Matroid Intersection under Minimum Rank Oracle

TL;DR

This work analyzes matroid intersection under a restricted minimum rank oracle, formalizing the challenge that the usual exchange-based augmenting-path methods cannot be applied directly. It introduces modified and consistent exchangeability graphs to emulate augmentations using only the minimum of the two matroid ranks, and shows unweighted tractability via these graphs, while the weighted case is tractable only in several special cases. A 2-SAT formulation enables efficient construction of almost consistent graphs, yielding polynomial-time solutions for certain structural conditions and a lexicographically maximal, near-optimal outcome. The paper also establishes NP-hardness for finding a fully consistent graph in general and proves hardness for polymatroid intersection under the same oracle, underscoring fundamental limits. Overall, the work delineates a nuanced boundary between tractable and intractable instances under the minimum rank oracle and offers algorithmic avenues for specific cases via 2-SAT and lexicographic strategies.

Abstract

In this paper, we consider the tractability of the matroid intersection problem under the minimum rank oracle. In this model, we are given an oracle that takes as its input a set of elements and returns as its output the minimum of the ranks of the given set in the two matroids. For the unweighted matroid intersection problem, we show how to construct a necessary part of the exchangeability graph, which enables us to emulate the standard augmenting path algorithm. For the weighted problem, the tractability is open in general. Nevertheless, we describe several special cases where tractability can be achieved, and we discuss potential approaches and the challenges encountered. On the positive side, we present a solution for the case where no circuit of one matroid is contained within a circuit of the other. Additionally, we propose a fixed-parameter tractable algorithm, parameterized by the maximum size of a circuit of one matroid. We also show that a lexicographically maximal common independent set can be found by the same approach, which leads to a nontrivial approximation ratio for finding a maximum-weight common independent set. On the negative side, we prove that the approach employed for the tractable cases above involves an NP-hard problem in the general case. We also show that if we consider the generalization to polymatroid intersection, even the unweighted problem is hard under the minimum rank oracle.

Figures (4)

• Figure 1: An illustration for Example \ref{['ex:LE-pair1']}.
• Figure 2: An illustration for Example \ref{['ex:LE-pair2']}. All the four situations are consistent.
• Figure 3: A shortest cheapest $S_I$--$T_I$ path $P$, an arc $a = (x, y) \in P \setminus {\tilde{A}}[I]$, and an arc $b = (y',x')$ taken as in Claim \ref{['cl:mate']} so that the index $j$ is minimized. Here, the segment of $L_j$ starting with $x'$ is enclosed by the dotted line. Other parts enclosed by solid lines are examples of segments of $L_{h}$ with $h < j$.
• Figure 4: The vertex gadgets for $u, w \in V$, together with the edge gadget for $e = \{u, w\} \in F$. White and black vertices represent elements in $E\setminus (I\cup S_I\cup T_I)$ and in $I$, respectively. The two parts with gray background are two evil LE-pairs $(X^u,Y^u)$ and $(X^w,Y^w)$ in the vertex gadgets, and the four parts enclosed by dotted lines are the LE-pairs $(X_i^{e,v}, Y_i^{e,v})$$(i \in \{1, 2\},~v \in \{u, w\})$ defined in the edge gadget. The illustrated realization of exchangeability arcs corresponds to assigning color $(1,1)$ to $u$ and $(1,2)$ to $w$, where we are omitting all pairs of opposite directed arcs $(x, y)$ and $(y, x)$ that actually exist between any pair of white and black vertices $x$ and $y$, respectively, not included in any LE-pairs mentioned above (e.g., $x = x_i^u$ and $y = y_j^w$$(i, j \in \{1, 2\})$, $x = x_1^{e,v}$ and $y = y_2^{v'}$$(v, v' \in \{u, w\})$, or $x = x_2^v$ and $y = y_i^e$$(v \in \{u, w\},~i \in \{1, 2\})$).

Theorems & Definitions (54)

• Theorem 2.1: Edmonds edmonds1970submodular
• Definition 2.2: Exchangeability Graphs
• Lemma 2.3
• Lemma 2.4: Unique Perfect Matching Lemma
• Lemma 2.5: cf. schrijver2003combinatorial
• Lemma 2.6: cf. frank2011connections
• Lemma 3.1
• proof
• Definition 3.2: Modified Exchangeability Graph
• Lemma 3.3