Reducing Matroid Optimization to Basis Search

TL;DR

The paper tackles parallel matroid optimization by addressing the adaptive–query tradeoff inherent in folklore reductions from optimization to basis search. It introduces a binary matroid–specific reduction that leverages a local optimality certificate on cocircuits and the lattice of flats to realize Borůvka‑style parallel tests, achieving $O(\log r\cdot\sqrt{n})$ adaptivity and $O(n r\log r)$ independence queries when combined with a $\mathcal{O}(\sqrt{n})$-adaptive basis search. The core idea is that a basis is optimal iff it comprises the minimum-weight points in some cocircuit of the dual matroid, and that modular-pair structure in binary matroids allows collision resolution via symmetric differences. The approach yields a near-optimal balance between adaptivity and query complexity in the sparse regime ($r\ll n$) and provides a framework (via lattice-theoretic and duality tools) for extending parallel matroid optimization techniques beyond the binary case. Overall, the work advances the design of parallel matroid algorithms by connecting cocircuit-based certificates, duality, and the lattice of flats to practical reduction strategies and complexity guarantees.

Abstract

Much energy has been devoted to developing a matroid's computational properties, yet parallel algorithm design for matroid optimization seems less understood. Specifically, the current state of the art is a folklore reduction from optimization to the search based on methods originating in [KUW88]. However, while this reduction adds only constant overhead in terms of \emph{adaptive complexity}, it imposes a high cost in \emph{query complexity}. In response, we present a new reduction from optimization to search within the class of \emph{binary matroids} which, when $n$ and $r$ take the size of the ground set and matroid rank respectively, implies a novel optimization algorithm terminating in $\mathcal{O}(\sqrt{n}\cdot\log r)$ parallel rounds using only $\mathcal{O}(rn\cdot\log r)$ independence queries. This is a significant improvement in query complexity when the matroid is sparse, meaning $r \ll n$, while trading off only a logarithmic factor of the rank in the adaptive complexity. At a technical level, our method begins by observing that a basis is optimal if and only if it is the set of points of minimum weight in any cocircuit. Importantly, this certificate reveals that simultaneous tests for \emph{local optimality} in cocircuits is a general paradigm for parallel matroid optimization. By combining this idea with connections between bases and cocircuits we obtain our reduction, whose efficiency follows by analyzing the lattice of flats. A primary goal of our study is initiating a finer understanding of parallel matroid optimization. And so, since many of our techniques begin with observations about general matroids and their flats, we hope that our efforts aid the future design of parallel matroid algorithms and applications of lattice theory thereof.

Figures (7)

• Figure 1: For graphic matroids, whose indepedent sets correspond to acyclic edge sets of a graph, the spanning trees (or forests if not connected) give the bases. Hence, the minimum weight spanning tree is the solution to an optimization instance over such a matroid.
• Figure 2: Shown above is a graph (left) and the lattices of flats of the associated graphic matroid (right). A basis is given by any spanning tree, i.e. $\{a,b,c\}$, $\{a,b,d\}$, or $\{a,c,d\}$. Observe, that among the rank 2 flats the three containing the edge $a$ are themselves independent, and the addition of any new edge forms a spanning tree. However, there is one rank 2 flat of cardinality 3, consisting of the triangle $\{b,c,d\}$. There are 3 independent sets spanning this triangle, and the addition of any edge but $a$ to one of these independent sets cannot increase the rank since a cycle forms.
• Figure 3: The Hasse diagram corresponding to the lattice of flats given by $\mathbb{F}_2^3 - 000$. Note, if the zero vector where included in this matroid it would be a loop. Linearly independent sets of vectors of a vector space form the most "full" linear matroid in some sense, and it follows that every matroid of rank 3 is representable on $\mathbb{F}_2$ is a minor of $\mathbb{F}_2^3 - 000$. Since the graphic matroid in Figure \ref{['fig:graphic-matroid']} is of rank 3, this concept should apply. In fact, one can work out that the lattice obtained from the deletion $\mathbb{F}_2^3\setminus \left\{000, 001, 011, 101\right\}$ is isomorphic to that in Figure \ref{['fig:graphic-matroid']}.
• Figure 4: An illustration of the evolution of a graph and edges selected by Borůvka's algorthm (shown in red dotted lines) when executed on the graph of Figure \ref{['fig:mst']}. The minimum weight spanning tree is computed in two steps, corresponding to the left and right subfigures.
• Figure 5: An illustration of the bonds corresponding to the incident edges of certain vertices in the graph corresponding to the execution of Borůvka's algorithm given by Figure \ref{['fig:boruvka']}. The left examines those edges corresponding to the upper right vertex in the first round, whereas the right examines those corresponding to the contracted vertex given by the selected $\{e_3, e_4\}$ edges in the second round. The induced components and their complements in the original graph are circled by a dashed line. The edges under consideration form maximal sized cut sets shown in red, i.e. a bond.

Theorems & Definitions (38)

• Theorem : Main Result
• Definition 3.1: Matroid
• Lemma 3.1: Folklore
• Remark 3.2
• Theorem 4.1: Folklore
• Theorem 5.1: white1971thesiswhite1987combinatorial
• Lemma 5.1: Folklore
• Definition 5.2: Free Sets
• Lemma 5.2
• proof