Approximate Multi-matroid Intersection via Iterative Refinement

Abstract

We introduce a new iterative rounding technique to round a point in a matroid polytope subject to further matroid constraints. This technique returns an independent set in one matroid with limited violations of the other ones. On top of the classical steps of iterative relaxation approaches, we iteratively refine/split involved matroid constraints to obtain a more restrictive constraint system, that is amenable to iterative relaxation techniques. Hence, throughout the iterations, we both tighten constraints and later relax them by dropping constraints under certain conditions. Due to the refinement step, we can deal with considerably more general constraint classes than existing iterative relaxation/rounding methods, which typically round on one matroid polytope with additional simple cardinality constraints that do not overlap too much.

We show how our rounding method, combined with an application of a matroid intersection algorithm, yields the first 2-approximation for finding a maximum-weight common independent set in 3 matroids. Moreover, our 2-approximation is LP-based, and settles the integrality gap for the natural relaxation of the problem. Prior to our work, no upper bound better than 3 was known for the integrality gap, which followed from the greedy algorithm. We also discuss various other applications of our techniques, including an extension that allows us to handle a mixture of matroid and knapsack constraints.

A. Linhares and C. Swamy—Research supported by NSERC grant 327620-09 and an NSERC DAS Award.

N. Olver—Supported by NWO VIDI grant 016.Vidi.189.087.

R. Zenklusen—Supported by Swiss National Science Foundation grant 200021_165866.

Appendices

A Impossibility of Achieving Small Additive Violations

We show that Theorem 2 for problem (1) cannot be strengthened to yield a basis of $M_0$ that has small additive violation for the matroid constraints of $M_1,\ldots ,M_k$, even when $k=2$.

We first define additive violation precisely. Given a matroid $M = (N,\mathcal {I})$ with rank function r, we say that a set $R \subseteq N$ is $\mu \text {-}additively\,independent$ in M if $|R|-r(R) \le \mu$; equivalently, we can remove at most $\mu$ elements from R to obtain an independent set in M. Unlike results for degree-bounded spanning trees, or matroidal degree-bounded MST [21], we show that small additive violation is not possible in polytime (assuming P $\ne$ NP) even for the special case of (1) where $k=2$, so we seek a basis of $M_0$ that is independent in $M_1,M_2$.

Theorem 8

Let $f(n) = O(n^{1-\varepsilon })$, where $\varepsilon > 0$ is a constant. Suppose we have a polytime algorithm $\mathcal {A}$ for (1) that returns a basis B of $M_0$ satisfying $|B|\le r_i(B)+f(|N|)$ for $i=1,2$ (where $r_i$ is the rank function of $M_i$). Then we can find in polytime a basis of $M_0$ that is independent in $M_1,M_2$.

The problem of finding a basis of $M_0$ that is independent in $M_1,M_2$ is NP-hard, as shown by an easy reduction from the directed Hamiltonian path problem. Thus, Theorem 8 shows that it is NP-hard to obtain an additive violation for problem (1) that is substantially better than linear violation.

Proof of Theorem

8. Choose t large enough so that $t > 2f(t|N|)$. Since $f(n) = O\bigl (n^{1-\varepsilon }\bigr )$, this is achieved by some $t={{\,\mathrm{poly}\,}}(|N|)$. For each $i \in \{0, 1, 2\}$, let $M'_i$ be the direct sum of t copies of $M_i$. Let $N'$ be the ground set of these matroids, which consists of t disjoint copies of N, which we label $N_1,\ldots ,N_t$.

Clearly, the instance $(M'_0, M'_1, M'_2)$ is feasible iff the original instance is feasible. Suppose that running $\mathcal {A}$ on the replicated instance yields a basis $R'$ of $M'_0$ that has the stated additive violation for the matroids $M'_1, M'_2$. Hence, there are two sets $Q_1,Q_2\subseteq R'$ with $|Q_1|,|Q_2|\le f(t|N|)$, such that $R'\setminus Q_i$ is independent in $M'_i$ for $i = 1, 2$. Hence, $R'\setminus (Q_1\cup Q_2)$ is independent in both $M'_1$ and $M'_2$. Because $|Q_1\cup Q_2| \le 2f(t|N|) < t$, we have by the pigeonhole principle that there is one $j\in [t]$ such that $(Q_1\cup Q_2)\cap N_j=\emptyset$. This implies that $R = R'\cap N_j = (R'\setminus (Q_1\cup Q_2))\cap N_j$, when interpreted on the ground set N, is independent in both $M_1$ and $M_2$. Moreover, the elements of R, when interpreted on the ground set N, are a basis in $M_0$ because $R'$ is a basis in $M'_0$. Hence, R is the desired basis without any violations.    $\square$

B Omitted Proofs

Proof of Corollary

3. Extend N by adding a set F of $r(N_0)$ additional elements with 0 weight, where r is the rank function of $M_0$. We modify $M_0$ to a matroid $\widehat{M}_0$ on the ground set $N_0\,\cup \,F$, given by the rank function . That is, $\widehat{M}_0$ is the union of $M_0$ with a free matroid on F, but then truncated to have rank $r(N_0)$. Let $P_{\widehat{\mathcal {B}}_0}$ be the matroid base polytope of $\widehat{M}_0$. It is now easy to see that if $x\in \mathbb R^{N_0\cup F}$ lies in $P_{\widehat{\mathcal {B}}_0}$, then $x\vert _{N_0}\in P_{\mathcal I_0}$. Moreover, we can extend $x\in \mathbb R^{N_0}$ with $x\in P_{\mathcal I_0}$ to $x'\in \mathbb R^{N_0\cup F}$ so that $x'\in P_{\widehat{\mathcal {B}}_0}$ and $x'\vert _{N_0}=x$. The corollary thus follows by applying Theorem 2 to $\widehat{M}_0, M_1, \ldots , M_k$.    $\square$

Proof sketch of Theorem

7. We first state the LP-relaxation ($\mathrm{LP}_\mathrm{matkn}$).

The algorithm leading to Theorem 7 is quite similar to Algorithm 1, and so is its analysis, and we highlight the main changes.

In the algorithm, whenever we contract an element e, for each knapsack constraint with $e\in N_i$, we now update $U_i\leftarrow U_i-c^i_e$ and drop e from $N_i$. After performing all possible deletions, contractions, and refinements, we now either drop a matroid $M'\in \mathcal M'$ in step 6 as before, or, we drop a knapsack constraint for some $i\in \{k+1,\ldots ,k+t\}$ if $|N_i|-x^*(N_i)\le q_i$.

To prove termination, we need only argue that we can always drop a matroid constraint, or a knapsack constraint in step 6 (modified as above). This follows from the same token-counting argument as in the proof of Lemma 6. Recall that if $Ax=b$ is a full-rank subsystem of ($\mathrm{LP}_\mathrm{matkn}$) consisting of linearly independent $x^*$-tight constraints, then we may assume that the rows of A corresponding to the $M_0$-constraints form a nested family $\mathcal {C}$. We define a token-assignment scheme, where each $e\in N$ supplies $x^*(e)$ tokens to the row of A corresponding to the smallest set in $\mathcal {C}$ containing e (if one exists), and $\bigl (1-x^*(e)\bigr )/q_{M'}$ to each row $A_{M'}$ coming from a matroid $M'\in \mathcal M$ in our collection whose ground set contains e. Additionally, every $e\in N$ now also supplies $\bigl (1-x^*(e)\bigr )/q_i$ tokens to each row of A originating from a knapsack constraint whose ground set contains e. Under this scheme, as before, given the constraint on our q-values, it follows that every $e\in N$ supplies at most 1 token unit. Also, as before, each row of A corresponding to an $M_0$ constraint receives at least 1 token unit. So either there is some row $A_{M'}$ coming from a matroid in $\mathcal M$ that receives strictly less than 1 token-unit, or there must be some row of A corresponding to a knapsack constraint that receives at most 1 token-unit; the latter case corresponds to a knapsack constraint i with $|N_i|-x^*(N_i)\le q_i$.

The proof of parts (i)–(iii) is exactly as before. To prove part (iv), consider the i-th knapsack constraint. Note that the only place where we possibly introduce a violation in the knapsack constraint is when we drop the constraint. If $x^*$ is the optimal solution just before we drop the constraint, then we know that $(c^i)^Tx^*\vert _{N_i}\le U_i$. (Note that $N_i$ and $U_i$ refer to the updated ground set and budget.) It follows that if S denotes the set of elements included from this residual ground set $N_i$, then the additive violation in the knapsack constraint is

$$\begin{aligned} c^i(S)-U_i\le c^i(N_i)-U_i\le \bigl (\max _{e\in N_i}c^i_e\bigr )\bigl (|N_i|-x^*(N_i)\bigr ) \le q_i\cdot \bigl (\max _{e\in N_i}c^i_e\bigr ). \end{aligned}$$

   $\square$