A Pivot Gray Code Listing for the Spanning Trees of the Fan Graph

Cameron, Ben; Grubb, Aaron; Sawada, Joe

doi:10.1007/978-3-030-89543-3_5

Cited by 5 publications

(3 citation statements)

References 19 publications

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“…A complete C implementation of our algorithms is available in the Appendix. A preliminary version of this paper appeared in COCOON 2021 [3].…”

Section: New Resultsmentioning

confidence: 99%

Pivot Gray Codes for the Spanning Trees of a Graph ft. the Fan

Cameron¹,

Grubb²,

Sawada³

2022

Preprint

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We consider the problem of listing all spanning trees of a graph G such that successive trees differ by pivoting a single edge around a vertex. Such a listing is called a "pivot Gray code", and it has more stringent conditions than known "revolving-door" Gray codes for spanning trees. Most revolving-door algorithms employ a standard edge-deletion/edge-contraction recursive approach which we demonstrate presents natural challenges when requiring the "pivot" property. Our main result is the discovery of a greedy strategy to list the spanning trees of the fan graph in a pivot Gray code order. It is the first greedy algorithm for exhaustively generating spanning trees using such a minimal change operation. The resulting listing is then studied to find a recursive algorithm that produces the same listing in O(1)-amortized time using O(n) space. Additionally, we present O(n)-time algorithms for ranking and unranking the spanning trees for our listing; an improvement over the generic O(n 3 )-time algorithm for ranking and unranking spanning trees of an arbitrary graph. Finally, we discuss how our listing can be applied to find a pivot Gray code for the wheel graph.

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“…A complete C implementation of our algorithms is available in the Appendix. A preliminary version of this paper appeared in COCOON 2021 [3].…”

Section: New Resultsmentioning

confidence: 99%

Pivot Gray Codes for the Spanning Trees of a Graph ft. the Fan

Cameron¹,

Grubb²,

Sawada³

2022

Preprint

Self Cite

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“…The greedy method has also been very useful in discovering new Gray codes for (generalized) permutations [SW16, CSW21], for spanning trees of special graphs [CGS21], and for spanning trees of arbitrary graphs and more generally bases of any matroid [MMW22]. Also, the permutation-based framework for combinatorial generation proposed by Hartung, Hoang, Mütze and Williams [HHMW22] relies on a simple greedy algorithm.…”

Section: Genlex Ordermentioning

confidence: 99%

Traversing combinatorial 0/1-polytopes via optimization

Merino,

Mütze

2023

2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)

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In this paper, we present a new framework that exploits combinatorial optimization for efficiently generating a large variety of combinatorial objects based on graphs, matroids, posets and polytopes. Our method relies on a simple and versatile algorithm for computing a Hamilton path on the skeleton of any 0/1-polytope conv(X), where X ⊆ {0, 1} n . The algorithm uses as a black box any algorithm that solves a variant of the classical linear optimization problem min{w • x | x ∈ X}, and the resulting delay, i.e., the running time per visited vertex on the Hamilton path, is only by a factor of log n larger than the running time of the optimization algorithm. When X encodes a particular class of combinatorial objects, then traversing the skeleton of the polytope conv(X) along a Hamilton path corresponds to listing the combinatorial objects by local change operations, i.e., we obtain Gray code listings.As concrete results of our general framework, we obtain efficient algorithms for generating all (c-optimal) bases and independent sets in a matroid; (c-optimal) spanning trees, forests, matchings, maximum matchings, and c-optimal matchings in a general graph; vertex covers, minimum vertex covers, c-optimal vertex covers, stable sets, maximum stable sets and c-optimal stable sets in a bipartite graph; as well as antichains, maximum antichains, c-optimal antichains, and c-optimal ideals of a poset. Specifically, the delay and space required by these algorithms are polynomial in the size of the matroid ground set, graph, or poset, respectively. Furthermore, all of these listings correspond to Hamilton paths on the corresponding combinatorial polytopes, namely the base polytope, matching polytope, vertex cover polytope, stable set polytope, chain polytope and order polytope, respectively.As another corollary from our framework, we obtain an O(t LP log n) delay algorithm for the vertex enumeration problem on 0/1-polytopes {x ∈ R n | Ax ≤ b}, where A ∈ R m×n and b ∈ R m , and t LP is the time needed to solve the linear program min{w • x | Ax ≤ b}. This improves upon the 25-year old O(t LP n) delay algorithm due to Bussieck and Lübbecke.

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“…This amazing rule works for any graph H, for any edge ordering that determines the indices of the characteristic vector, for any initial spanning tree, and for any tie-breaking rule, it always yields a genlex listing, and it also generalizes straightforwardly to bases of a matroid (however, this Gray code is in general not cyclic). P50 Cameron, Grubb and Sawada [CGS21] asked whether there is a Gray code for spanning trees of a graph H that uses edge exchanges, with the additional requirement that the removed and added edge have a vertex of H in common. They refer to a listing satisfying this stronger requirement as a pivot Gray code, and they constructed such a Gray code for the family of fan graphs, which are obtained by joining a single vertex to all vertices of a path.…”

Section: Spanning Trees Cycles and Generalizations To Matroidsmentioning

confidence: 99%

Combinatorial Gray Codes — an Updated Survey

Mütze

2023

Electron. J. Combin.

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A combinatorial Gray code for a class of objects is a listing that contains each object from the class exactly once such that any two consecutive objects in the list differ only by a `small change'. Such listings are known for many different combinatorial objects, including bitstrings, combinations, permutations, partitions, triangulations, but also for objects defined with respect to a fixed graph, such as spanning trees, perfect matchings or vertex colorings. This survey provides a comprehensive picture of the state-of-the-art of the research on combinatorial Gray codes. In particular, it gives an update on Savage's influential survey [C. D. Savage. A survey of combinatorial Gray codes. SIAM Rev., 39(4):605--629, 1997.], incorporating many more recent developments. We also elaborate on the connections to closely related problems in graph theory, algebra, order theory, geometry and algorithms, which embeds this research area into a broader context. Lastly, we collect and propose a number of challenging research problems, thus stimulating new research endeavors.

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A Pivot Gray Code Listing for the Spanning Trees of the Fan Graph

Cited by 5 publications

References 19 publications

Pivot Gray Codes for the Spanning Trees of a Graph ft. the Fan

Pivot Gray Codes for the Spanning Trees of a Graph ft. the Fan

Traversing combinatorial 0/1-polytopes via optimization

Combinatorial Gray Codes — an Updated Survey

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