Brokkelkamp, R.

de Vries, M.

van Venetië, R.

Westerdiep, J.

English

We describe tdULL, an algorithm for computing treedepth decompositions of minimal depth. An implementation was submitted to the exact track of PACE 2020. tdULL is a branch and bound algorithm branching on inclusion-minimal separators

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UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl)UvA-DARE (Digital Academic Repository)PACE solver description: tdULLBrokkelkamp, R.; van Venetië, R.; de Vries, M.; Westerdiep, J.DOI10.4230/LIPIcs.IPEC.2020.29Publication date2020Document VersionFinal published versionPublished in15th International Symposium on Parameterized and Exact ComputationLicenseCC BYLink to publicationCitation for published version (APA):Brokkelkamp, R., van Venetië, R., de Vries, M., & Westerdiep, J. (2020). PACE solverdescription: tdULL. In Y. Cao, & M. Pilipczuk (Eds.), 15th International Symposium onParameterized and Exact Computation: IPEC 2020, December 14–18, 2020, Hong Kong,China (Virtual Conference) [29] (Leibniz International Proceedings in Informatics; Vol. 180).Schloss Dagstuhl - Leibniz-Zentrum für Informatik.https://doi.org/10.4230/LIPIcs.IPEC.2020.29General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s)and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an opencontent license (like Creative Commons).Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, pleaselet the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the materialinaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letterto: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. Youwill be contacted as soon as possible.Download date:10 Mar 2023PACE Solver Description: tdULLRuben BrokkelkampCentrum Wiskunde & Informatica (CWI), The Netherlandsruben.brokkelkamp@cwi.nlRaymond van VenetiëKorteweg–de Vries Institute, University of Amsterdam, The Netherlandsr.vanvenetie@uva.nlMees de VriesUniversity of Amsterdam, The Netherlandsmeesdevries@protonmail.comJan WesterdiepKorteweg–de Vries Institute, University of Amsterdam, The Netherlandsj.h.westerdiep@uva.nlAbstractWe describe tdULL, an algorithm for computing treedepth decompositions of minimal depth. Animplementation was submitted to the exact track of PACE 2020. tdULL is a branch and boundalgorithm branching on inclusion-minimal separators.2012 ACM Subject Classification Mathematics of computing → Graph algorithms; Theory ofcomputation → Algorithm design techniques; Theory of computation → Graph algorithms analysisKeywords and phrases PACE 2020, treedepth, treedepth decomposition, vertex ranking, minimalseparators, branch and boundDigital Object Identifier 10.4230/LIPIcs.IPEC.2020.29Supplementary Material The source code can be found on https://github.com/mjdv/tdULL andhttps://doi.org/10.5281/zenodo.38814721 IntroductionThe treedepth of an undirected graph is a measure of the complexity of the graph. Informally,it measures how resistant the graph is to being disconnected by removing vertices. All graphshave a treedepth between 1 and their number of vertices. Star graphs Kn,1, which canbe completely disconnected by removing a single vertex, have treedepth 2. Trees have atreedepth at most logarithmic in their size. Complete graphs Kn have full treedepth of n. Ingeneral, computing the treedepth of a graph is NP-complete.The PACE 2020 challenge consists of implementing an algorithm that is capable ofcomputing treedepth. In the exact track, to which the solver tdULL was submitted, thegoal was to compute the exact treedepth of 100 graphs from an unknown set, with a timelimit of 30 minutes per graph. In order to test implementations, a set of 100 different butrepresentative graphs was published at the start of the contest.There are many equivalent definitions of treedepth. The following is useful for ourpurposes. For S a set of vertices of G, we write G \ S for the graph obtained from G byremoving the vertices from S and any incident edges. For a graph G, we write cc(G) for theset of its connected components.© Ruben Brokkelkamp, Raymond van Venetië, Mees de Vries, and Jan Westerdiep;licensed under Creative Commons License CC-BY15th International Symposium on Parameterized and Exact Computation (IPEC 2020).Editors: Yixin Cao and Marcin Pilipczuk; Article No. 29; pp. 29:1–29:4Leibniz International Proceedings in InformaticsSchloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany29:2 PACE Solver Description: tdULLI Definition 1 (Treedepth). The treedepth td(G) of a connected graph G = (V, E) is recurs-ively defined astd(G) :=1 if |V | = 1,minv∈VmaxH∈cc(G\{v})1 + td(H) else.The minimizing vertex v in the recursion can be taken as the root of a tree, with the connectedcomponents of G \ {v} its children; this way, the computation of treedepth gives rise to atree, the treedepth decomposition of G. The depth of this tree is the treedepth of G.If removal of v does not cause the graph to be disconnected, the next (recursive) step isagain the removal of a single vertex, until enough vertices have been removed to disconnectthe graph. This leads to the following alternative definition. A set S of vertices of a connectedgraph G is called a separator of G if G \ S is disconnected. Such a set S is called inclusionminimal if there is no S′ ( S which separates G. Write sep(G) for the inclusion-minimalseparators of a graph G.I Definition 2 (Treedepth). The treedepth td(G) of a connected graph G = (V, E) is recurs-ively defined astd(G) :=|V | if G∼= K|V |,minS∈sep(G)maxH∈cc(G\S)|S|+ td(H) else.With this definition, tdULL can be described in one sentence as a branch-and-bound algorithmbased on Definition 2, with heuristics, clever data structures and exact special cases used forspeed-up. The advantage of recurring on separators rather than on single vertices is avoidingduplicate branches: if we recur on a separator of size n, a vertex-based recursion may haven! branches leading to that same point.2 Algorithm2.1 Branch and boundTo prune the search tree we use branch and bound. If we have found a treedepth decompositionof depth d for a particular graph, then any further search for a treedepth decomposition ofthat graph need not continue with any attempt that will have treedepth at least d. Similarly,if we have picked a separator S, and for H ∈ cc(G \ S) we have that td(H) = d, then thereis no need for us to find a treedepth decomposition for H ′ ∈ cc(G \ S) of depth lower than d,since it will not make the depth of the full decomposition any lower.To this end, the main component of tdULL is a function treedepth, whose argumentsare a graph G, as well as a search lower bound (SLB) and a search upper bound (SUB). Theinterval between these two bounds is the interval of treedepths dat are still relevant. Thefunction treedepth returns a lower bound l and an upper bound u on the treedepth of G,for which at least one of the following three things is true:u < SLB: the treedepth of G is so low we do not need the exact value;l > SUB: the treedepth of G is so high we do not need the exact value;u = l: the treedepth of G is equal to u = l.The main loop of treedepth generates the separators of the graph G, and recursively callsitself on the components left after removing the separator. The search bounds allow us toskip computation that is provably not going to improve the treedepth of the graph.R. Brokkelkamp, R. van Venetië, M. de Vries, and J. Westerdiep 29:32.1.1 Lower boundsUpper bounds on the treedepth of the graph G can be found directly, by completing branchesof the recursion. Lower bounds are harder to find: the only sure way to find a lower bound isto recur on all possible separators, and conclude that a smaller treedepth cannot be realized.To skip as many of the branches as possible, it is therefore important that we obtaingood lower bounds as quickly as possible. Our main tool is the fact that the treedepth of agraph is minor monotone: if H is a graph that can be obtained from G by removing verticesand edges and contracting edges, then td(H) ≤ td(G). We apply this principle in a numberof ways.Any lower bound on the treedepth of a subgraph returned by the recursion is also a lowerbound on td(G). In rare cases, this lower bound may actually be equal to the treedepth ofG, allowing us to short-circuit the rest of the computation.We pick specific subgraphs of G of which we compute the treedepth. The most importantof these is a core: we take the vertex of lowest degree of G, and iteratively remove all verticesfrom G that have at most that degree. In the resulting graph every vertex has higher degree.Heuristically, this core should be the “toughest subset” of G, and thus provide good lowerbounds.If this core is empty, then we compute a contraction of the graph G instead, and try tofind its treedepth to use as a lower bound. Specifically, we contract the edge between a vertexof minimal degree and a neighboring vertex for which the overlap of common neighbors isminimal. We experimented with other contraction strategies, but they proved less effective.2.2 SeparatorsIn order to use the recursion suggested by Definition 2, we need to be able to generateinclusion-minimal separators of the graph G. Analysis of the runtime of tdULL suggests thatmost of the runtime is spent on this generation process.We essentially use the algorithm from [1]. This is an algorithm that produces minimalseparators, which are subtly different from inclusion-minimal separators. A separator S iscalled minimal if there are vertices v, w ∈ G \ S such that v, w are in different componentsafter removing S, and there are no S′ ( S which separate v, w. It is easy to check whethera minimal separator S is also inclusion minimal: S is inclusion minimal if and only if eachvertex in S has an edge to each connected component of G\S. Thus we can use the algorithmfor minimal separators, and filter out those which are not inclusion minimal.We do not generate all separators of a graph at once, but in batches of 10,000. Thisoccasionally helps us to avoid having to compute all the separators, by finding both an optimaldecomposition and an optimal lower bound early. We tried several batch size strategies, bothfixed and dynamic, and this one worked best.A batch of separators is not tried in arbitrary order: we sort the separators by the size ofthe largest component that remains after removing the separator. This prioritizes separatorsthat appear to be efficient at disconnecting the graph, which are heuristically more likely tolead to optimal solutions. We tried several sorting strategies, this one appeared to work best.If G has a leaf (degree one vertex), then the one neighbor of that leaf forms an inclusion-minimal separator by itself. If a graph has a leaf attached to (nearly) every other vertex, ourinclusion-minimal separators are (almost) the same as the vertices of degree greater than one.Then the recursion turns into the one from Definition 1, which is much less efficient. To avoidthis problem, we actually compute separators on the graph with all leaves removed. Sinceleaves are never useful as the root of a treedepth decomposition, these separators suffice.IPEC 202029:4 PACE Solver Description: tdULL2.3 Special casesThere are a few cases in which we can quickly find an optimal treedepth decomposition of agraph G. We can easily recognize these cases and apply the faster direct algorithm.G ∼= Kn for some n;G ∼= Cn for some n;G is a tree (with the algorithm described in [2]).2.4 CacheRemoving the same set of vertices in a different order results in the same graph. To avoiddouble work we store all lower and upper bounds on the treedepth of subgraphs in a cache.To quickly add, update and retrieve items from the cache we make use of a SetTrie datastructure, as described in [3]. Furthermore, this data structure allows for efficient retrieval ofsubsets and hence can be used to compute lower bounds: because td(H) ≤ td(G) if H ⊆ G,we can use a lower bound on the treedepth of H as a lower bound on the treedepth of G.This cache also works with contractions: every vertex created by a contraction has acanonical representation as the set of vertices which are contracted. When a new combinationof vertices is used for a contraction, it gets a new global index to use in the SetTrie cache.3 DiscussionThe above algorithm is the result of a trial-and-error process, where a big subset of the publicinstances was used to compare versions. Comparing the number of cases solved by us and bythe submissions above us on both the public and private test sets, it seems we may havesuffered from some overfitting: we made algorithmic choices that performed better on thepublic instances of the problem than on the hidden instances.Many ideas did not make it to the final algorithm. One notable omission is using thesymmetry of a graph. Neither using graph automorphisms to reduce the number of separatorsnor trying to use a cache that works up to isomorphism yielded any improvement.tdULL spends a lot of time on generating separators. If a graph has many – in theworst case, there may be exponentially many – this pushes us quickly over the time limit.Small improvements such as only using separators of the graph where all leaves are removedcertainly helped, but we we have not found a way to substantially reduce the number ofseparators. Switching from a single vertex recursion to a separator-based recursion has beenthe biggest performance improvement, though.Another improvement which yielded big performance improvements was finding bettersubgraphs with which to compute lower bounds. Even when a graph has a lot of separators,if for some separator the upper bound from the recursion matches the lower bound foundbefore, the algorithm finishes fast because of an early exit.References1 Anne Berry, Jean-Paul Bordat, and Olivier Cogis. Generating all the minimal separators of agraph. In Graph-Theoretic Concepts in Computer Science, pages 167–172, Berlin, Heidelberg,1999. Springer Berlin Heidelberg.2 Ananth V. Iyer, H. Donald Ratliff, and G. Vijayan. Optimal node ranking of trees. Inf.Process. Lett., 28(5):225–229, August 1988. doi:10.1016/0020-0190(88)90194-9.3 Iztok Savnik. Index data structure for fast subset and superset queries. In InternationalConference on Availability, Reliability, and Security, pages 134–148. Springer, 2013.

PACE solver description: tdULL

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PACE solver description: tdULL

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