Optimal matroid bases with intersection constraints: valuated matroids, M-convex functions, and their applications

Abstract

For two matroids $M_1$ and $M_2$ with the same ground set V and two cost functions $w_1$ and $w_2$ on $2^V$, we consider the problem of finding bases $X_1$ of $M_1$ and $X_2$ of $M_2$ minimizing $w_1(X_1)+w_2(X_2)$ subject to a certain cardinality constraint on their intersection $X_1 \cap X_2$. For this problem, Lendl et al. (Matroid bases with cardinality constraints on the intersection, arXiv:1907.04741v2, 2019) discussed modular cost functions: they reduced the problem to weighted matroid intersection for the case where the cardinality constraint is $|X_1 \cap X_2|\le k$ or $|X_1 \cap X_2|\ge k$; and designed a new primal-dual algorithm for the case where the constraint is $|X_1 \cap X_2|=k$. The aim of this paper is to generalize the problems to have nonlinear convex cost functions, and to comprehend them from the viewpoint of discrete convex analysis. We prove that each generalized problem can be solved via valuated independent assignment, valuated matroid intersection, or $\mathrm {M}$-convex submodular flow, to offer a comprehensive understanding of weighted matroid intersection with intersection constraints. We also show the NP-hardness of some variants of these problems, which clarifies the coverage of discrete convex analysis for those problems. Finally, we present applications of our generalized problems in the recoverable robust matroid basis problem, combinatorial optimization problems with interaction costs, and matroid congestion games.

Appendices

A Algorithm description and complexity analyses

In this section, we describe the full behavior of the algorithm for $(\mathrm{V}_{\ge k})$. We then analyse the time complexities of our algorithms for $(\mathrm{V}_{\ge k})$ and $(\mathrm{V}_{= k})$ to prove Theorem 3.

Suppose that, for some nonnegative integer $i < k$, we have at hand an optimal solution $(X_1, X_2)$ for $(\mathrm{V}_{\ge i})$ and its optimality witness $(p_1, p_2, F)$; recall that the optimality condition has been described in Lemma 2. We then find an optimal solution for $(\mathrm{V}_{\ge i+1})$ and its optimality witness by utilizing an auxiliary digraph $\vec {G}=(V_1 \cup V_2 \cup \{s, t\}, A)$ endowed with a nonnegative arc-length function $\ell$ on A defined from $(X_1, X_2)$ and $(p_1, p_2, F)$ as follows. Here s and t are new vertices, which play roles as a source vertex and sink vertex, respectively. The arc set A is defined by

$$\begin{aligned} A := \vec {E} \cup \vec {F} \cup A_1 \cup A_2 \cup S \cup T, \end{aligned}$$

where

$$\begin{aligned} \vec {E}&:= \{ (v^1, v^2) \mid v \in V \},\\ \vec {F}&:= \{ (v^2, v^1) \mid v \in F \},\\ A_1&:= \{ (u^1, v^1) \mid u^1 \in X_1,\ v^1 \in V_1 {\setminus } X_1,\ X_1 {\setminus } \{u^1\} \cup \{v^1\} \in {{\,\mathrm{dom}\,}}\omega _1 \},\\ A_2&:= \{ (v^2, u^2) \mid u^2 \in X_2,\ v^2 \in V_2 {\setminus } X_2,\ X_2 {\setminus } \{u^2\} \cup \{v^2\} \in {{\,\mathrm{dom}\,}}\omega _2 \},\\ S&:= \{ (s, v^1) \mid v^1 \in X_1 {\setminus } X_2 \},\\ T&:= \{ (v^2, t) \mid v^2 \in X_2 {\setminus } X_1 \}. \end{aligned}$$

Define the arc-length function $\ell :A\rightarrow \mathbf {R}$ by

$$\begin{aligned} \ell (a) := {\left\{ \begin{array}{ll} (\omega _1 - p_1)(X_1 {\setminus } \{u^1\} \cup \{v^1\}) - (\omega _1 - p_1)(X_1) &{} \text {if}\quad a = (u^1, v^1) \in A_1,\\ (\omega _2 + p_2)(X_2 {\setminus } \{u^2\} \cup \{v^2\}) - (\omega _2 + p_2)(X_2) &{} \text {if}\quad a = (v^2, u^2) \in A_2,\\ 0 &{} \text {if}\quad a \in A {\setminus } (A_1 \cup A_2). \end{array}\right. } \end{aligned}$$

We remark that $\ell$ is nonnegative since $X_1$ and $X_2$ are minimizers of $\omega _1 - p_1$ and $\omega _2 + p_2$, respectively.

The augmenting path algorithm for $(\mathrm{V}_{\ge k})$ runs in the auxiliary digraph as follows:

• Step 1: Let $X_1$ and $X_2$ be the minimizers of $\omega _1$ and $\omega _2$, respectively.

  • If $|X_1 \cap X_2| \ge k$, then output $(X_1,X_2)$ and stop.

  • Otherwise, let $k' := |X_1 \cap X_2| < k$ and let $p_1, p_2$ be potential functions defined by $p_1(v^1) = p_2(v^2) = 0$ for all $v \in V$. Then $(X_1, X_2)$ is an optimal solution for $(\mathrm{V}_{\ge k'})$ with optimality witness $(p_1, p_2, X_1 \cap X_2)$.

• Step 2: While $i=|X_1 \cap X_2| < k$, do the following. Suppose here that $(X_1, X_2)$ is an optimal solution for $(\mathrm{V}_{\ge i})$ with optimality witness $(p_1, p_2, X_1 \cap X_2)$. Let $\vec {G}$ be the auxiliary digraph for $(X_1, X_2)$ with $(p_1, p_2, X_1 \cap X_2)$.

  • If there is no s-t path in $\vec {G}$, then output “$(\mathrm{V}_{\ge k})$ is infeasible” and stop.

  • Otherwise, find a shortest s-t path P in $\vec {G}$ with respect to the arc length $\ell$ such that the number of edges is smallest among the shortest s-t paths. For $x \in V_1 \cup V_2 \cup \{t\}$, let d(x) be the length of the shortest s-x path with respect to $\ell$ in $\vec {G}$, where $d(x) := +\infty$ if there is no s-x path. Update $X_1$, $X_2$, $p_1$, and $p_2$ by

    $$\begin{aligned} X_1 \&\leftarrow \ X_1 {\setminus } (P \cap X_1) \cup (P \cap (V_1 {\setminus } X_1)),\\ X_2 \&\leftarrow \ X_2 {\setminus } (P \cap X_2) \cup (P \cap (V_2 \setminus X_2)),\\ p_1(v^1) \&\leftarrow \ p_1(v^1) + \min \{d(v^1), d(t) \} \quad \text {for}\quad v^1 \in V_1,\\ p_2(v^2) \&\leftarrow \ p_2(v^2) - \min \{ d(v^2), d(t) \} \quad \text {for}\quad v^2 \in V_2. \end{aligned}$$

    Then, the resulting $X_1$, $X_2$, $p_1$, and $p_2$ satisfy that $|X_1 \cap X_2| = i + 1$ and $(X_1,X_2)$ is an optimal solution for $(\mathrm{V}_{\ge i+1})$ with optimality witness $(p_1, p_2, X_1 \cap X_2)$.

• Step 3: Output $(X_1, X_2)$ and stop.

We are ready to prove Theorem 3. We first see the time complexity of the algorithm for $(\mathrm{V}_{\ge k})$. We can obtain minimizers of $\omega _1$ and $\omega _2$ in $O(|V| r \gamma )$ time [5], and hence Step 1 can be done in $O(|V| r \gamma )$ time. Consider each iteration in Step 2. We can construct the auxiliary graph $\vec {G}$ in $O(|V|r\gamma )$ time. Indeed, the number of edges in $\vec {G}$ is O(|V|r): $|\vec {E}|$ is O(|V|), $|\vec {F}|$, |S|, and |T| are O(|r|), $|A_1|$ is $O(r_1(|V| - r_1)) = O(|V|r)$, and $|A_2|$ is $O(r_2(|V| - r_2)) = O(|V|r)$, where $r_1$ and $r_2$ are the ranks of $\omega _1$ and $\omega _2$, respectively. Furthermore the arc-length function $\ell$ can be computed in $O((|A_1| + |A_2|) \gamma ) = O(|V|r \gamma )$ time. Then we can compute the length d(x) of the shortest s-x path for each $x \in V_1 \cup V_2 \cup \{s\}$ in $O(|V|r + |V| \log |V|)$ time by using Dijkstra’s algorithm with Fibonacci heaps [10] (see also [38, Section 7.4]). The update of $(X_1, X_2)$ and $(p_1, p_2)$ requires O(|V|) time. Therefore each iteration in Step 2 takes $O(|V|r\gamma + |V| \log |V|)$ time. Since the number of iterations is at most k, Step 2 can be done in $O(|V|r k \gamma + |V| k \log |V|)$ time. Thus the running-time of the algorithm for $(\mathrm{V}_{\ge k})$ is $O(|V|r k \gamma + |V| k \log |V|)$.

We then consider the time complexity of the algorithm for $(\mathrm{V}_{= k})$. Let $X_1$ and $X_2$ be minimizers of $\omega _1$ and $\omega _2$, respectively, which can be obtained in $O(|V| r \gamma )$ time. If $|X_1 \cap X_2| \le k$, then the algorithm for $(\mathrm{V}_{= k})$ is the same as that for $(\mathrm{V}_{\ge k})$, and hence the running-time is $O(|V|r k \gamma + |V| k \log |V|)$. If $|X_1 \cap X_2| > k$, we solve $(\mathrm{V}_{\ge r_1 - k})$ for $\omega _1$ and $\overline{\omega _2}$. By the same argument as for the algorithm for $(\mathrm{V}_{\ge k})$, one can see that the running-time of $(\mathrm{V}_{\ge r_1 - k})$ is $O(|V|r (r_1-k) \gamma + |V| (r_1-k) \log |V|)$. Altogether, $(\mathrm{V}_{= k})$ can be solved in $O(|V|r^2 \gamma + |V| r \log |V|)$ time.

B Anothor solution to $(\mathrm{V}_{= k})$

This section provides another solution to $(\mathrm{V}_{= k})$, which is mentioned at the end of Sect. 3.1. We first solve $(\mathrm{V}_{\le k})$ and $(\mathrm{V}_{\ge k})$. The former is a special case of $(\mathrm{V}_{\mathcal {I}}^n)$ in which $n = 2$ and $\mathcal {I}$ is the independent set family of the uniform matroid with rank k, and can be solved in $O(|V|r^2 \gamma + |V| r \log |V|)$ time by Theorem 4. The latter can also be solved in $O(|V|r^2 \gamma + |V| r \log |V|)$ time by Theorem 3 (or the reduction to valuated matroid intersection). If the output optimal solution $(X_1, X_2)$ of $(\mathrm{V}_{\le k})$ satisfies $|X_1 \cap X_2| = k$ or that of $(X_1', X_2')$ of $(\mathrm{V}_{\ge k})$ satisfies $|X_1' \cap X_2'| = k$, then we are done.

Otherwise, we have that $X_1$ and $X_1'$ are minimizers of $\omega _1$, and $X_2$ and $X_2'$ are those of $\omega _2$. Indeed, suppose, to the contrary, that $X_1$ is not a minimizer of $\omega _1$. Then there are $u \in X_1$ and $v \in V {\setminus } X_1$ such that $\omega _1(X_1 {\setminus } \{u\} \cup \{v\}) < \omega _1(X_1)$. By $|X_1 \cap X_2| < k$, it follows that $|(X_1 {\setminus } \{u\} \cup \{v\}) \cap X_2| \le k$. Therefore, $(X_1 {\setminus } \{u\} \cup \{v\}, X_2)$ is feasible and $\omega _1(X_1 {\setminus } \{u\} \cup \{v\}) + \omega _2(X_2) < \omega _1(X_1) + \omega _2(X_2)$, which contradicts the optimality of $(X_1, X_2)$. By the same argument, we can conclude that $X_1'$ is also a minimizer of $\omega _1$ and $X_2, X_2'$ are minimizers of $\omega _2$.

Let $p = |X_1' {\setminus } X_1|$ and $q= |X_2' {\setminus } X_2|$. Since the set of minimizers of a valuated matroid forms a base family of some matroid, there is a sequence $(X_1^0, X_1^1, \dots , X_1^p)$ of minimizers of $\omega _1$ such that $X_1^0 = X_1$ and $|X_1' {\setminus } X_1^{i}| = p-i$ for each $i = 0,1,\dots , p$. Note that $|X_1^{i-1} \cap X_2| - 1 \le |X_1^i \cap X_2| \le |X_1^{i-1} \cap X_2| + 1$. If $|X_1^i \cap X_2| = k$ holds for some $i \in [p]$, then output $(X_1^i, X_2)$. Otherwise, we have that $|X_1' \cap X_2| < k$. Then we can similarly consider a sequence $(X_2^0, X_2^1, \dots , X_2^q)$ of minimizers of $\omega _2$, where $X_2^0 = X_2$ and $|X_2' \setminus X_2^{j}| = q-j$ for each $j = 0,1,\dots ,q$. It then follows from $|X_1' \cap X_2| < k$ and $|X_1' \cap X_2'| > k$ that there is $X_2^j$ with $|X_1' \cap X_2^j| = k$. We thus output $(X_1' ,X_2^j)$ as an optimal solution.

C The algorithm for $(\mathrm{W}_{=k})$ by Lendl et al.

The primal-dual algorithm of Lendl et al. [24] is described as follows. Let $X_1 \in \mathrm{argmin}_{X \in \mathcal {B}_1} w_1(X)$ and $X_2 \in \mathrm{argmin}_{X \in \mathcal {B}_2} w_2(X)$, respectively.

Case 1 ($|X_1 \cap X_2| \le k$)::

Let $k' := |X_1 \cap X_2|$ and let $q_1, q_2$ be potential functions defined by $q_1(v^1) = q_2(v^2) = 0$ for all $v \in V$. While $|X_1 \cap X_2| < k$, do the following. Let $\vec {G}$ be the auxiliary graph for $(X_1, X_2)$ with its LPT optimality witness $(q_1, q_2)$.

• Suppose that there is a zero length s-t path in $\vec {G}$ with respective to $\ell$. Then let P be a zero length s-t path with the smallest number of edges, and update $X_1$ and $X_2$ by

  $$\begin{aligned} X_1&\leftarrow X_1 \setminus (P \cap X_1) \cup (P \cap (V_1 \setminus X_1)),\\ X_2&\leftarrow X_2 \setminus (P \cap X_2) \cup (P \cap (V_2 \setminus X_2)). \end{aligned}$$

• Suppose that there is no zero length s-t path in $\vec {G}$. Let R be the set of vertices reachable from s with a zero length path, and $\delta := \min \{ \ell (a) \mid a \in A_1 \cup A_2,\ \ell (a) > 0 \}$. If $\delta = +\infty$, then output “$(\mathrm{W}_{=k})$ is infeasible.” Otherwise update $q_1$ and $q_2$ by

  $$\begin{aligned} q_1(v^1)&\leftarrow {\left\{ \begin{array}{ll} q_1(v^1) + \delta &{} \text {if } v^1 \in V_1 \cap R,\\ q_1(v^1) &{} \text {if } v^1 \in V_1 \setminus R, \end{array}\right. }\\ q_2(v^2)&\leftarrow {\left\{ \begin{array}{ll} q_2(v^2) &{} \text {if } v^2 \in V_2 \cap R,\\ q_2(v^2) - \delta &{} \text {if } v^2 \in V_2 \setminus R. \end{array}\right. } \end{aligned}$$

Case 2 ($|X_1 \cap X_2| > k$)::

Let $r_1$ be the rank of $M_1 = (V, \mathcal {B}_1)$, and let $\overline{w_2} := - w_2$ and $\overline{M_2} := (V, \{ V \setminus B \mid B \in \mathcal {B}_2 \})$. Then apply Case 1 to $(\mathrm{W}_{= r_1 - k})$ with $(w_1, \overline{w}_2; M_1, \overline{M_2})$.

We discuss the difference of this algorithm and our algorithm in the updating procedures. The algorithm of Lendl et al. only considers the zero length edges in the auxiliary digraph $\vec {G}$ in updating a solution or potential functions, while our algorithm considers all edges in $\vec {G}$ to find the shortest paths with respect to $\ell$. Furthermore, as mentioned in Sect. 3.2, in the algorithm of Lendl et al., the update phases of a solution and of potential functions are completely separated, while our algorithm simultaneously updates a solution and potential functions. That is, for one update of a solution, the algorithm of Lendl et al. requires at most |V| updates of potential functions, while our algorithm requires only one update of potential functions.

The difference in the running-times of the algorithms follows from the above arguments. Here we remark that, in $(\mathrm{W}_{\ge k})$, we can compute the edge length $\ell (a)$ of a in constant time for each a. In the algorithm of Lendl et al., one update of a solution takes $O(|V|^2 r)$ time, since each update of potential functions requires O(|V|r) time. Hence the time complexity of their algorithm is $O(|V|^2 r^2)$. On the other hand, our algorithm requires $O(|V|r + |V| \log |V|)$ time for one update of a solution. Thus, the running-time of our algorithm is $O(|V|r^2 + |V| r \log |V|)$ time by Theorem 3 with $\gamma = O(1)$.