On maximizing monotone or non-monotone k-submodular functions with the intersection of knapsack and matroid constraints

Abstract

A k-submodular function is a generalization of a submodular function. The definition domain of a k-submodular function is a collection of k-disjoint subsets instead of simple subsets of ground set. In this paper, we consider the maximization of a k-submodular function with the intersection of a knapsack and m matroid constraints. When the k-submodular function is monotone, we use a special analytical method to get an approximation ratio $\frac{1}{m+2}(1-e^{-(m+2)})$ for a nested greedy and local search algorithm. For non-monotone case, we can obtain an approximate ratio $\frac{1}{m+3}(1-e^{-(m+3)})$.

1 Introduction

Given a ground set G containing n elements and $k\in N_+$, refer $(X_1,\ldots ,X_k)$ as k-disjoint subsets, with $X_i\subseteq G$, $\forall i\in [k]$ and $X_i\cap X_j=\emptyset$, $\forall i\ne j\in [k]$; write $(k+1)^{G}$ as the family of k disjoint subsets. Define join and meet operations for any $\textbf{x}=(X_1,\ldots ,X_k)$ and $\textbf{y}=(Y_1,\ldots ,Y_k)$ in $(k+1)^{G}$, that is,

$$\begin{aligned}{} & {} \textbf{x}\sqcup \textbf{y}:=(X_1\cup Y_1\setminus (\bigcup _{i\ne 1}X_i\cup Y_i),\ldots ,X_k\cup Y_k\setminus (\bigcup _{i\ne k}X_i\cup Y_i)),\\{} & {} \textbf{x}\sqcap \textbf{y}:= (X_1\cap Y_1,\ldots ,X_k\cap Y_k).\end{aligned}$$

The join operation removes some points with different positions in $\textbf{x}$ and $\textbf{y}$, that is, points v with $v\in X_i$, $v\in Y_j$, $\forall i\ne j\in [k]$. And the meet operation is just an intersection operation of sets.

A function $f:(k+1)^{G}\rightarrow R$ is said to be k-submodular (Huber and Kolmogorov 2012) if

$$\begin{aligned} f(\textbf{x})+f(\textbf{y}) \ \ge \ f(\textbf{x}\sqcup \textbf{y})+f(\textbf{x}\sqcap \textbf{y}), \end{aligned}$$

for any $\textbf{x}$ and $\textbf{y}$ in $(k+1)^{G}$. The k-submodular function is a generalization of a submodular function. Note that the definition domain of k-submodular function is a collection of k disjoint subsets instead of simple subsets. When $k=1$, a k-submodular function becomes a submodular function.

1.1 Related work

There have been many research results on monotone submodular maximization problem. Nemhauser et al. (1978) firstly achieved a greedy $(1-1/e)$-approximation algorithm under a cardinality constraint, which was known as a tight bound. Later, Sviridenko (2004) designed a combinatorial $(1-1/e)$ approximate algorithm under a knapsack constraint. For this problem, Ene and Nguyen (2019) also offered an approximate ratio of $(1-1/e-\varepsilon )$ by using multilinear extention function, which only needed approximate linear running time. With a matroid constraint, Calinescu et al. (2011) got an approximate ratio of $(1-1/e)$, by using the continuous greedy method and pipage rounding technique. Filmus and Ward (2014) designed a combination algorithm using local search technique, which also achieved an approximate ratio of $(1-1/e)$. More recently, Sarpatwar et al. (2019) contributed an algorithm with an approximate ratio of $\frac{1-e^{-(m+1)}}{m+1}$ combining the greedy algorithm and local search techniques for maximization problem of submodular function subject to the intersection of a knapsack and m matroid constraints. For maximizing non-monotone submodular functions, Lee et al. (2010) presented a $(\frac{1}{m+2+\frac{1}{m}+\varepsilon })$ approximation algorithm under m matroid constraints, and a $(\frac{1}{5}-\varepsilon )$ approximation algorithm under m knapsack constraints. Feldman et al. (2011) and Chekuri et al. (2014) studied constant factor approximation algorithms to maximize a multilinear extension of the submodular function over a down-closed polytope, respectively. The fractional solution could be rounded with contention resolution schemes. For more references on submodular maximization, see Bian et al. (2017); Calinescu et al. (2011); Ene and Nguyen (2019); Feldman and Naor (2013); Filmus and Ward (2014); Huang et al. (2022); Liu et al. (2022b); Sviridenko (2004); Yoshida (2019).

As a generalization of submodular function, the k-submodular function still has diminishing marginal benefits, where the definition domain is extended from the collection of simple subsets to the collection of k disjoint subsets. Many practical applications can be attributed to the k-submodular maximization problem. Ohsaka and Yoshida (2015) studied influence maximization with k topics and sensor placement with k sensors both based on k-submodular maximization with a size constraint. Rafiey and Yoshida (2020) applied k-submodular maximization to facility location.

In recent years, many researches on k-submodular maximization has sprung up. For k-submodular maximization without monotonicity assumption, Ward and Zivny (2014) studied the unconstrained problem and gave a deterministic greedy algorithm and a randomized greedy algorithm achieving the approximate ratio of 1/3 and $\frac{1}{1+a}$ with $a=\max \{1, \sqrt{\frac{k-1}{4}}\}$, respectively. Later, the approximation ratio was improved to 1/2 by Iwata et al. (2016). And Oshima (2021) also contributed a $\frac{k^2+1}{2k^2+1}$-approximate algorithm. For monotone k-submodular maximization, Ward and Zivny (2014) showed a 1/2-approximate algorithm without constraint, and then it was improved to $k/(2k-1)$ by Iwata et al. (2016), which is asymptotically tight. Ohsaka and Yoshida (2015) introduced a construction method between current solution and optimal solution to obtain a 1/2-approximate ratio, for a total size constraint. Using the similar construction method, a 1/2-approximate ratio could be also achieved by Sakaue (2017) for a matroid constraint. Tang et al. (2022) contributed a $\frac{1}{2}(1-e^{-1})$-approximate algorithm with a knapsack constraint. Xiao et al. found that this result could be improved to $\frac{1}{2}(1-e^{-2})$. Recently, Liu et al. (2022a) designed a nested greedy and local search $\frac{1}{2(m+1)}(1-e^{-(m+1)})$-approximation algorithm for monotone k-submodular maximization subject to the intersection of a knapsack and m matroid constraints.

1.2 Our contributions

In this paper, we consider the k-submodular maximization subject to the intersection of a knapsack and m matroid constraints, and discuss the results in monotone and non monotone cases respectively. The main contributions of this paper are as follows:

• We improve the approximate ratio from $\frac{1}{2(m+1)}(1-e^{-(m+1)})$ in Liu et al. (2022a) to $\frac{1}{m+2}(1-e^{-(m+2)})$ for monotone k-submodular maximization problem with the intersection of a knapsack and m matroid constraints. In the theoretical analysis of the algorithm, we no longer rely on the conclusion of the greedy algorithm for unconstrained k-submodular maximization problem, and use the properties of k-submodular function to get the new result. Note that our result will be $\frac{1}{3}(1-e^{-3})$ when $m=1$, it improves the result $\frac{1}{4}(1-e^{-2})$ in Liu et al. (2022a) with the intersection of a knapsack and a matroid constraint.

• We extend the approximation algorithm to non-monotone case. By increasing the number of enumeration points in the algorithm and using the pairwise monotone property, we achieve a $\frac{1}{m+3}(1-e^{-(m+3)})$ approximate ratio. It is easy to know that we have a $\frac{1}{4}(1-e^{-4})$ approximate ratio for the non-monotone k-submodular maximization problem with the intersection of a knapsack and a matroid constraint.

1.3 Organization

Organize our paper as follows: In Sect. 2, we introduce notations, properties and some basic results about k-submodular function. In Sect. 3, we give and explain the nested greedy and local search algorithm. In Sects. 4 and 5, we present our theoretical analysis and show the main results for monotone case and non-monotone case, respectively.

2 Preliminaries

2.1 k-Submodular function

In this paper, we set $k\ge 2$ and $k\in N_+$, because k-submodular function is submodular function when $k=1$. For any two k disjoint subsets $\textbf{x},~\textbf{y}\in (k+1)^{G}$, we need to introduce a remove operation and a partial order, i.e.

$$\begin{aligned}\textbf{x}\setminus \textbf{y}:= (X_1\setminus Y_1,\ldots ,X_k\setminus Y_k),\end{aligned}$$

$\textbf{x}\preceq \textbf{y}$, if $X_i\subseteq Y_i, \forall i\in [k]$.

Define one-item $\textbf{1}_{v,i}:=(X_1,\dots ,X_k)$, where $X_i=\{v\}$ and $X_{j\ne i}=\emptyset$, and empty-item $\textbf{0}:=(\emptyset ,\dots ,\emptyset )$. Denote the support set $U(\textbf{x}):=\bigcup _{i=1}^{k}X_i$.

Given a function $f:(k+1)^G\rightarrow R$, for any $\textbf{x}\in (k+1)^G$, $v\in G\setminus U(\textbf{x})$ and $i\in [k]$, it is said to be monotone if its marginal gain satisfies:

$$\begin{aligned} \begin{aligned} f_{\textbf{x}}(\textbf{1}_{v,i})=f(\textbf{x}\sqcup \textbf{1}_{v,i})-f(\textbf{x})\ge 0. \end{aligned} \end{aligned}$$

From Ohsaka and Yoshida (2015), f is pairwise monotone if

$$\begin{aligned} f_{\textbf{x}}(\textbf{1}_{v,i})+f_{\textbf{x}}(\textbf{1}_{v,j})\ge 0, \end{aligned}$$

for any $\textbf{x}\in (k+1)^G$, $v\in G\setminus U(\textbf{x})$ and $i\ne j\in [k]$. And f is orthant submodular, if

$$\begin{aligned} f_{\textbf{x}}(\textbf{1}_{v,i})\ge f_{\textbf{y}}(\textbf{1}_{v,i}), \end{aligned}$$

for $\textbf{x}\preceq \textbf{y}\in (k+1)^G$, $v\in G{\setminus } U(\textbf{y})$ and $i\ne j\in [k]$. As below, a k-submodular function has a well-known equivalent definition (Ward and Zivny 2014).

Definition 1

A function $f:(k+1)^G\rightarrow R$ is k-submodular iff it is pairwise monotone and orthant submodular.

Obviously, the monotonicity of f implies pairwise monotonicity. For a monotone function $f:(k+1)^G\rightarrow R$, the k-submodularity is equivalent to the orthant submodularity. In addition, a k-submodular function also has the following useful property (Ohsaka and Yoshida 2015).

Lemma 1

Given a k-submodular function f, we have

$$\begin{aligned}f(\textbf{y})-f(\textbf{x})\le \sum \limits _{\textbf{1}_{v,i}\preceq \textbf{y}\backslash \textbf{x}}f_{\textbf{x}}(\textbf{1}_{v,i}),\end{aligned}$$

for any $\textbf{x}, \textbf{y}\in (k+1)^G$ and $\textbf{x}\preceq \textbf{y}$.

Given a fixed k disjoint subsets $\textbf{y}\in (k+1)^G$, define a family of k disjoint subsets $D(\textbf{y}):=\{\textbf{x}\in (k+1)^G~|~\textbf{y}\preceq \textbf{x}\}$. In the later analysis, we need to construct a function $g(\textbf{x}): D(\textbf{y})\rightarrow R$ by temporarily hiding $\textbf{y}$. In order to maintain the regularity, we can set a k-submodular function $g(\textbf{x})=f(\textbf{x})-f(\textbf{y})$, which is still a k-submodular function.

Lemma 2

Given a k-submodular function $f: (k+1)^{G}\rightarrow R$ and $\textbf{y}\in (k+1)^G$, then $g(\textbf{x})=f(\textbf{x})-f(\textbf{y}): D(\textbf{y})\rightarrow R$ is a k-submodular function and $g(\textbf{y})=0$.

2.2 Knapsack and matroid constraints

Given $\mathcal {L}\subseteq 2^G$, a pair $(G,\mathcal {L})$ is an independence system if $(\mathcal {M}1)$ and $(\mathcal {M}2)$ hold, and a set A is an independence set if $A\in \mathcal {L}$. Further, the independence system $(G,\mathcal {L})$ is said to be a matroid if $(\mathcal {M}3)$ holds.

Definition 2

Given $\mathcal {L}\subseteq 2^G$ and a pair $\mathcal {M}=(G,\mathcal {L})$ is a matroid if

$(\mathcal {M}1)$: $\emptyset \in \mathcal {L}$.

$(\mathcal {M}2)$: $A\subseteq B$ and $B\in \mathcal {L}$ $\Longrightarrow$ $A\in \mathcal {L}$.

$(\mathcal {M}3)$: $A,B\in \mathcal {L}$ and $\mid A\mid >\mid B\mid$ $\Longrightarrow$ $\exists ~ v\in A\backslash B$, s.t. $B\cup \{v\}\in \mathcal {L}$.

For $m\in N_+$ and each $j\in [m]$, $\mathcal {L}_j$ is a collection of independent sets, and $\mathcal {M}_j = (G, \mathcal {L}_j)$ is a matroid. Given a nonnegative bound B, and for each element $v\in G$, there is a nonnegative weight $w_v$. Without losing generality, we assume that $w_v$ and B are integers. Otherwise, we can always enlarge them to integers in the same proportion. Let $w_{\textbf{x}}=\sum \limits _{v\in U(\textbf{x})}w_v$. The k-submodular maximization problem with the intersection of a knapsack and m matroid constraints is

$$\begin{aligned} \max _{\textbf{x}\in (k+1)^G}\{f(\textbf{x})\mid w_{\textbf{x}}\le B ~\textrm{and} ~ U(\textbf{x})\in \bigcap _{j=1}^{m}\mathcal {L}_j \}. \end{aligned}$$

(1)

For any $A\in G$, we use $[A]^{m}$ to express a collection of subsets of A, whose size does not exceed m. Given an independence set $A\in \bigcap _{j=1}^{m}\mathcal {L}_j$ and a pair $(\bar{a},b)$ with $\bar{a}\in [A]^{m}$ and $b\in G\backslash A$, we refer the pair $(\bar{a},b)$ as a m-swap $(\bar{a},b)$ if $(A\backslash \bar{a})\cup \{b\}\in \bigcap _{j=1}^{m}\mathcal {L}_j$. The next lemma ensures that there exists some m-swap $(\bar{a},b)$ between two independence sets. The detailed proof of Lemma 3 is given by Sarpatwar et al. (2019).

Lemma 3

Assume two independence sets $A,B\in \bigcap _{j=1}^{m}\mathcal {L}_j$, then we can construct a mapping $y: B\backslash A\rightarrow [A\backslash B]^{m}$, such that $(A\backslash \bar{a})\cup \{b\} \in \bigcap _{j=1}^{m}\mathcal {L}_j$ with $b\in B \backslash A$, $\bar{a}\in [A\backslash B]^{m}$, and each element $a\in A\backslash B$ appears in mapping y no more than m times.

In the later theoretical proof, the following Lemma 4 (Nemhauser et al. 1978) needs to be used.

Lemma 4

Given two fixed $P,D\in N_+$ and a sequence of nonnegative real numbers $\{\gamma _i\}_{i\in [P]}$, then we have

$$\begin{aligned}{} & {} \frac{\sum _{i=1}^P \gamma _i}{\min _{t\in [P]}(\sum _{i=1}^{t-1}\gamma _i+D\gamma _t)}\nonumber \\{} & {} \quad \ge 1-(1-\frac{1}{D})^P \ge 1-e^{-P/D}. \end{aligned}$$

(2)

3 Algorithm overview

3.1 Greedy algorithm

Firstly, we introduce a Greedy Algorithm (f, G) from Ward and Zivny (2014). By Definition 1, k-submodularity of f implies pairwise monotonicity, that is, $f_{\textbf{x}}(\textbf{1}_{v,i})+f_{\textbf{x}}(\textbf{1}_{v,j})\ge 0$ for any $\textbf{x}\in (k+1)^G$, $v\notin U(\textbf{x})$ and $i\ne j\in [k]$. It means that there are no two positions $i\ne j\in [k]$ such that $f_{\textbf{x}}(\textbf{1}_{v,i})<0$ and $f_{\textbf{x}}(\textbf{1}_{v,j})<0$ both hold. For k-submodular maximization problem without constraint, there is always an optimal solution $\textbf{x}^*$ satisfying $U(\textbf{x}^*)=G$. In Greedy Algorithm (f, G), we enter a set G and give a fixed order to the points in G, that is $G=\{v_1,\dots ,v_{|G|}\}$. Each current solution $\textbf{x}_l$ is obtained by $\textbf{x}_{l-1}$ adding $v_{l}\in G\backslash U(\textbf{x}_{l-1})$ with a greedy position $i_{l}\in [k]$ for $l=1,\ldots ,|G|$.

3.2 Nested greedy and local search algorithm KM-KM

Next, we present a nested greedy and local search algorithm for problem (1), which is inspired by Liu et al. (2022a). For simplicity, we call it KM-KM. If the objective function f is monotone, we choose $\lambda =2$ in KM-KM. Otherwise, we need to choose $\lambda \ge \frac{(m+1)(m+3)}{m+2+e^{-(m+3)}}$, because of the proof of the approximate ratio.

KM-KM starts with $\textbf{x}^\lambda \preceq \textbf{x}^*$ obtained by enumerating with the largest marginal profits, where $\textbf{x}^*$ is an optimal solution of problem (1). If $|U(\textbf{x}^*)|\le \lambda$, we can find $\textbf{x}^*$ by enumerating $\textbf{x}\in (k+1)^G$ with $|U(\textbf{x})|\le |U(\textbf{x}^*)|$. Therefore, we only consider the case when $|U(\textbf{x}^*)|$ is greater than $\lambda$. For a positive integer $t\ge \lambda$, we define t-th iteration as the process when KM-KM finds a suitable m-swap $(\bar{a}^t,b^t)$ to update $\textbf{x}^{t}$. Clearly $|(U(\textbf{x}^{t}\backslash \textbf{x}^\lambda )\backslash \bar{a}^t)\cup \{b^t\}|=|U(\textbf{x}^{t+1}\backslash \textbf{x}^\lambda )|$. If the current m-swap $(\bar{a}^t,b^t)$ satisfies all the conditions in line 11, KM-KM performs line 12-18 and breaks loop 9-19 to update $S^m$ in line 8. In line 12 of KM-KM, we consider the elements in $(U(\textbf{x}^{t}\backslash \textbf{x}^\lambda )\backslash \bar{a}^t)\cup \{b^t\}$, and add them to Greedy Algorithm in the same order as in KM-KM. For $l\in \{1,\dots ,|(U(\textbf{x}^{t}\backslash \textbf{x}^\lambda )\backslash \bar{a}^t)\cup \{b^t\}|\}$, Greedy Algorithm $(f(\tilde{\textbf{x}}^{t+1}\sqcup \textbf{x}^\lambda ), ~(U(\textbf{x}^{t}\backslash \textbf{x}^\lambda )\backslash \bar{a}^t)\cup \{b^t\})$ reorders the positions i of points $v_l\in (U(\textbf{x}^{t}\backslash \textbf{x}^\lambda )\backslash \bar{a}^t)\cup \{b^t\}$. Define $\tilde{\textbf{x}}^{t+1}_l$ as the current solution, such that $\tilde{\textbf{x}}^{t+1}_l=\tilde{\textbf{x}}^{t+1}_{l-1}\sqcup \textbf{1}_{v_l,i_l}$. If current m-swap $(\bar{a}^t,b^t)$ violates any conditions in line 11, KM-KM will remove it and continue to pick the next m-swap. Finally, KM-KM breaks all loops when $S^m=\emptyset$ in line 9 and return $\textbf{x}^t$. We define the time when KM-KM outputs $\textbf{x}^t$ as T and $T\ge \lambda +1$.

3.3 A construction method for analysis

In order to give an approximate ratio analysis, we introduce a construction method based on Algorithm 2. Mark $\textbf{x}^*$ as an optimal solution of problem (1).

Given a fixed iteration step $t\ge \lambda +1$ in KM-KM and $l\in \{1, \dots ,|U(\textbf{x}^t{\setminus } \textbf{x}^\lambda )|\}$. Define $\textbf{x}^t_l=\tilde{\textbf{x}}^t_l\sqcup \textbf{x}^\lambda$, then $\textbf{x}^t_{|U(\textbf{x}^t\backslash \textbf{x}^\lambda )|}=\textbf{x}^t$. We further construct two sequences $\{\textbf{o}_{l-1/2}^t\}$ and $\{\textbf{o}_{l}^t\}$ such that $\textbf{o}_{l-1/2}^t =(\textbf{x}^*~\sqcup ~ \textbf{x}_{l}^t)~\sqcup ~\textbf{x}_{l-1}^t$, $\textbf{o}_{l}^t =(\textbf{x}^*~\sqcup ~\textbf{x}_{l}^t)~\sqcup ~\textbf{x}_l^t$ and $\textbf{o}_{l=0}^t=\textbf{x}^*$. Note that $\textbf{x}^t_{l-1}\preceq \textbf{x}^t_l\preceq \textbf{o}_{l}^t$, $\textbf{o}_{l-1/2}^t\preceq \textbf{o}_{l}^t$ and $U(\textbf{o}^t_{|U(\textbf{x}^t\backslash \textbf{x}^\lambda )|})\backslash U(\textbf{x}^t)=U(\mathbf {x^*})\backslash U(\textbf{x}^t)$.

By Lemma 2, define a k-submodular function $g(\textbf{x})=f(\textbf{x})-f(\textbf{x}^\lambda ):D(\textbf{x}^\lambda )\rightarrow R_+$. The construction method has the following conclusions. The detailed proofs of them are shown in the Appendix.

Lemma 5

Given a fixed iteration step $t\ge \lambda +1$ in KM-KM and an optimal solution $\textbf{x}^*$ for problem (1), we have : 

(i) when the objective function f is monotone,

$$\begin{aligned}{} & {} g(\textbf{o}_{l-1}^t)-g(\textbf{o}_{l}^t)\le g(\textbf{x}^t_{l})-g(\textbf{x}^t_{l-1}), \end{aligned}$$

(3)

$$\begin{aligned}{} & {} g(\textbf{x}^*)\le g(\textbf{o}^t_{|U(\textbf{x}^t\backslash \textbf{x}^\lambda )|})+g(\textbf{x}^t). \end{aligned}$$

(4)

(ii) when the objective function f is non-monotone,

$$\begin{aligned}{} & {} g(\textbf{o}_{l-1}^t)-g(\textbf{o}_{l}^t)\le 2[g(\textbf{x}^t_{l})-g(\textbf{x}^t_{l-1})], \end{aligned}$$

(5)

$$\begin{aligned}{} & {} g(\textbf{x}^*)\le g(\textbf{o}^t_{|U(\textbf{x}^t\backslash \textbf{x}^\lambda )|})+2g(\textbf{x}^t). \end{aligned}$$

(6)

4 Analysis for monotone k-submodular maximization with a knapsack and m matroid constraints

In this section, we will explain in detail how to obtain the approximate ratio for problem (1). Our framework of proof is inspired by Sviridenko (2004); Sarpatwar et al. (2019); Liu et al. (2022a). To simplify the process of analyzing approximate ratio, we give several lemmas. The detailed proofs of them are shown in the Appendix.

Lemma 6

Given a fixed iteration step $t\ge \lambda +1$ in KM-KM and an optimal solution $\textbf{x}^*$ for problem (1), there exists a mapping y : 

$$\begin{aligned} U(\textbf{o}^t_{|U(\textbf{x}^t\backslash \textbf{x}^\lambda )|})\backslash U(\textbf{x}^t)\rightarrow [U(\textbf{x}^t)\backslash U(\textbf{x}^*)]^m \end{aligned}$$

such that $(U(\textbf{x}^t)\backslash \bar{y}(b))\cup \{b\}\in \bigcap _{j=1}^{m}\mathcal {L}_j$, for $~b\in U(\textbf{o}^t_{|U(\textbf{x}^t\backslash \textbf{x}^\lambda )|})\backslash U(\textbf{x}^t)$, $\bar{y}(b)\in [U(\textbf{x}^t)\backslash U(\textbf{x}^*)]^m$, and each element $a\in U(\textbf{x}^t)\backslash U(\textbf{x}^*)$ appears in mapping y no more than m times. Then we have

$$\begin{aligned} \begin{aligned} g(\textbf{o}^t_{|U(\textbf{x}^t\backslash \textbf{x}^\lambda )|})&\le \sum \limits _{\textbf{1}_{b,i}\preceq (\textbf{o}^t_{|U(\textbf{x}^t\backslash \textbf{x}^\lambda )|}\backslash \textbf{x}^t)} [g((\textbf{x}^t\backslash \bigsqcup _{y(b)\in \bar{y}(b)}\textbf{1}_{y(b),j})\sqcup \textbf{1}_{b,i})\\&\quad -g(\textbf{x}^t\backslash \bigsqcup _{y(b)\in \bar{y}(b)}\textbf{1}_{y(b),j})] +g(\textbf{x}^t) \end{aligned}\end{aligned}$$

(7)

and

$$\begin{aligned} \sum \limits _{\textbf{1}_{b,i}\preceq (\textbf{o}^t_{|U(\textbf{x}^t\backslash \textbf{x}^\lambda )|}\backslash \textbf{x}^t)} [g(\textbf{x}^t)-g(\textbf{x}^t\backslash \bigsqcup _{y(b)\in \bar{y}(b)}\textbf{1}_{y(b),j})] \le mg(\textbf{x}^t). \end{aligned}$$

(8)

for $\textbf{1}_{y(b),j}\preceq \textbf{x}^t\backslash \textbf{x}^\lambda$.

Let us assume that there exists a m-swap $(\bar{y}(b),b)$ with respect to $\textbf{x}^T$ and $\textbf{x}^*$ satisfying $w_{\textbf{x}^T}-w_{\bar{y}(b)}+ w_b > B$, when KM-KM runs. Let $t^*+1$ be the iteration which appears a m-swap $(\bar{y}(b^{t^*}),b^{t^*})$ in $S^m(U(\textbf{x}^{t^*}))\setminus \{m-$swap $(\bar{a},b)~|~\bar{a}\cap U(\textbf{x}^\lambda )\ne \emptyset \}$ violating $w_{\textbf{x}^{t^*}}-w_{\bar{y}(b^{t^*})}+ w_{b^{t^*}} \le B$, with $b^{t^*}\in U(\textbf{x}^*)\backslash U(\textbf{x}^{t^*})$ and $\bar{y}(b^{t^*})\in [(U(\textbf{x}^t)\backslash U(\textbf{x}^*))]^m$, for the first time.

Lemma 7

Considering the current solution $\textbf{x}^{t^*}$ and the m-swap $(\bar{y}(b^{t^*}),b^{t^*})$ mentioned above, we have

$$\begin{aligned} \begin{aligned}&f((\textbf{x}^{t^*}\setminus \bigsqcup _{y(b^{t^*})\in \bar{y}(b^{t^*})}\textbf{1}_{y(b^{t^*}),j^{t^*}})\sqcup \textbf{1}_{b^{t^*},i^{t^*}})-f(\textbf{x}^{t^*}) \le \frac{1}{2}\cdot f(\textbf{x}^\lambda ), \end{aligned} \end{aligned}$$

(9)

where $\textbf{1}_{y(b^{t^*}),j^{t^*}}\preceq \textbf{x}^{t^*}\backslash \textbf{x}^\lambda$, if f is monotone.

Lemma 8

Given $t\in \{\lambda +1,\dots ,t^*\}$ in KM-KM for problem (1), we have

$$\begin{aligned}{} & {} \sum \limits _{\textbf{1}_{b,i}\preceq (\textbf{o}^t_{|U(\textbf{x}^t\backslash \textbf{x}^\lambda )|}\backslash \textbf{x}^t)} [g((\textbf{x}^t\backslash \bigsqcup _{y(b)\in \bar{y}(b)}\textbf{1}_{y(b),j})\sqcup \textbf{1}_{b,i})-g(\textbf{x}^t)]\nonumber \\{} & {} \quad \le (B-w_{\textbf{x}^\lambda })\rho _{t}, \end{aligned}$$

(10)

for $\textbf{1}_{y(b),j}\preceq \textbf{x}^t\backslash \textbf{x}^\lambda$.

Lemma 9

Given $t\in \{\lambda +1,\dots ,t^*\}$ in KM-KM, $\alpha , \beta$, r are positive constants satisfying $1-\frac{1}{\alpha }(1-e^{-\beta })-r\ge 0$ and $\textbf{x}^*$ be an optimal solution of problem (1). If

$$\begin{aligned} g(\textbf{x}^*)\le \alpha [g(\textbf{x}^t)+\frac{(B-w_{\textbf{x}^\lambda })}{\beta }\rho _{t}] \end{aligned}$$

and

$$\begin{aligned} f((\textbf{x}^{t^*}\setminus \bigsqcup _{y(b^{t^*})\in \bar{y}(b^{t^*})}\textbf{1}_{y(b^{t^*}),j^{t^*}})\sqcup \textbf{1}_{b^{t^*},i^{t^*}})-f(\textbf{x}^{t^*}) \le r\cdot f(\textbf{x}^\lambda ) \end{aligned}$$

hold, we have

$$\begin{aligned} f(\textbf{x}^{t^*})\ge \frac{1}{\alpha }(1-e^{-\beta })f(\textbf{x}^*). \end{aligned}$$

(11)

Theorem 1

If the objective function f is monotone for problem (1), we can obtain a $\frac{1}{m+2}(1-e^{-(m+2)})$-approximate solution in KM-KM by setting $\lambda =2$.

Proof

When there is no qualified m-swap$(\bar{a},b)\in S^m$, KM-KM will break all loops and output $\textbf{x}^T$. Using Lemma 3 between $U(\textbf{x}^t)$ and $U(\textbf{x}^*)$, for a fixed $t\ge \lambda$, there exists a mapping y :  $U(\mathbf {x^*})\backslash U(\textbf{x}^t)\rightarrow [U(\textbf{x}^t)\backslash U(\textbf{x}^*)]^m$ such that $(U(\textbf{x}^t)\backslash \bar{y}(b))\cup \{b\}\in \bigcap _{j=1}^{m}\mathcal {L}_j$, for $~b\in U(\mathbf {x^*})\backslash U(\textbf{x}^t)$ and $\bar{y}(b)\in [U(\textbf{x}^t)\backslash U(\textbf{x}^*)]^m$. Thus, there are some m-swaps $(\bar{y}(b),b)$ with respect to $\textbf{x}^t$ and $\textbf{x}^*$.

When $t=T$, according to whether the conditions in line  11 of KM-KM are violated, consider dividing m-swaps (y(b), b) with respect to $\textbf{x}^T$ and $\textbf{x}^*$ into two cases.

Case 1: Considering the m-swaps $(\bar{y}(b),b)$ with respect to $\textbf{x}^T$ and $\textbf{x}^*$, they were all rejected just due to $\rho (\bar{y}(b),b)\le 0$ instead of knapsack constraint.

Due to our assumption about the m-swaps, we get

$$\begin{aligned} \begin{aligned} g((\textbf{x}^T\backslash \bigsqcup _{y(b)\in \bar{y}(b)}\textbf{1}_{y(b),j})\sqcup \textbf{1}_{b,i})\le g(\textbf{x}^T). \end{aligned} \end{aligned}$$

(12)

Since f is monotone, we combine formula (4) in Lemma 5 and formula (7) in Lemma 6, then use formula (12) and formula (8) in Lemma 6 to get $g(\textbf{x}^*)\le (m+2)g(\textbf{x}^T)$. Finally, we have $f(\textbf{x}^*)\le (m+2)f(\textbf{x}^T)-(m+1)f(\textbf{x}^\lambda )\le (m+2)f(\textbf{x}^T)$ due to nonnegativity of f. Therefore, we find a $\frac{1}{m+2}$-approximate solution in case 1, if f is monotone.

Case 2: Considering the m-swaps $(\bar{y}(b),b)$ with respect to $\textbf{x}^T$ and $\textbf{x}^*$, there exists at least one satisfying $w_{\textbf{x}^t}-w_{\bar{y}(b)}+ w_b > B$.

For a fixed $t\ge \lambda$, KM-KM selects a qualified m-swap$(\bar{a}^t,b^t)$ to update $\textbf{x}^{t}$ in each t-th iteration. In $t^*+1$ iteration, KM-KM checks m-swap $(\bar{y}(b^{t^*}),b^{t^*})$, where $b^{t^*}\in U(\textbf{x}^*)\backslash U(\textbf{x}^{t^*})$ and $\bar{y}(b^{t^*})\in [(U(\textbf{x}^t)\backslash U(\textbf{x}^*))]^m$, in line 11 and removed it due to $w_{\textbf{x}^{t^*}}-w_{\bar{y}(b^{t^*})}+ w_{b^{t^*}} > B$, for the first time. Define $\rho _t:=\rho (\bar{a}^t,b^t)$ for $t\in \{\lambda ,\dots ,t^*-1\}$ and

$$\begin{aligned} \rho _{t^*}:=\frac{f((\textbf{x}^{t^*}\setminus \bigsqcup _{y(b^{t^*})\in \bar{y}(b^{t^*})}\textbf{1}_{y(b^{t^*}),j^{t^*}})\sqcup \textbf{1}_{b^{t^*},i^{t^*}})-f(\textbf{x}^{t^*})}{w_{b^{t^*}}}. \end{aligned}$$

When $t\in \{\lambda +1,\ldots ,t^{*}\}$, we combine formula (4) in Lemma 5 and formula (7) in Lemma 6, then rewrite formula (7) in Lemma 6 as below

$$\begin{aligned} \begin{aligned}&g((\textbf{x}^t\backslash \bigsqcup _{y(b)\in \bar{y}(b)}\textbf{1}_{y(b),j})\sqcup \textbf{1}_{b,i}) -g((\textbf{x}^t\backslash \bigsqcup _{y(b)\in \bar{y}(b)}\textbf{1}_{y(b),j}))\\&=[g((\textbf{x}^t\backslash \bigsqcup _{y(b)\in \bar{y}(b)}\textbf{1}_{y(b),j})\sqcup \textbf{1}_{b,i})-g(\textbf{x}^t)]\\&+[g(\textbf{x}^t)-g((\textbf{x}^t\backslash \bigsqcup _{y(b)\in \bar{y}(b)}\textbf{1}_{y(b),j}))]. \end{aligned} \end{aligned}$$

(13)

Using formula (8) in Lemma 6 and Lemma 8, we can get $g(\textbf{x}^*)\le (m+2)[g(\textbf{x}^t)+\frac{(B-w_{\textbf{x}^\lambda })}{m+2}\rho _{t}].$ By formula (9) in Lemma  7, we set $r=\frac{1}{2}$ in Lemma 9. Therefore, $f(\textbf{x}^{t^*})\ge \frac{1}{m+2}(1-e^{-(m+2)})f(\textbf{x}^*)$ holds immediately. So we get the approximate ratio of $\frac{1}{m+2}(1-e^{-(m+2)})$ in case 2, if f is monotone. $\square$

As above, we show a $\frac{1}{m+2}(1-e^{-(m+2)})$-approximate ratio for monotone k-submodular maximization with a knapsack and m matroid constraints. Due to our conclusion, we improve the approximate ratio of monotone k-submodular maximization with a knapsack and m matroid constraints (Liu et al. 2022a) from $\frac{1}{2(m+1)}(1-e^{-(m+1)})$ to $\frac{1}{m+2}(1-e^{-(m+2)})$.

When $m=1$, i.e. monotone k-submodular maximization with a knapsack and a matroid constraints, we have the corresponding conclusion as below. It also improves the result $\frac{1}{4}(1-e^{-2})$ in Liu et al. (2022a).

Corollary 1

If the objective function f is monotone for problem (1) with $m=1$, we can obtain a $\frac{1}{3}(1-e^{-3})$-approximate solution in KM-KM by setting $\lambda =2$.

5 Analysis for non-monotone k-submodular maximization with a knapsack and m matroid constraints

In this section, we further study non-monotone k-submodular maximization with a knapsack and m matroids constraints. In fact, the impact of monotonicity of f is not reflected in Lemmas 6, 8,  9. So we only need to give the following Lemma 10. Using Lemmas 6, 8, 9 10, we can get an approximate ratio $\frac{1}{m+3}(1-e^{-(m+3)})$.

Lemma 10

Considering the current solution $\textbf{x}^{t^*}$ and the m-swap $(\bar{y}(b^{t^*}),b^{t^*})$ as in Lemma 7, we have

$$\begin{aligned} \begin{aligned}&f((\textbf{x}^{t^*}\setminus \bigsqcup _{y(b^{t^*})\in \bar{y}(b^{t^*})}\textbf{1}_{y(b^{t^*}),j^{t^*}})\sqcup \textbf{1}_{b^{t^*},i^{t^*}})-f(\textbf{x}^{t^*}) \le \frac{m+1}{\lambda }\cdot f(\textbf{x}^\lambda ), \end{aligned} \end{aligned}$$

(14)

where $\textbf{1}_{y(b^{t^*}),j^{t^*}}\preceq \textbf{x}^{t^*}\backslash \textbf{x}^\lambda$.

Theorem 2

If the objective function f is non-monotone for problem (1), we can obtain a $\frac{1}{m+3}(1-e^{-(m+3)})$-approximate solution in KM-KM by setting $\lambda \ge \frac{(m+1)(m+3)}{m+2+e^{-(m+3)}}$.

Proof

When $t=T$, similar to Theorem 1 in Sect. 4, we consider dividing the m-swaps (y(b), b) with respect to $\textbf{x}^T$ and $\textbf{x}^*$ into two cases.

Case 1: Considering the m-swaps $(\bar{y}(b),b)$ with respect to $\textbf{x}^T$ and $\textbf{x}^*$, they were all rejected just due to $\rho (\bar{y}(b),b)\le 0$ instead of knapsack constraint.

Combine formula (6) in Lemma 5 and formula (7) in Lemma 6, then use formula (12) and formula (8) in Lemma 6 to get $g(\textbf{x}^*)\le (m+3)g(\textbf{x}^T)$. Finally, we have $f(\textbf{x}^*)\le (m+3)f(\textbf{x}^T)-(m+2)f(\textbf{x}^\lambda )\le (m+3)f(\textbf{x}^T)$ due to nonnegativity of f. Therefore, we find a $\frac{1}{m+3}$-approximate solution in case 1, if f is non-monotone in problem (1).

Case 2: Considering the m-swaps $(\bar{y}(b),b)$ with respect to $\textbf{x}^T$ and $\textbf{x}^*$, there exists at least one satisfying $w_{\textbf{x}^t}-w_{\bar{y}(b)}+ w_b > B$.

When $t\in \{\lambda +1,\ldots ,t^{*}\}$, we combine formula (6) in Lemma 5, formula (7) in Lemma 6 and formula(13). Then use formula (8) in Lemma 6 and Lemma 8, we can get $g(\textbf{x}^*)\le (m+3)[g(\textbf{x}^t)+\frac{(B-w_{\textbf{x}^\lambda })}{m+3}\rho _{t}].$ By formula (14) in Lemma  10, we set $r=\frac{m+1}{\lambda }$ in Lemma 9. Therefore, $f(\textbf{x}^{t^*})\ge \frac{1}{m+3}(1-e^{-(m+3)})f(\textbf{x}^*)$ holds immediately. So we get the approximate ratio of $\frac{1}{m+3}(1-e^{-(m+3)})$ in case 2, if f is non-monotone in problem (1). $\square$

As above, we show a $\frac{1}{m+3}(1-e^{-(m+3)})$-approximate ratio for non-monotone k-submodular maximization with a knapsack and m matroid constraints. Due to our conclusion, we extend monotone k-submodular maximization with a knapsack and m matroid constraints (Liu et al. 2022a) to non-monotone case.

When $m=1$, i.e. non-monotone k-submodular maximization with a knapsack and a matroid constraints, we have the corresponding conclusion as below.

Corollary 2

If the objective function f is non-monotone for problem (1) with $m=1$, we can obtain a $\frac{1}{4}(1-e^{-4})$-approximate solution in KM-KM by setting $\lambda =3$.

6 Conclusions

In our paper, based on a nested greedy and local search algorithm KM-KM (Liu et al. 2022a) and a construction method (Nguyen and Thai 2020), we improve the approximate ratio for problem (1) (Liu et al. 2022a) from $\frac{1}{2(m+1)}(1-e^{-(m+1)})$ to $\frac{1}{m+2}(1-e^{-(m+2)})$ by enumerating $\lambda =2$ items with the largest marginal profits in the optimal solution. The conclusion can get $\frac{1}{3}(1-e^{-3})$-approximate ratio for problem (1) with $m=1$. Furthermore, we extend the conclusion to non-monotone case and get the approximate ratio $\frac{1}{m+3}(1-e^{-(m+3)})$ for problem (1) by enumerating $\lambda \ge \frac{(m+1)(m+3)}{m+2+e^{-(m+3)}}$ items with $\lambda \in N_+$. The conclusion can get $\frac{1}{4}(1-e^{-4})$ -approximate ratio for problem (1) with $m=1$. And we need to enumerate $\lambda =3$ items with the largest marginal profits in the optimal solution.

Appendix

Proof of Lemma 5:

When k-submodular function f is monotone, the conclusions are as follows. Due to monotonicity of f, $f_{\textbf{x}^t_{l-1}}(\textbf{1}_{v_l.i_l})\ge 0$ holds in Greedy Algorithm. By the definition of g, we have $g_{\textbf{x}^t_{l-1}}(\textbf{1}_{v_l.i_l})\ge 0$. For $v_l$ in l-th iteration of Greedy Algorithm, we compare the position $i_l$ with $\textbf{1}_{v_l,i_l}\preceq \textbf{x}^t_l$ and $i_*$ with $\textbf{1}_{v_l,i_*}\preceq \textbf{x}^*$.

If $v_l\in U(\textbf{x}^*)$ with $i_*=i_l$, then $\textbf{o}_{l-1}^t=\textbf{o}_l^t$. Therefore, we have

$$\begin{aligned} \begin{aligned} g(\textbf{o}_{l-1}^t)-g(\textbf{o}_l^t)=0\le g(\textbf{x}^t_{l})-g(\textbf{x}^t_{l-1}). \end{aligned} \end{aligned}$$

(15)

If $v_l\in U(\textbf{x}^*)$ with $i_*\ne i_l$, then $\textbf{o}_{l-1}^t=\textbf{o}_{l-1/2}^t\sqcup \textbf{1}_{v_l,i_*}$ and $\textbf{o}_{l}^t=\textbf{o}_{l-1/2}^t\sqcup \textbf{1}_{v_l,i_l}$. By monotonicity of f, greedy choice of Greedy Algorithm and orthant submodularity, we get

$$\begin{aligned} \begin{aligned} g(\textbf{o}_{l-1}^t)-g(\textbf{o}_l^t)&\le g(\textbf{o}_{l-1}^t)-g(\textbf{o}_{l-1/2}^t)\\&= g_{\textbf{o}_{l-1/2}^t}(\textbf{1}_{v_l,i_*})\\&\le g_{\textbf{x}^t_{l-1}}(\textbf{1}_{v_l,i_*})\\&\le g(\textbf{x}^t_{l})-g(\textbf{x}^t_{l-1}). \end{aligned} \end{aligned}$$

(16)

If $v_l\notin U(\textbf{x}^*)$, then $\textbf{o}_{l-1/2}^t=\textbf{o}_{l}^t=\textbf{o}_{l-1}^t\sqcup \textbf{1}_{v_l,i_l}$, we have

$$\begin{aligned} \begin{aligned} g(\textbf{o}_{l-1}^t)-g(\textbf{o}_l^t)\le 0\le g(\textbf{x}^t_{l})-g(\textbf{x}^t_{l-1}). \end{aligned} \end{aligned}$$

(17)

In summary, we have

$$\begin{aligned} \begin{aligned} g(\textbf{o}_{l-1}^t)-g(\textbf{o}_l^t)\le g(\textbf{x}^t_{l})-g(\textbf{x}^t_{l-1}). \end{aligned} \end{aligned}$$

(18)

Sum it for l from 1 to $|U(\textbf{x}^t\backslash \textbf{x}^\lambda )|$ and get

$$\begin{aligned} \begin{aligned} g(\textbf{x}^*)-g(\textbf{o}^t_{|U(\textbf{x}^t\backslash \textbf{x}^\lambda )|})&= \sum ^{|U(\textbf{x}^t\backslash \textbf{x}^\lambda )|}_{l=1} [g(\textbf{o}_{l-1}^t)-g(\textbf{o}_l^t)]\\&\le \sum ^{|U(\textbf{x}^t\backslash \textbf{x}^\lambda )|}_{l=1} g(\textbf{x}_{l}^t)-g(\textbf{x}_{l-1}^t)\\&=g(\textbf{x}^t). \end{aligned} \end{aligned}$$

(19)

When k-submodular function f is non-monotone, the conclusion will change as below. Due to pairwise monotonicity, $f_{\textbf{x}^t_{l-1}}(\textbf{1}_{v_l.i_l})\ge 0$ holds in Greedy Algorithm. By the definition of g, we have $g_{\textbf{x}^t_{l-1}}(\textbf{1}_{v_l.i_l})\ge 0$.

If $v_l\in U(\textbf{x}^*)$ with $i_*=i_l$, we have

$$\begin{aligned} \begin{aligned} g(\textbf{o}_{l-1}^t)-g(\textbf{o}_l^t)=0\le 2[g(\textbf{x}^t_{l})-g(\textbf{x}^t_{l-1})]. \end{aligned} \end{aligned}$$

(20)

If $v_l\in U(\textbf{x}^*)$ with $i_*\ne i_l$, we get

$$\begin{aligned} \begin{aligned} g(\textbf{o}_{l-1}^t)-g(\textbf{o}_l^t)&=g_{\textbf{o}_{l-1/2}^t}(\textbf{1}_{v_l,i_*})+g_{\textbf{o}_{l-1/2}^t}(\textbf{1}_{v_l,i'})\\&-[g_{\textbf{o}_{l-1/2}^t}(\textbf{1}_{v_l,i_l})+g_{\textbf{o}_{l-1/2}^t}(\textbf{1}_{v_l,i'})]\\&\le g_{\textbf{o}_{l-1/2}^t}(\textbf{1}_{v_l,i_*})+g_{\textbf{o}_{l-1/2}^t}(\textbf{1}_{v_l,i'})\\&\le 2g_{\textbf{x}^t_{l-1}}(\textbf{1}_{v_l,i_l})\\&= 2[g(\textbf{x}^t_{l})-g(\textbf{x}^t_{l-1})] \end{aligned} \end{aligned}$$

(21)

for any $i'\in [k]$ with $i'\ne i_l$. Due to pairwise monotonicity, we get the first inequality. By greedy choice of Greedy Algorithm and orthant submodularity, the second holds.

If $v_l\notin U(\textbf{x}^*)$, similar to above, we have

$$\begin{aligned} \begin{aligned} g(\textbf{o}_{l-1}^t)-g(\textbf{o}_l^t)&=g_{\textbf{o}_{l-1}^t}(\textbf{1}_{v_l,i'}) -[g_{\textbf{o}_{l-1}^t}(\textbf{1}_{v_l,i'})+g_{\textbf{o}_{l-1}^t}(\textbf{1}_{v_l,i_l})]\\&\le g_{\textbf{o}_{l-1}^t}(\textbf{1}_{v_l,i'})\\&\le g_{\textbf{o}_{l-1}^t}(\textbf{1}_{v_l,i_l})\\&\le g_{\textbf{x}^t_{l-1}}(\textbf{1}_{v_l,i_l})\\&\le 2[g(\textbf{x}^t_{l})-g(\textbf{x}^t_{l-1})]. \end{aligned} \end{aligned}$$

(22)

In summary, we have

$$\begin{aligned} \begin{aligned} g(\textbf{o}_{l-1}^t)-g(\textbf{o}_l^t)\le 2[g(\textbf{x}^t_{l})-g(\textbf{x}^t_{l-1})]. \end{aligned} \end{aligned}$$

(23)

Sum it for l from 1 to $|U(\textbf{x}^t\backslash \textbf{x}^\lambda )|$ and get

$$\begin{aligned} \begin{aligned} g(\textbf{x}^*)-g(\textbf{o}^t_{|U(\textbf{x}^t\backslash \textbf{x}^\lambda )|})&= \sum ^{|U(\textbf{x}^t\backslash \textbf{x}^\lambda )|}_{l=1} [g(\textbf{o}_{l-1}^t)-g(\textbf{o}_l^t)]\\&\le \sum ^{|U(\textbf{x}^t\backslash \textbf{x}^\lambda )|}_{l=1} 2[g(\textbf{x}_{l}^t)-g(\textbf{x}_{l-1}^t)]\\&=2g(\textbf{x}^t). \end{aligned} \end{aligned}$$

(24)

Proof of Lemma 6:

Due to $\textbf{o}^t_{|U(\textbf{x}^t\backslash \textbf{x}^\lambda )|}=(\textbf{x}^*\sqcup \textbf{x}^t)\sqcup \textbf{x}^t$, we have $U(\textbf{o}^t_{|U(\textbf{x}^t\backslash \textbf{x}^\lambda )|})\backslash U(\textbf{x}^t)= U(\textbf{x}^*)\backslash U(\textbf{x}^t)$ and $\textbf{x}^t\preceq \textbf{o}^t_{|U(\textbf{x}^t\backslash \textbf{x}^\lambda )|}$. By Lemma 3 between $U(\textbf{x}^*)$ and $U(\textbf{x}^t)$, there exists a mapping y : 

$$\begin{aligned} U(\textbf{o}^t_{|U(\textbf{x}^t\backslash \textbf{x}^\lambda )|})\backslash U(\textbf{x}^t)\rightarrow [U(\textbf{x}^t)\backslash U(\textbf{x}^*)] \end{aligned}$$

such that $(U(\textbf{x}^t)\backslash \bar{y}(b))\cup \{b\}\in \bigcap _{j=1}^{m}\mathcal {L}_j$, for $~b\in U(\textbf{o}^t_{|U(\textbf{x}^t\backslash \textbf{x}^\lambda )|})\backslash U(\textbf{x}^t)$, $\bar{y}(b)\in [U(\textbf{x}^t)\backslash U(\textbf{x}^*)]^m$, and each element $a\in U(\textbf{x}^t)\backslash U(\textbf{x}^*)$ appears in mapping y no more than m times. Using Lemma 1 and the mapping $y:b\rightarrow \bar{y}(b)$, we get

$$\begin{aligned} \begin{aligned} g(\textbf{o}^t_{|U(\textbf{x}^t\backslash \textbf{x}^\lambda )|})&\le \sum \limits _{\textbf{1}_{b,i}\preceq (\textbf{o}^t_{|U(\textbf{x}^t\backslash \textbf{x}^\lambda )|}\backslash \textbf{x}^t)} [g(\textbf{x}^t\sqcup \textbf{1}_{b,i})-g(\textbf{x}^t)]+g(\textbf{x}^t) \\&\le \sum \limits _{\textbf{1}_{b,i}\preceq (\textbf{o}^t_{|U(\textbf{x}^t\backslash \textbf{x}^\lambda )|}\backslash \textbf{x}^t)} [g((\textbf{x}^t\backslash \bigsqcup _{y(b)\in \bar{y}(b)}\textbf{1}_{y(b),j})\sqcup \textbf{1}_{b,i})\\&-g(\textbf{x}^t\backslash \bigsqcup _{y(b)\in \bar{y}(b)}\textbf{1}_{y(b),j})] +g(\textbf{x}^t) \end{aligned} \end{aligned}$$

(25)

for $\textbf{1}_{y(b),j}\preceq \textbf{x}^t\backslash \textbf{x}^\lambda$.

Then we give the proof of second inequality as follows.

For fixed iteration step $t\ge \lambda +1$ in KM-KM, the ground set of Greedy Algorithm(f, G) is $G=\{v_1,\dots ,v_{|U(\textbf{x}^t\backslash \textbf{x}^\lambda )|}\}=U(\textbf{x}^t\backslash \textbf{x}^\lambda )$ in a fixed order as we mentioned earlier.

Each $b\in U(\textbf{o}^t_{|U(\textbf{x}^t\backslash \textbf{x}^\lambda )|})\backslash U(\textbf{x}^t)$ will be mapped to $\bar{y}(b)\in [U(\textbf{x}^t)\backslash U(\textbf{x}^*)]^m$. Let $p=|\bar{y}(b)|\in [m]$ for each such b, then write $\bar{y}(b)=\{v_{q_1},\dots ,v_{q_p}\}$, where $1\le q_1<\dots q_p\le m$.

By our settings, we have

$$\begin{aligned} \begin{aligned}&\sum \limits _{\textbf{1}_{b,i}\preceq (\textbf{o}^t_{|U(\textbf{x}^t\backslash \textbf{x}^\lambda )|}\backslash \textbf{x}^t)} [g(\textbf{x}^t)-g(\textbf{x}^t\backslash \bigsqcup _{y(b)\in \bar{y}(b)}\textbf{1}_{y(b),j})]\\&\quad =\sum \limits _{\textbf{1}_{b,i}\preceq (\textbf{o}^t_{|U(\textbf{x}^t\backslash \textbf{x}^\lambda )|}\backslash \textbf{x}^t)} [g(\textbf{x}^t)-g(\textbf{x}^t\backslash \bigsqcup _{l={q_1}}^{q_p}\textbf{1}_{v_l,j_l})]\\&\quad \le \sum \limits _{\textbf{1}_{b,i}\preceq (\textbf{o}^t_{|U(\textbf{x}^t\backslash \textbf{x}^\lambda )|}\backslash \textbf{x}^t)} \sum _{r=q_1}^{q_p}[g(\textbf{x}^\lambda \sqcup (\bigsqcup _{l=1}^{r}\textbf{1}_{v_l,j_l})) -g(\textbf{x}^\lambda \sqcup (\bigsqcup _{l=1}^{r-1}\textbf{1}_{v_l,j_l}))]\\&\quad \le m\cdot \sum _{r=1}^{|U(\textbf{x}^t\backslash \textbf{x}^\lambda )|} [g(\textbf{x}^\lambda \sqcup (\bigsqcup _{l=1}^{r}\textbf{1}_{v_l,j_l})) -g(\textbf{x}^\lambda \sqcup (\bigsqcup _{l=1}^{r-1}\textbf{1}_{v_l,j_l}))]\\&\quad =m\cdot [g(\textbf{x}^t)-g(\textbf{x}^\lambda )]\\&\quad \le m\cdot g(\textbf{x}^t) \end{aligned} \end{aligned}$$

(26)

for $\textbf{1}_{y(b),j}\preceq \textbf{x}^t\backslash \textbf{x}^\lambda$. The first inequality is due to orthant submodularity. As we mentioned, each element $a\in U(\textbf{x}^t)\backslash U(\textbf{x}^*)$ appears in mapping y no more than m times. In Greedy Algorithm(f, G), all marginal gains $g_{\textbf{x}}(\textbf{1}_{v_l,j_l})\ge 0$ for non-monotone or monotone k-submodular function f input. Therefore, we get the second inequality. The third inequality needs nonnegativity of g.

Proof of Lemma 7:

For problem (1), input a monotone k-submodular function f and $\lambda =2$ in KM-KM. In the fixed $t^*+1$ iteration, considering the current solution $\textbf{x}^{t^*}$ and the m-swap $(\bar{y}(b^{t^*}),b^{t^*})$, we have

$$\begin{aligned} \begin{aligned}&f((\textbf{x}^{t^*}\setminus \textbf{1}_{y(b^{t^*}),j^{t^*}})\sqcup \textbf{1}_{b^{t^*},i^{t^*}})-f(\textbf{x}^{t^*})\\&\quad \le f((\textbf{x}^{t^*}\setminus \textbf{1}_{y(b^{t^*}),j^{t^*}})\sqcup \textbf{1}_{b^{t^*},i^{t^*}})-f(\textbf{x}^{t^*}\setminus \textbf{1}_{y(b^{t^*}),j^{t^*}})\\&\quad \le f((\textbf{x}^{1}\sqcup \textbf{1}_{b^{t^*},i^{t^*}})-f(\textbf{x}^{1})\\&\quad \le f(\textbf{x}^2)-f(\textbf{x}^1). \end{aligned} \end{aligned}$$

Using the monotonicity of f, we get the first inequality. Then the second is due to orthant submodularity. Because we greedily choose $\textbf{x}^t$ for $t\in \{1,2\}$, the third inequality holds. Similarly, we have

$$\begin{aligned} \begin{aligned} f((\textbf{x}^{t^*}\setminus \textbf{1}_{y(b^{t^*}),j^{t^*}})\sqcup \textbf{1}_{b^{t^*},i^{t^*}})-f(\textbf{x}^{t^*}) \le f(\textbf{x}^1)-f(\textbf{x}^0). \end{aligned} \end{aligned}$$

Combining the above two formulas, we have

$$\begin{aligned} f((\textbf{x}^{t^*}\setminus \textbf{1}_{y(b^{t^*}),j^{t^*}})\sqcup \textbf{1}_{b^{t^*},i^{t^*}})-f(\textbf{x}^{t^*}) \le \frac{1}{2}f(\textbf{x}^{2})=\frac{1}{2}f(\textbf{x}^\lambda ). \end{aligned}$$

(27)

Proof of Lemma 8:

Given a fixed $t\in \{\lambda ,\dots ,t^*\}$, by greedy choice of t-th iteration and the assumption about $t^*+1$, we have

$$\begin{aligned} \frac{f(\textbf{x}^t\backslash \bigsqcup _{y(b)\in \bar{y}(b)}\textbf{1}_{y(b),j}\sqcup \textbf{1}_{b,i})-f(\textbf{x}^t)}{w_{b}}\le \rho _{t}, \end{aligned}$$

(28)

for m-swaps $(\bar{y}(b),b)$ with $~b\in U(\textbf{o}^t_{|U(\textbf{x}^t\backslash \textbf{x}^\lambda )|})\backslash U(\textbf{x}^t)$ and $\bar{y}(b)\in [U(\textbf{x}^t)\backslash U(\textbf{x}^*)]^m$. Due to $U(\textbf{o}^t_{|U(\textbf{x}^t\backslash \textbf{x}^\lambda )|})\backslash U(\textbf{x}^t)=U(\mathbf {x^*})\backslash U(\textbf{x}^t)$, we have

$$\begin{aligned} \sum \limits _{\textbf{1}_{b,i}\preceq (\textbf{o}^t_{|U(\textbf{x}^t\backslash \textbf{x}^\lambda )|}\backslash \textbf{x}^t)} w_b \le B-w_{\textbf{x}^\lambda }. \end{aligned}$$

(29)

Combining the above formula, we get

$$\begin{aligned} \begin{aligned}&\sum \limits _{\textbf{1}_{b,i}\preceq (\textbf{o}^t_{|U(\textbf{x}^t\backslash \textbf{x}^\lambda )|}\backslash \textbf{x}^t)} [g((\textbf{x}^t\backslash \bigsqcup _{y(b)\in \bar{y}(b)}\textbf{1}_{y(b),j})\sqcup \textbf{1}_{b,i})-g(\textbf{x}^t)]\\&\quad = \sum \limits _{\textbf{1}_{b,i}\preceq (\textbf{o}^t_{|U(\textbf{x}^t\backslash \textbf{x}^\lambda )|}\backslash \textbf{x}^t)} [f((\textbf{x}^t\backslash \bigsqcup _{y(b)\in \bar{y}(b)}\textbf{1}_{y(b),j})\sqcup \textbf{1}_{b,i})-f(\textbf{x}^t)]\\&\quad \le (B-w_{\textbf{x}^\lambda })\rho _{t}, \end{aligned} \end{aligned}$$

(30)

for $\textbf{1}_{y(b),j}\preceq \textbf{x}^t\backslash \textbf{x}^\lambda$.

Proof of Lemma 9:

We introduce a framework of proof inspired by Sarpatwar et al. (2019) to get

$$\begin{aligned} \begin{aligned}&\frac{g((\textbf{x}^{t^*}\setminus \bigsqcup _{y(b^{t^*})\in \bar{y}(b^{t^*})}\textbf{1}_{y(b^{t^*}),j^{t^*}})\sqcup \textbf{1}_{b^{t^*},i^{t^*}})}{g(\textbf{x}^*)} \ge \frac{1}{\alpha }(1-e^{-\beta }). \end{aligned} \end{aligned}$$

(31)

Let $B_\lambda =0$ and $B_{t}=\sum _{\tau =\lambda +1}^{t}w_{b^\tau }$, for any $t\in \{\lambda +1,\ldots ,t^{*}+1\}$. Define $B'= B_{t^{*}+1}=B_{t^{*}}+w_{b^{t^*}}$ and $B''= B-w_{\textbf{x}^\lambda }$. By the assumption of case 2, we have $B'> B\ge B''$. For $j = 1,\ldots , B'$, we define $\gamma _j=\rho _{t-1}$ when $j = B_{t-1}+1,\ldots , B_t$. Note that $g(\textbf{x}^\tau )-g(\textbf{x}^{\tau -1})=w_{b^{\tau -1}}\rho _{\tau -1}$, using the above definition, we obtain that

$$\begin{aligned} g(\textbf{x}^{t})=\sum \limits _{\tau =\lambda +1}^{t}[g(\textbf{x}^\tau )-g(\textbf{x}^{\tau -1})] =\sum \limits _{\tau =\lambda +1}^{t}w_{b^{\tau -1}}\rho _{\tau -1} =\sum \limits _{j=1}^{B_t}\gamma _j, \end{aligned}$$

(32)

for each $t\in \{\lambda +1,\ldots ,t^{*}\}$, and

$$\begin{aligned} \begin{aligned} g((\textbf{x}^{t^*}\setminus \bigsqcup _{y(b^{t^*})\in \bar{y}(b^{t^*})}\textbf{1}_{y(b^{t^*}),j^{t^*}})\sqcup \textbf{1}_{b^{t^*},i^{t^*}})&=\sum \limits _{\tau =\lambda +1}^{t^*+1}[g(\textbf{x}^t)-g(\textbf{x}^{t-1})]\\&=\sum \limits _{\tau =\lambda +1}^{t^*+1}w_{b^{\tau -1}}\rho _{\tau -1} =\sum \limits _{j=1}^{B'}\gamma _j. \end{aligned}\end{aligned}$$

(33)

Using $g(\textbf{x}^*)\le \alpha [g(\textbf{x}^t)+\frac{(B-w_{\textbf{x}^\lambda })}{\beta }\rho _{t}]$ and (32), we have the following equalities

$$\begin{aligned} \begin{aligned} g(\textbf{x}^*)&\le \alpha \min \limits _{t\in \{\lambda +1,\ldots ,t^*\}}\{ g(\textbf{x}^{t})+\frac{B''}{\beta }\rho _{t}\}\\&\le \alpha \min \limits _{t\in \{\lambda +1,\ldots ,t^*\}}\{\sum \limits _{j=1}^{B_t}\gamma _j+\frac{B''}{\beta }\gamma _{B_{t+1}}\}. \end{aligned} \end{aligned}$$

(34)

From (33), (34) and Lemma 4, we obtain that

$$\begin{aligned} \begin{aligned}&\frac{g((\textbf{x}^{t^*}\setminus \bigsqcup _{y(b^{t^*})\in \bar{y}(b^{t^*})}\textbf{1}_{y(b^{t^*}),j^{t^*}})\sqcup \textbf{1}_{b^{t^*},i^{t^*}})}{g(\textbf{x}^*)}\\&\quad \ge \frac{\sum _{j=1}^{B'}\gamma _j}{\alpha \min \limits _{t\in \{\lambda +1,\ldots ,t^*\}}\{\sum \limits _{j=1}^{B_t}\gamma _j+\frac{B''}{\beta }\gamma _{B_{t+1}}\}}\\&\quad =\frac{\sum _{j=1}^{B'}\gamma _j}{\alpha \min \limits _{t\in \{1,\ldots ,B'\}}\{\sum \limits _{j=1}^{t-1}\gamma _j+\frac{B''}{\beta }\gamma _{t}\}}\\&\quad \ge \frac{1}{\alpha }(1-(1-\frac{\beta }{B''})^{B'})\\&\quad \ge \frac{1}{\alpha }(1-e^{-\frac{\beta B'}{B''}})\\&\quad \ge \frac{1}{\alpha }(1-e^{-\beta }). \end{aligned} \end{aligned}$$

(35)

Using $1-\frac{1}{\alpha }(1-e^{-\beta })-r\ge 0$ and (35), we have

$$\begin{aligned} \begin{aligned} f(\textbf{x}^{t^*})&=f(\textbf{x}^\lambda )+g(\textbf{x}^{t^*})\\&= f(\textbf{x}^\lambda )+g((\textbf{x}^{t^*}\setminus \bigsqcup _{y(b^{t^*})\in \bar{y}(b^{t^*})}\textbf{1}_{y(b^{t^*}),j^{t^*}})\sqcup \textbf{1}_{b^{t^*},i^{t^*}})\\&-[g((\textbf{x}^{t^*}\setminus \bigsqcup _{y(b^{t^*})\in \bar{y}(b^{t^*})}\textbf{1}_{y(b^{t^*}),j^{t^*}})\sqcup \textbf{1}_{b^{t^*},i^{t^*}})-g(\textbf{x}^{t^*})]\\&= f(\textbf{x}^\lambda )+g((\textbf{x}^{t^*}\setminus \bigsqcup _{y(b^{t^*})\in \bar{y}(b^{t^*})}\textbf{1}_{y(b^{t^*}),j^{t^*}})\sqcup \textbf{1}_{b^{t^*},i^{t^*}})\\&-[f((\textbf{x}^{t^*}\setminus \bigsqcup _{y(b^{t^*})\in \bar{y}(b^{t^*})}\textbf{1}_{y(b^{t^*}),j^{t^*}})\sqcup \textbf{1}_{b^{t^*},i^{t^*}})-f(\textbf{x}^{t^*})]\\&\ge f(\textbf{x}^\lambda )+\frac{1}{\alpha }(1-e^{-\beta })g(\textbf{x}^*)-rf(\textbf{x}^\lambda )\\&= \frac{1}{\alpha }(1-e^{-\beta })f(\textbf{x}^*)+(1-\frac{1}{\alpha }(1-e^{-\beta })-r)f(\textbf{x}^\lambda )\\&\ge \frac{1}{\alpha }(1-e^{-\beta })f(\textbf{x}^*). \end{aligned} \end{aligned}$$

(36)

Proof of Lemma 10:

Recall our settings in proof of Lemma 6: For a fixed iteration step $t\ge \lambda +1$ in KM-KM, the ground set of Greedy Algorithm(f, G) is $G=\{v_1,\dots ,v_{|U(\textbf{x}^t\backslash \textbf{x}^\lambda )|}\}=U(\textbf{x}^t\backslash \textbf{x}^\lambda )$ in a fixed order as we mentioned earlier. Each $b\in U(\textbf{o}^t_{|U(\textbf{x}^t\backslash \textbf{x}^\lambda )|})\backslash U(\textbf{x}^t)$ will be mapped to $\bar{y}(b)\in [U(\textbf{x}^t)\backslash U(\textbf{x}^*)]^m$. Let $p=|\bar{y}(b)|\in [m]$ for each such b, then write $\bar{y}(b)=\{v_{q_1},\dots ,v_{q_p}\}$, where $1\le q_1<\dots q_p\le m$. We have

$$\begin{aligned} \begin{aligned}&f((\textbf{x}^{t^*}\setminus \bigsqcup _{y(b^{t^*})\in \bar{y}(b^{t^*})}\textbf{1}_{y(b^{t^*}),j^{t^*}})\sqcup \textbf{1}_{b^{t^*},i^{t^*}})-f(\textbf{x}^{t^*})\\ =&~~~~~f((\textbf{x}^{t^*}\setminus \bigsqcup _{l={q_1}}^{q_p}\textbf{1}_{v_l,j_l})\sqcup \textbf{1}_{b^{t^*},i^{t^*}})-f(\textbf{x}^{t^*})\\ =&-[f(\textbf{x}^{t^*}\setminus \textbf{1}_{v_{q_1},j_{q_1}}\sqcup \textbf{1}_{v_{q_1},j_{q_1}})-f(\textbf{x}^{t^*}\setminus \textbf{1}_{v_{q_1},j_{q_1}})]\\&-[f((\textbf{x}^{t^*}\setminus \bigsqcup _{l={q_1}}^{q_{2}}\textbf{1}_{v_l,j_l})\sqcup \textbf{1}_{v_{q_2},j_{q_2}})-f(\textbf{x}^{t^*}\setminus \bigsqcup _{l={q_1}}^{q_{2}}\textbf{1}_{v_l,j_l})]\\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\dots \\&-[f((\textbf{x}^{t^*}\setminus \bigsqcup _{l={q_1}}^{q_{p}}\textbf{1}_{v_l,j_l})\sqcup \textbf{1}_{v_{q_p},j_{q_p}})-f(\textbf{x}^{t^*}\setminus \bigsqcup _{l={q_1}}^{q_{p}}\textbf{1}_{v_l,j_l})]\\&+f((\textbf{x}^{t^*}\setminus \bigsqcup _{l={q_1}}^{q_p}\textbf{1}_{v_l,j_l})\sqcup \textbf{1}_{b^{t^*},i^{t^*}})-f(\textbf{x}^{t^*}\setminus \bigsqcup _{l={q_1}}^{q_{p}}\textbf{1}_{v_l,j_l})\\ \le&~~~~[f(\textbf{x}^{t^*}\setminus \textbf{1}_{v_{q_1},j_{q_1}}\sqcup \textbf{1}_{v_{q_1},j'})-f(\textbf{x}^{t^*}\setminus \textbf{1}_{v_{q_1},j_{q_1}})]\\&+[f((\textbf{x}^{t^*}\setminus \bigsqcup _{l={q_1}}^{q_{2}}\textbf{1}_{v_l,j_l})\sqcup \textbf{1}_{v_{q_2},j'})-f(\textbf{x}^{t^*}\setminus \bigsqcup _{l={q_1}}^{q_{2}}\textbf{1}_{v_l,j_l})]\\&\dots \\&+[f((\textbf{x}^{t^*}\setminus \bigsqcup _{l={q_1}}^{q_{p}}\textbf{1}_{v_l,j_l})\sqcup \textbf{1}_{v_{q_p},j'})-f(\textbf{x}^{t^*}\setminus \bigsqcup _{l={q_1}}^{q_{p}}\textbf{1}_{v_l,j_l})]\\&+f((\textbf{x}^{t^*}\setminus \bigsqcup _{l={q_1}}^{q_p}\textbf{1}_{v_l,j_l})\sqcup \textbf{1}_{b^{t^*},i^{t^*}})-f(\textbf{x}^{t^*}\setminus \bigsqcup _{l={q_1}}^{q_{p}}\textbf{1}_{v_l,j_l})\\ \le&~~ \sum _{l=q_1}^{q_p}[f(\textbf{x}^{\tau -1}\sqcup \textbf{1}_{v_l,j'})-f(\textbf{x}^{\tau -1})]+[f((\textbf{x}^{\tau -1}\sqcup \textbf{1}_{b^{t^*},i^{t^*}})-f(\textbf{x}^{\tau -1})]\\ \le&~~(p+1) [f(\textbf{x}^{\tau })-f(\textbf{x}^{\tau -1})]\\ \le&~~(m+1) [f(\textbf{x}^{\tau })-f(\textbf{x}^{\tau -1})]\\ \end{aligned} \end{aligned}$$

(37)

for $\tau \in {1,\dots ,\lambda }$. The first inequality is due to pairwise monotonicity, that is, $-f_{\textbf{x}}((v,i))\le f_{\textbf{x}}((v,j))$ for $i\ne j\in [k]$. The second is due to orthant submodularity. Because we greedily choose $\textbf{x}^{t}$ for $t\in \{1,\dots ,\lambda \}$, the third inequality holds. Combining the above $\lambda$ formulas, we have

$$\begin{aligned} f((\textbf{x}^{t^*}\setminus \textbf{1}_{y(b^{t^*}),j^{t^*}})\sqcup \textbf{1}_{b^{t^*},i^{t^*}})-f(\textbf{x}^{t^*}) \le \frac{m+1}{\lambda }f(\textbf{x}^{\lambda }). \end{aligned}$$

(38)