \sum _{i=1}^m\sum _{S:S\subseteq Vand{v}_j\in S}{x}_{i,S}+\sum _{S:S\subseteq Vand{v}_j\in S}{z}_S\ge 1,\hfill \\ & & \forall j=1,2,\dots ,n,\hfill \\ & & x_{i,S},z_S\in \{0,1\},\forall i=1,2,\dots ,m,\forall S\subseteq V.\hfill \end{array}$$

(1)

where $f_1(·),\dots ,f_m(·)$ and $\pi (·)$ are polymatriod functions as mentioned above.

Clearly, if $\pi \left(S\right)=+\infty$ for any $S\ne \mathsf{Ø}$, ILP(1) is exactly the SLB problem considered in [22]. If $\pi \left(S\right)={\sum }_{j:v_j\in S}\pi \left(\left\{v_j\right\}\right)$ and $f_i\left(S\right)={\sum }_{j:v_j\in S}f\left(\left\{v_j\right\}\right)$ for any $S\ne \mathsf{Ø}$ and $i=1,2,\dots ,m$, ILP(1) is exactly multiprocessor scheduling with rejection considered in [14]. Therefore, the SLB with submodular penalties generalizes both the problems in [14,22]. We further categorize the SLB with submodular penalties depending on whether the $f_1(·),\dots ,f_m(·)$ are identical to each other (homogeneous), i.e., $f_i\left(S\right)=f\left(S\right)$ for any $i,S$, or not (heterogeneous).

As mention above, SLB with submodular penalties generalizes the SLB problem [22,23]. Thus,

Lemma 1.

[23] The submodular load balancing with submodular penalties is hard to approximate to a factor of $\mathit{o}\left(\sqrt{n/logn}\right)$ even for the homogeneous setting.

If $m\ll n$, we have

Corollary 1.

[22] For any $\epsilon >0$, the submodular load balancing with submodular penalties cannot be approximated to a factor of $(1-\epsilon )m$ for any $m=\mathit{o}\left(\sqrt{n/logn}\right)$ with polynomial number of queries even for the homogeneous setting.

In the remainder of this paper, we propose a $(m+1)$-approximation algorithm using the Lovász extension for the problem under the heterogeneous setting. Then, we design a combination $\mathcal{O}\left(\sqrt{n}\right)$-approximation algorithm for the problem under the homogeneous setting.

3. Heterogeneous Setting

In this section, we first introduce the definition of Lovász extension to show that the optimal relaxation solution of (1) can be solved in polynomial time. Then, round the optimal relaxation solution to a feasible solution of the SLB with submodular penalties under the heterogeneous setting.

Let $V=\{v_1,v_2,\dots ,v_n\}$. For each vector $\mathit{x}\in {\mathbb{R}}_{\ge 0}^n$, it is represented as

$$\begin{array}{c}\hfill \mathit{x}=(x\left(1\right),x\left(2\right),\dots ,x\left(n\right))\in {\mathbb{R}}_{\ge 0}^n.\end{array}$$

For each subset $S\subseteq V$, let ${\mathit{I}}_S=(I_S\left(1\right),I_S\left(2\right),\dots ,I_S\left(n\right))\in {\mathbb{R}}_{\ge 0}^n$ be the characteristic vector of set S, where

$$\begin{array}{c}\hfill I_S\left(j\right)=\left\{\begin{array}{cc}& 1,\mathrm{if}{v}_j\in S,\hfill \\ & 0,\mathrm{otherwise}.\hfill \end{array}\right.\end{array}$$

Definition 1.

(Lovász extension [32] The Lovász extension of a normalized submodular function $f(·)$ is a function $\widehat{f}(·):{\mathbb{R}}_{\ge 0}^n\to {\mathbb{R}}_{\ge 0}$ satisfying:

1. For each subset $S\subseteq V$, $\widehat{f}\left({\mathit{I}}_S\right)=f\left(S\right)$.

2. For any vector $\mathit{x}=(x\left(1\right),x\left(2\right),\dots ,x\left(n\right))\in {\mathbb{R}}_{\ge 0}^n$

$$\begin{array}{ccc}\hfill \widehat{f}\left(\mathit{x}\right)=& max& \sum _{j=1}^n{\alpha }_j·x\left(j\right)\hfill \\ & s.t.& \sum _{v_j\in S}{\alpha }_j\le f\left(S\right),\forall S\subset V,\hfill \\ & & {\alpha }_j\ge 0,\hfill \end{array}$$

(2)

The following theorem reveals the relationship between set function and convexity.

Theorem 1.

[32] A set function $f(·)$ is submodular if and only if its Lovász extension $\widehat{f}(·)$ is a convex function.

Without loss of generality, we assume that $x\left(1\right)\ge x\left(2\right)\ge \cdots \ge x\left(n\right)$, and let $S_j=\{v_1,v_2,\dots ,v_j\}$ for any $j=1,2,\dots ,n$ and $S_0=\mathsf{Ø}$. From Lemma 3.1 in [33], we can construct an optimal solution ${\alpha }^{*}=({\alpha }_1^{*},{\alpha }_2^{*},\dots ,{\alpha }_n^{*})$ of (2), where

$$\begin{array}{c}\hfill {\alpha }_j^{*}=f\left(S_j\right)-f\left(S_{j-1}\right).\end{array}$$

Consider a feasible partition $(A_1,\dots ,A_m,R)$. For each element $v_j\in V$, we introduce a binary variable $z\left(j\right)$, where $z\left(j\right)=1$ if and only if $v_j$ is rejected. For every element $v_j$ and every $A_i$, we introduce a binary variable $x_i\left(j\right)$ where $x_i\left(j\right)=1$ if and only if $v_j\in A_i$. For each $A_i$, by the definition of Lov$\acute{a}$sz extension, we have

$$\begin{array}{c}\hfill {\widehat{f}}_i\left({\mathit{I}}_{A_i}\right)=f_i\left(A_i\right)=\sum _{S:S\subseteq V}{f}_i\left(S\right)x_{i,S},\end{array}$$

where $x_{i,S}=1$ if and only if $S=A_i$. Similarly, for the rejected set R, we have

$$\begin{array}{c}\hfill \widehat{\pi }\left({\mathit{I}}_R\right)=\pi \left(R\right)=\sum _{S:S\subseteq V}\pi \left(S\right)z_S,\end{array}$$

where ${\mathit{I}}_R$ is the characteristic vector of R and $z_S=1$ if and only if $S=R$. Therefore, ILP(1) is equivalent to the following convex programming:

$$\begin{array}{ccc}& & min(\underset{i}{max}{\widehat{f}}_i\left({\mathit{x}}_i\right)+\widehat{\pi }\left(\mathit{z}\right))\hfill \\ \hfill \mathit{s}.\mathit{t}.& & \sum _{i=1}^m{x}_i\left(j\right)+z\left(j\right)\ge 1,\forall j=1,2,\dots ,n,\hfill \\ & & x_i\left(j\right),z\left(j\right)\in \{0,1\},\forall i=1,2,\dots ,m\hfill \\ & & and\forall j=1,2,\dots ,n,\hfill \end{array}$$

The relaxed version is

$$\begin{array}{ccc}\hfill & & min(\underset{i}{max}{\widehat{f}}_i\left({\mathit{x}}_i\right)+\widehat{\pi }\left(\mathit{z}\right))\hfill \\ \hfill \mathit{s}.\mathit{t}.& & \sum _{i=1}^m{x}_i\left(j\right)+z\left(j\right)\ge 1,\forall j=1,2,\dots ,n,\hfill \\ & & x_i\left(j\right),z\left(j\right)\in [0,1],\forall i=1,2,\dots ,m\hfill \\ & & and\forall j=1,2,\dots ,n.\hfill \end{array}$$

(3)

Let $({\mathit{x}}_1^{*},{\mathit{x}}_2^{*},\dots ,{\mathit{x}}_m^{*},{\mathit{z}}^{*})$ be the optimal solution of the convex relaxation (3), which can be solved in polynomial time by the ellipsoid method. We design a $(m+1)$-approximation algorithm by the rounding technique, which is described as follows.

Theorem 2.

Algorithm 1 achieves a worst-case guarantee of $m+1$.

Algorithm 1: Lovász rounding algorithm.

Proof. 

For the vector ${\mathit{z}}^{*}$, without loss of generality, we assume

$$\begin{array}{c}\hfill z^{*}\left(1\right)\ge z^{*}\left(2\right)\ge \cdots \ge z^{*}\left(n\right).\end{array}$$

For each element $v_j\in V$, if $z^{*}\left(j\right)\ge \frac{1}{m+1}$, $v_j$ is rejected and $v_j\in R$, according to Algorithm 1. It implies that $I_R\left(j\right)=1$, and

$$\begin{array}{c}\hfill z^{*}\left(j\right)\ge \frac{1}{m+1}=\frac{1}{m+1}{I}_R\left(j\right).\end{array}$$

If $z^{*}\left(j\right)<\frac{1}{m+1}$, we have $I_R\left(j\right)=0$. Thus, for any case, we have

$$\begin{array}{c}\hfill z^{*}\left(j\right)\ge \frac{1}{m+1}{I}_R\left(j\right),\forall j=1,2,\dots ,n.\end{array}$$

(4)

By the definition of Lov$\acute{a}$sz extension, we have

$$\begin{array}{ccc}\hfill \widehat{\pi }\left({\mathit{z}}^{*}\right)& =& \sum _{j=1}^{n-1}(z^{*}\left(j\right)-z^{*}\left(j+1\right))\pi \left(S_j\right)+z^{*}\left(n\right)\pi \left(S_n\right)\hfill \\ & =& z^{*}\left(1\right)\pi \left(S_1\right)+\sum _{j=2}^n{z}^{*}\left(j\right)(\pi \left(S_j\right)-\pi \left(S_{j-1}\right))\hfill \\ & \ge & \frac{1}{m+1}{I}_R\left(1\right)\pi \left(S_1\right)\hfill \\ & & +\sum _{j=2}^n\frac{1}{m+1}{I}_R\left(j\right)(\pi \left(S_j\right)-\pi \left(S_{j-1}\right))\hfill \\ & =& \widehat{\pi }\left(\frac{1}{m+1}{\mathit{I}}_R\right)\hfill \end{array}$$

where $S_j=\{v_1,v_2,\dots ,v_j\}$ for any $j=1,2,\dots ,n$, the inequality follows from inequality (4) and the fact that $\pi (·)$ is the nondecreasing, and the last equality follows from the definition of $\widehat{\pi }$. Therefore, we have

$$\begin{array}{ccc}\hfill \pi \left(R\right)& =& \widehat{\pi }\left({\mathit{I}}_R\right)=(m+1)\widehat{\pi }\left(\frac{1}{m+1}{\mathit{I}}_R\right)\hfill \\ & \le & (m+1)\widehat{\pi }\left({\mathit{z}}^{*}\right).\hfill \end{array}$$

(5)

For any $v_j\in V\setminus R$, we have $z^{*}\left(j\right)<\frac{1}{m+1}$, by the choice of Algorithm 1. Therefore,

$$\begin{array}{c}\hfill \underset{i}{max}{x}_i^{*}\left(j\right)\ge \frac{1}{m}\sum _{i=1}^m{x}_i^{*}\left(j\right)\ge \frac{1}{m}(1-z^{*}\left(j\right))>\frac{1}{m+1},\end{array}$$

where the second inequality follows from that ${\mathit{x}}_i^{*}=(x_i^{*}\left(1\right),\dots ,x_i^{*}\left(n\right))$ is the optimal solution of the convex relaxation (3). Similarly, for every $i=1,2,\dots ,m$, we have

$$\begin{array}{ccc}\hfill f_i\left(A_i\right)& =& {\widehat{f}}_i\left({\mathit{I}}_{A_i}\right)=(m+1){\widehat{f}}_i\left(\frac{1}{m+1}{\mathit{I}}_{A_i}\right)\hfill \\ & \le & (m+1){\widehat{f}}_i\left({\mathit{x}}_i^{*}\right).\hfill \end{array}$$

(6)

Therefore, we have

$$\begin{array}{ccc}& & \underset{i}{max}{f}_i\left(A_i\right)+\pi \left(R\right)\hfill \\ & \le & (m+1)\underset{i}{max}{\widehat{f}}_i\left({\mathit{x}}_i^{*}\right)+(m+1)\widehat{\pi }\left({\mathit{z}}^{*}\right)\hfill \\ & =& (m+1)·(\underset{i}{max}{\widehat{f}}_i\left({\mathit{x}}_i^{*}\right)+\widehat{\pi }\left({\mathit{z}}^{*}\right))\hfill \\ & \le & (m+1)·OPT,\hfill \end{array}$$

where the first inequality follows from inequalities (5) and (6), and $OPT$ is the optimal value of the ILP (1). □

4. Homogeneous Setting

In this section, we present two approximation algorithms to solve the SLB with submodular penalties under the homogeneous setting. We begin by presenting an m-approximation combinational algorithm based on the greedy technique. Then, we give a $\lceil \frac{n}{m}\rceil +1$-approximation combinational algorithm based on the average partition of the elements into $A_1,\dots ,A_m,R$. Combining these two approximation algorithms, we can design a $\mathcal{O}\left(\sqrt{n}\right)$-approximation combinational.

For the SLB with submodular penalties under the homogeneous setting, we have that $f_1(·),\dots ,f_m(·)$ are identical. For convenience, we use polymatriod load function $f(·)$ to represent each load function $f_1(·),\dots ,f_m(·)$, i.e., for any $i=1,\dots ,m$,

$$f\left(S\right)=f_i\left(S\right),\forall S\subseteq V.$$

For this problem, we propose a combinatorial m-approximation algorithm using an auxiliary set function $w(·)$ with the ground set V, where

$$\begin{array}{c}\hfill w\left(S\right)=f(V\setminus S)+\pi \left(S\right),\forall S\subseteq V.\end{array}$$

(7)

We propose the detailed m-approximation combinational algorithm as follows.

Theorem 3.

Algorithm 2 achieves a worst-case guarantee of m in polynomial time.

Algorithm 2: Algorithm based on the greedy technique.

Proof. 

Since $f(·)$ and $\pi (·)$ are polymatriod, for any set $S,T\subseteq V$, we have

$$\begin{array}{ccc}& & w\left(S\right)+w\left(T\right)\hfill \\ & =& f(V\setminus S)+\pi \left(S\right)+f(V\setminus T)+\pi \left(T\right)\hfill \\ & \ge & f\left(\right(V\setminus S)\cup (V\setminus T\left)\right)+f\left(\right(V\setminus S)\cap (V\setminus T\left)\right)\hfill \\ & & +\pi (S\cup R)+\pi (S\cap R)\hfill \\ & =& f(V\setminus (S\cap T\left)\right)+\pi (S\cap R)\hfill \\ & & +f(V\setminus (S\cup T\left)\right)+\pi (S\cup R)\hfill \\ & =& w(S\cap T)+w(S\cup T),\hfill \end{array}$$

where the inequality follows from the submodularity of $f(·)$ and $\pi (·)$. It implies that $w(·)$ is submodular. By modifying the method in [34] slightly, ${min}_{S:S\subseteq V}w\left(S\right)$ and corresponding solution R can be computed within polynomial time. It is easy to verify other processes can be implemented in polynomial time.

Let ($A_1^{*},\dots ,A_m^{*},R^{*}$) be the optimal partition for the SLB with submodular penalties under the homogeneous setting. Following from the definition of R, we have

$$\begin{array}{ccc}\hfill f(V\setminus R)+\pi \left(R\right)& =& w\left(R\right)\le w\left(R^{*}\right)\hfill \\ & =& f(V\setminus R^{*})+\pi \left(R^{*}\right).\hfill \end{array}$$

(8)

Therefore, the objective value of $(A_1,A_2,\dots ,A_m,R)$ produced by Algorithm 2 is

$$\begin{array}{ccc}\hfill \underset{i}{max}f\left(A_i\right)+\pi \left(R\right)& \le & f(V\setminus R)+\pi \left(R\right)\hfill \\ & \le & f(V\setminus R^{*})+\pi \left(R^{*}\right)\hfill \\ & \le & \sum _{i}f\left(A_i^{*}\right)+\pi \left(R^{*}\right)\hfill \\ & \le & m(\underset{i}{max}f\left(A_i^{*}\right)+\pi \left(R^{*}\right))\hfill \\ & =& mOPT,\hfill \end{array}$$

where the first inequality follows since $f(·)$ is nondecreasing, the second inequality follows from the inequality (8), and the third inequality follows from the submodularity of $f(·)$. □

Then, we present an $(\lceil \frac{n}{m}\rceil +1)$-approximation algorithm for the SLB with submodular penalties under the homogeneous setting. Consider the following decision version of the problem: given an instance I and target load ‘a’ and a target penalty ‘r’, does there exist a partition with its load at most a and its rejected penalty at most r. On input $(I,a,r)$, the algorithm decision procedure would output ‘Fail’ or ‘$(A_1,\dots ,A_m,R)$’, depending on, whether there was such a partition. The details of the proposed algorithm are shown as follows.

Let ($A_1^{*},\dots ,A_m^{*},R^{*}$) be the optimal partition for the SLB with submodular penalties under the homogeneous setting and

$$a^{*}=\underset{i}{max}f\left(A_i^{*}\right)\mathrm{and}{r}^{*}=\pi \left(R^{*}\right)$$

Theorem 4.

For $a=a^{*}$ and $r=r^{*}$, Algorithm 3 achieves a feasible partition with objective value at most $(\lceil \frac{n}{m}\rceil +1)(a^{*}+r^{*})$.

Algorithm 3: Algorithm based on the average rule.

Proof. 

Let ($A_1,\dots ,A_m,R$) be the output partition by Algorithm 3 and

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}& A_L=\{v_j\in A|\pi \left(\left\{v_j\right\}\right)>r^{*}\};\hfill \\ & R_L=\{v_j\in R|f\left(\left\{v_j\right\}\right)>a^{*}\}.\hfill \end{array}\right.\end{array}$$

For each element $v_j\in R_L$, we have $v_j\in R^{*}$, otherwise ${max}_{i}f\left(A_i^{*}\right)>a^{*}$. It implies that $R_L\subseteq R^{*}$ and

$$\pi \left(R_L\right)\le \pi \left(R^{*}\right)=r^{*}$$

that $\pi (·)$ is nondecreasing.

For each $v_j\in A_L$, $v_j$ is the partition to a set among $A_1,\dots ,A_m$ with the minimum number of elements. Moreover, for each $v_j\in V\setminus (A_L\cup R_L)$ is the partition to a set among $A_1,\dots ,A_m,R$ with the minimum number of elements. Thus, we claim

$$\begin{array}{ccc}\hfill \underset{i}{max}\left|A_i\right|& \le & \lceil \frac{|A_L|}{m}\rceil +\lceil \frac{|V\setminus (A_L\cup R_L\left)\right|}{m+1}\rceil \hfill \\ & \le & \lceil \frac{|A_L|}{m}+\frac{|V\setminus (A_L\cup R_L\left)\right|}{m+1}\rceil +1\hfill \\ & \le & \lceil \frac{n}{m}\rceil +1\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \left|R\right|& \le & |R_L|+\lceil \frac{|V\setminus (A_L\cup R_L\left)\right|}{m+1}\rceil \hfill \\ & \le & |R_L|+\lceil \frac{n}{m+1}\rceil \hfill \\ & \le & |R_L|+\lceil \frac{n}{m}\rceil .\hfill \end{array}$$

Therefore, the objective value of $(A_1,A_2,\dots ,A_m,R)$ produced by Algorithm 3 is

$$\begin{array}{ccc}& & \underset{i}{max}f\left(A_i\right)+\pi \left(R\right)\hfill \\ & \le & \underset{i}{max}\sum _{v_j\in A_i}f\left(\left\{v_j\right\}\right)+\pi \left(R_L\right)+\sum _{v_j\in R\setminus R_L}\pi \left(\left\{v_j\right\}\right)\hfill \\ & \le & \underset{i}{max}|A_i|a^{*}+r^{*}+|R\setminus R_L|r^{*}\hfill \\ & \le & (\lceil \frac{n}{m}\rceil +1)(a^{*}+r^{*}),\hfill \end{array}$$

where the first inequality follows from the submodularity of $f(·)$ and the second inequality follows from $f\left(\left\{v_j\right\}\right)\le a^{*}$ for $v_j\in V\setminus R_L$ and $\pi \left(\left\{v_j\right\}\right)\le r^{*}$ for $v_j\in V\setminus A_L$. □

Using the m-approximation algorithm, we can easily find lower and upper bounds on ‘a’ and ‘r’. Applying binary search in this interval, we can find the $a=a^{*}$ and $r=r^{*}$ in time polynomial time. In other words, we can design a $\lceil \frac{n}{m}\rceil +1$-approximation combinational algorithm using Algorithm 3.

Combining the above two approximation algorithms, select the minimum objective value the partition between two output partitions produced by Algorithms 2 and 3, we have

Theorem 5.

There exists a combinatorial $min\{m,\lceil \frac{n}{m}\rceil +1\}=\mathcal{O}\left(\sqrt{n}\right)$-approximation algorithm for the SLB with submodular penalties under the homogeneous setting.

5. Conclusions and Future Work

In this paper, we consider the submodular load balancing (SLB) with submodular penalties under two settings: Heterogenous setting and homogenous setting. We design a Lovász rounding algorithm achieving a worst-case guarantee of $m+1$ under the heterogenous setting and a $\mathcal{O}\left(\sqrt{n}\right)$-approximation combinatorial algorithm under the homogenous setting.

In Corollary 1, we present a low bound showing that there is no m approximation algorithm even for the homogeneous setting for the SLB with submodular penalties. Therefore, there is a gap of 1 between Lovász rounding algorithm and this lower bound. Is it possible to either improve our algorithm or find a better low bound for the SLB with submodular penalties?

For the SLB with submodular penalties, our lower bounds show the impossibility of constant approximations to solve it. This means that to obtain better algorithms for specific applications, one has to resort to more restricted models, avoiding the full generality of arbitrary polymatriod functions. Thus, the SLB with submodular penalties under some restriction (such as curvature) is also a meaningful research direction.