Matroid-Constrained Vertex Cover

Let $G=(V,E)$ be a graph. A weight $w(e)$ is associated with each edge $e\in E$. By convention, we set $n=|V|$ and $m=|E|$. For a vertex $v\in V$ we define $\delta(v)$ the set of edges that are incident to $v$. The degree of a vertex $v\in V$, denoted $\deg(v)$, is the size of $\delta(v)$, and we define the weighted degree of a vertex $v\in V$ as the sum $\deg_{w}(v)=\sum_{e\in\delta(v)}w(e)$. For two sets of vertices $S,T\subseteq V$ in a graph $G$, we denote $E_{G}(S,T)=\sum_{e\in E,e\cap S\neq\emptyset,e\cap T\neq\emptyset}w(e)$ the sum of the weights of the edges that have at least one endpoint in $S$ and at least one endpoint in $T$. Then $E_{G}(S,S)$, abbreviated $E_{G}(S)$, denotes the sum of the weights of the edges that are covered by $S$ (i.e. having at least one of its endpoints in $S$).

Let $\mathcal{M}=(V,\mathcal{I})$ be a matroid on the ground set $V$. Recall that $\mathcal{M}=(V,\mathcal{I})$ is a matroid if the following three conditions hold: (1) $\emptyset\in\mathcal{I}$, (2) if $X\subseteq Y\in\mathcal{I}$, then $X\in\mathcal{I}$, and (3) if $X,Y\in\mathcal{I},|Y|>|X|$, there exists an element $e\in Y\backslash X$ so that $X\cup\{e\}\in\mathcal{I}$. The sets in $\mathcal{I}$ are the independent sets and the rank $k$ of the matroid $\mathcal{M}$ is defined as $\max_{X\in\mathcal{I}}|X|$. For more details about matroids, we refer the reader to [41]. In this paper, given a set $S\subseteq V$ and $v\in V$, we will denote $S\cup\{v\}$ by $S+v$ and $S\backslash\{v\}$ by $S-v$ for conciseness.

The problem that we consider in this paper is to choose an independent set of vertices $S\in\mathcal{I}$, with the objective of maximizing $E_{G}(S)$, namely, the total weight of the edges covered by $S$.

Let us put our problem in a larger picture. When the given matroid $\mathcal{M}$ is a uniform matroid (see below for a formal definition), our problem reduces to the max $k$-vertex-cover problem, where we want to choose $k$ arbitrary vertices so as to maximize the total weight of covered edges. This is a classical problem with a long history: the greedy heuristic is known to give $1-1/e$ approximation as shown by Hochbaum and Pathria [26]. Ageev and Sviridenko [1] propose an LP-based approach and the technique of pipage rounding to obtain $3/4$ approximation. Using SDP, Feige and Langberg [15] improve this ratio to $3/4+\delta$ for some small constant $\delta>0$. The current best approximation ratio is $0.92$, achieved by Manurangsi [34]. For some special cases of the problem, different ratios are also obtained, e.g., see [4, 24, 25]. On the hardness side, to our knowledge, the best inapproximability ratio is due to Austrin and Stankovic [2], which is $0.929$.

The max $k$-vertex-cover has also been studied through the lens of fixed-parameterized-tractability. Guo et al. [22] show the problem to be $W[1]$-hard with $k$ as parameter, thus showing the unlikelihood of getting an exact solution in FPT time. Nonetheless, Marx [36] shows that it is possible to get a near-optimal solution in FPT time. Precisely, he gives an FPT approximation scheme (FPT-AS), that delivers a $(1-\varepsilon)$-approximate solution in $(k/\varepsilon)^{O(k^{3}/\varepsilon)}n^{O(1)}$ time. This running time is later improved by Gupta et al. [23] and Manurangsi [34].

Here we recall the definition of an FPT-AS [36]:

In our case, the instances are made of a graph and a matroid, and the parameter of an instance is the rank $k$ of its matroid.

Regarding the more general case of an arbitrary matroid of rank $k$, one can obtain $3/4$ approximation in polynomial time by combining known techniques.¹¹1Ageev and Sviridenko [1] show that, for the case of a uniform matroid, the optimal fractional solution $x^{*}$ of the LP has at least $3/4$ of the optimal value. They then use the pipage rounding to transform it into an integral solution with value no less than $x^{*}$. The same LP approach can be generalized for arbitrary matroids. The optimal fractional solution can be obtained by the Ellipsoid algorithm: even though the linear program to describe the independent sets of an arbitrary matroid may use exponentially many constraints, we can design a separation oracle using an algorithm of Cunningham [13]. What remains is just the pipage rounding with a general matroid—this is already known to be do-able by Calinescu et al. [8]. We thank Pasin Manurangsi for communicating to us this method. This is also a special case of maximizing a submodular function (more precisely, a coverage function) under matroid constraint, for which a $1-1/e$ approximation can be achieved in polynomial time [7, 20]. In this work, we try to do better than this ratio for some special cases of matroids, in the context of fixed-parameter algorithms. We also show that the ideas developed here can be applied in the streaming setting [39]. In streaming, maximizing a submodular function under a general matroid constraint has received much attention recently [9, 11, 16]. Then we also show how local search can be used to obtain simple algorithms for this problem and what are the limitations of local search algorithms.

Let us recall some definitions. A uniform matroid of rank $k$ is a matroid where the independent sets are the sets $S$ of cardinality at most $k$. A partition matroid is a matroid where we are given a partition $V_{1},\dots,V_{r}$ of the ground set $V$ and bounds $k_{1},\dots,k_{r}$ such that a set $S$ is independent if for all $1\leq i\leq r$, $|S\cap V_{i}|\leq k_{i}$. A laminar matroid is given as a laminar family $V_{1},\dots V_{r}$ of $V$, i.e. given $V_{i}\neq V_{j}$, then either $V_{i}\cap V_{j}=\emptyset$, or $V_{i}\subset V_{j}$, or $V_{j}\subset V_{i}$, along with bounds $k_{1},\dots,k_{r}$; a set $S\subseteq V$ is independent if for all $1\leq i\leq r$, $|S\cap V_{i}|\leq k_{i}$. Finally, a transversal matroid is given in a family $V_{1},\dots,V_{k}\subseteq V$, where $V_{i}$s are not necessarily disjoint, and a set $S=\{u_{1},\cdots,u_{t}\}$ is independent if and only if for each element $u_{i}$, there exists a distinct $\phi(i)$ so that $u_{i}\in V_{\phi(i)}$. These simple types of matroid have been extensively studied in a large variety of contexts.

A uniform matroid is a special case of a partition matroid, which is again a special case of a laminar or a transversal matroid. However, laminar matroids and transversal matroids are not inclusive of each other [40]. Transversal matroids were introduced in the 60s, by Edmonds and Fulkerson [14] and by Mirsky and Perfect [38]. They unified many results in transversal theory and are generally considered an important class of matroids, e.g., see [43]. Laminar matroids receive much attention recently in the community of theoretical computer science, especially in the context of matroid secretary problem, e.g., see [3, 10, 17, 28, 42]. Our FPT-AS involves these kinds of matroids.

Note that the result for transversal matroids has since been improved in [29] to get a kernel of size $\frac{k}{\varepsilon}$. Besides, by a straightforward modification of our proofs in Section 2 (see Appendix A), we can show the following corollary.

Put slightly differently, when $G$ is a hypergraph with edge size at most $\eta$, we can obtain a $(1-\varepsilon)$ approximation in $\left(\frac{\eta}{\varepsilon}\right)^{O(k)}n^{O(1)}$ or $\left(\frac{\eta}{\varepsilon}+k\right)^{O(k)}n^{O(1)}$ time, depending on the type of matroid. To see the interest of this corollary, we recall that recently Manurangsi [35] showed that if $\eta$ is unbounded, one cannot obtain an approximation ratio better than $1-1/e+\varepsilon$, assuming GAP-ETH, in FPT time (where the matroid rank $k$ is the parameter). This result holds even for the simplest uniform matroid. Thus Corollary 3 implies that one can circumvent this lower bound by introducing another parameter $\eta$, even for more general matroids.

Our algorithm is inspired by that of Manurangsi [34] for the case of uniform matroid. So let us briefly summarize his approach: an approximate kernel²²2In the rest of the paper, we will just say kernel, dropping the adjective. Roughly speaking, a $(1-\varepsilon)$-approximate kernel is a kernel which contains a $(1-\varepsilon)$-approximate solution of the original problem. The interest for this kind of kernel has risen recently [18, 33]. $V^{\prime}$ is first extracted from $V$, where $V^{\prime}$ is simply made of the $k/\varepsilon$ vertices with the largest weighted degrees. Let $O$ be an optimal solution. Apparently, a vertex of $O$ is either part of the kernel $V^{\prime}$, or its weighted degree is dominated by all vertices in $V^{\prime}\backslash O$. To recover the optimal value, we can potentially use the vertices in $V^{\prime}\backslash O$ to replace the vertices in $O\backslash V^{\prime}$. However, there is a risk in doing this: an edge among the vertices in $V^{\prime}\backslash O$ can be double-counted, if both of its endpoints are chosen to replace the vertices in $O\backslash V^{\prime}$. To circumvent this issue, Manurangsi uses a random sampling argument to show that in expectation such double counting is negligible. Therefore, by the averaging principle, there exists a $(1-\varepsilon)$-approximate solution in the kernel $V^{\prime}$, which can be found using brute force.

To generalize the approach of Manurangsi for more general matroids, one has to answer the quintessential question: how does one guarantee that the sampled vertices, along with $O\cap V^{\prime}$, are independent in $\mathcal{M}$? To overcome this difficulty, we introduce a new technique. We take the union of some number $\tau$ of matroids $\mathcal{M}$. Such a union is still a matroid, which we denote as $\tau\mathcal{M}$. We then apply a greedy algorithm on $\tau\mathcal{M}$ (based on non-increasing weighted degrees) to construct an independent set $V^{\prime}$ in $\tau\mathcal{M}$. We show that such a set $V^{\prime}$ is “robust” (see Definition 6) in the sense that we can sample vertices from $V^{\prime}$ so that they, along with $O\cap V^{\prime}$, are always independent and in expectation cover edges of weight at least $(1-\varepsilon)$ times that of $O$.

We note that the value of $\tau$ automatically gives an upper bound on the kernel size $V^{\prime}$, which is $\tau k$. Theorem 8 shows the required scale of $\tau$, depending on the type of the given matroid. We leave as an open question whether, for matroids more general than considered in the paper, a larger $\tau$ can always yield the kernel.

In Section 4 we consider the problem in the semi-streaming model [39]. In that context, the edges in $E$ arrive over time but we have only limited space (for instance, $O(n\cdot polylog(n))=o(m)$) and cannot afford to store all edges in $E$. In this context we can also obtain a $(1-\varepsilon)$ approximation using $O(\frac{nk}{\varepsilon})$ space in a single pass.³³3Here we assume that the matroid is given in the form of an oracle, which can be accessed by the algorithm freely—this is a standard assumption in the streaming setting when matroids are involved. The idea of using (parameterized) kernels for streaming algorithms has recently been introduced, for instance in [12, 37]. We also show that an FPT-streaming algorithm can be derived from our ideas to get a $(1-\varepsilon)$ approximation for a special form of maximization of a coverage function with bounded frequency (see Theorem 16, Remark 17 and Appendix B for details).

In the second part of the paper, we consider general matroids. As mentioned above, we aim for a ratio better than $1-1/e$. Combining known techniques involving LP, $3/4$ approximation can be achieved in polynomial time, as we mentioned earlier. However, the aforementioned LP-based approach involves quite heavy machinery, and is likely to be slow and hard to implement. Here we seek for simple approaches. A natural candidate is the local search technique. However, if we try to optimize according to the original objective function $E_{G}(S)$ (in the literature, this is called oblivious local search [19]), we show in Appendix C that one cannot achieve better than $1/2$ approximation, even if we allow large exchanges.

What we do is an adaptation of a technique of Filmus and Ward [19], originally designed for a general coverage function. Their idea is to optimize a special potential function, which takes the form of a linear sum of the covered elements, but with the modification that elements covered multiple times have larger coefficients.

See Section 5.1 for this result. Here we have fast running time thanks to the fact that small exchanges (of 1 element) are enough to establish the desired ratio. In Section 5.2, we show that by a slight modification, we can improve the approximation ratio to $3/4$ in FPT time–precisely, in $O(k!\cdot n^{O(1)})$ time. Thus if $k=O(\log n/\log\log n)$, we still use only polynomial time.

We next consider the more general case where two matroids are imposed on the graph vertices. A solution is feasible only if it is a common independent set in both matroids.

This result is presented in Section 5. To see the interest of Theorem 5, observe that it implies that when the submodular function is a coverage function of frequency 2, we can obtain an approximation ratio significantly better than $1/2-\varepsilon$ [6, 31], which is known for a general submodular function, under the constraint of two matroids. The only other special case where this ratio is improved upon that we are aware of, is when the submodular function is a weighted rank function [32], where the ratio of exactly $1/2$ is attained.

Our main tool is a characterization theorem of Lee et al. [31] based on a certain exchange graph [41, Section 41.2], originally used for matroid intersection. The much higher running time is due to the fact that here we are obliged to use larger exchanges (of up to $\theta(p)$ elements).

Considering the closeness of the approximation ratios with regard to one or two matroids, it is tempting to ask if one can do better than $2/3$ if there is just one matroid—maybe by trying larger exchanges.

We show in Section 5.3 that using the same type of potential functions [19] (linear sum with modified coefficients based on the number of times an edge is covered), $2/3$ approximation is the best possible even if we allow larger $O(1)$ exchanges.

Besides, another naturally related question is whether one can do better when the graphs are simpler, for instance, when the vertex degree is bounded by some constant $d$. Again we show in Section 5.3 that the same type of potential functions cannot achieve better than $2/3+\theta(1/d)$ approximation, even if large exchanges are allowed.

These examples seem to indicate a certain limitation on the power of currently known local search techniques. Other than the approach of [19, 21], we are unaware of any other type of local search for coverage (or general submodular) functions. To do better than $2/3$ (say up to $3/4$), a significantly different new idea may be required.