We present an exact solver for the DFVS, submitted for the exact track of the Parameterized Algorithms and Computational Experiments challenge (PACE) in 2022. The solver heavily relies on data reduction (known from the literature and new reduction rules). The instances are then further processed by integer linear programming approaches. We implemented the algorithm in the scope of a student project at the University of Bremen

Bergenthal, Moritz

Dirks, Jona

Freese, Thorben

Gahde, Jakob

Gerhard, Enna

Grobler, Mario

Siebertz, Sebastian

English

Dagstuhl Research Online Publication Server

PACE Solver Description: GraPA-JAVA∗Moritz Bergenthal !Universität Bremen, GermanyJona Dirks !Universität Bremen, GermanyThorben Freese !Universität Bremen, GermanyJakob Gahde !Universität Bremen, GermanyEnna Gerhard !Universität Bremen, GermanyMario Grobler !Universität Bremen, GermanySebastian Siebertz !Universität Bremen, GermanyAbstractWe present an exact solver for the Directed Feedback Vertex Set Problem (DFVS), submitted forthe exact track of the Parameterized Algorithms and Computational Experiments challenge (PACE)in 2022. The solver heavily relies on data reduction (known from the literature and new reductionrules). The instances are then further processed by integer linear programming approaches. Weimplemented the algorithm in the scope of a student project at the University of Bremen.2012 ACM Subject Classification Theory of computation → Parameterized complexity and exactalgorithmsKeywords and phrases complexity theory, parameterized complexity, linear programming, java,directed feedback vertex set, PACE 2022Digital Object Identifier 10.4230/LIPIcs.IPEC.2022.30Supplementary MaterialSoftware (Source Code Release): https://doi.org/10.5281/zenodo.6647003Software (Public Git Repository): https://gitlab.informatik.uni-bremen.de/grapa/java/fptg-library, archived at swh:1:dir:fd00e212eda2d4eab270dadc83923224551db839Software (Public Git Repository): https://gitlab.informatik.uni-bremen.de/grapa/java/maxcliqueenumeration, archived at swh:1:dir:a4cde68f2e0de0e5b873eb5a02dc975c4a37fbb1Software (Public Git Repository): https://gitlab.informatik.uni-bremen.de/grapa/java/pace-2022-dfvs-solver, archived at swh:1:dir:a80285858dd46e917d4f0c89fd13948177c6a048PreliminariesWe use standard notation for directed graphs as in [2], with u, v ∈ V (G) being two vertices,and (u, v) ∈ E(G) being an edge from vertex u to vertex v. We use the term directed edgefor any edge (u, v) ∈ E(G) if (v, u) ̸∈ E(G) and undirected edge (or 2-cycle) otherwise.1 Solver overviewOn a high level, our algorithm works as follows. It uses three steps to find a minimum DFVSfor an input graph G:1. Parse the input and apply the most basic data reduction rules.2. Apply advanced data reduction rules iteratively (details are presented in Section 2).3. Solve the remaining instance, possibly iteratively (details are presented in Section 3).∗ This is a brief description of one of the highest ranked solvers of PACE Challenge 2022. It has beenmade public for the benefit of the community and was selected based on the ranking. PACE encouragespublication of work building on the ideas presented in this description in peer-reviewed venues.© Moritz Bergenthal, Jona Dirks, Thorben Freese, Jakob Gahde, Enna Gerhard, Mario Grobler, andSebastian Siebertz;licensed under Creative Commons License CC-BY 4.017th International Symposium on Parameterized and Exact Computation (IPEC 2022).Editors: Holger Dell and Jesper Nederlof; Article No. 30; pp. 30:1–30:4Leibniz International Proceedings in InformaticsSchloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany30:2 PACE Solver Description: GraPA-JAVATable 1 Data reduction rules.Name Note Source1. Delete Loops Implicitly applied Folklore, rule 1 of [6]2. Remove connected to n − 1 Special case of rule 53. In-/out-degree 0 Included in CRR Rule 3 of [6], rule 5 of [1]4. In-/out-degree 1 Included in CRR Rule 4 of [6], rule 6 of [1]5. Dominating 2-cycle Superset of both prede-cessors and successorsAdapted from rule 6 of [4]6. Strongly connected compon-entsPrevious rules inbetween One step of [5] at a time7. Delete unnecessary edges Partially included in CRR Special case of rule 108. Contract isolated paths oflength 3 (lines)Possible when not creatingnew induced cyclesAdapted from rule 7 of [4]9. Contract degree three Possible when not creatingnew induced cyclesAdapted from special case ofrule 8 of [4]10. Delete non-induced edges NewWe only need to consider induced edges. For every edge, we start a BFS and donot visit a vertex if a cycle is closed. For this, we track disallowed predecessorsthat get reduced if an alternative path exists. It is not exhaustive, as theremight be cases where two alternative paths would be closed.11. Pick dominating vertices on2-cyclesNewIf a vertex dominates all predecessors/successors of the other vertex on anundirected edge with undirected edges, we can include it in the solution.12. Crown reduction Special case of connected toall in footAdapted from rule 3 of [4]13. Three hitting set Only applies in Section 3.3 NewIf all cycles running through a vertex are at most 3-cycles, we can replace themwith a hitting set gadget. If all vertices of the internal clique are selected, weare allowed to push one outwards, hitting the cycle14. Any hitting set Only applies in Section 3.3 NewApplied to a cycle that is isolated except for at most one edge, in that casecreate a gadget as in rule 132 Data reduction rulesWe apply the data reduction rules presented in Table 1. The rules are applied iteratively.Each rule is applied exhaustively. After a successful rule application, we return to the firstrule.We apply commonly known rules for DFVS (Rules 1, 3, 4, 6). The most local rules areapplied recursively in a rule that we call Combined Recursive Reduction (CRR). As DFVS isa generalization of the Vertex Cover problem, we were able to adapt several rules designedfor Vertex Cover (Rules 2, 5, 8, 9, 12). Additionally, we have found several rules generalazingknown rules from the literature, as well as several completely new rules not known from theliterature. Many of these rules rely on the observation that we only need to hit inducedcycles when solving a DFVS instance, as all other cycles will be hit in that case as well. Thecorrectness of all new rules will be presented in a companion paper.M. Bergenthal et al. 30:33 Exact solvingAfter applying the data reduction rules, depending on the relative number of undirectededges, we employ different solving strategies. If less than half of the edges are undirectededges, we immediately resort to the iterative addition of cycles as a constraint for a hittingset ILP formulation, explained in Section 3.1.In mixed or largely undirected graphs, we resort to an ILP formulation that models theproblem as finding a topological order (Section 3.2) with additional hints to improve theinternal lower bounds of the solver. ILPs are solved with SCIP. If the ILP solver does notterminate within twelve minutes, the attempt will be terminated. Instead, we compute anexact solution to the Vertex Cover problem. If this is not a solution to DFVS, we do notreturn a solution. Details are presented in Section 3.3.3.1 Iterative cycle hitting set ILPOver time, we generate a set K of induced cycles. We initialize K with all induced cycles oflength 2, 3 and 4, and add longer disjoint cycles that are packed greedily until no furtherdisjoint cycles remain. We then solve the ILP (Algorithm 1) and interpret the chosen variablesas vertices to remove from the graph. If no cycle remains, we have obtained the optimalsolution, otherwise, we greedily compute a new cycle packing on the remainder of the graphand add all these cycles to K.Algorithm 1 Hitting set ILP formulation.foreach vi ∈ V (G) add variable x(vi) = xi ∈ X with constraint xi ∈ [0, 1]foreach cycle K ∈ K add constraint∑vi∈K x(vi) ≥ 1minimize∑xi∈X xi3.2 Topological order ILPA graph is acyclic if and only if its vertices can be ordered as v1, . . . , vn such that every edge(vi, vj) satisfies i < j. We use this observation for the following ILP formulation (Algorithm 2).For the undirected subcomponents of the graph, we compute a set of maximum cliques Cusing [3]. This external solver does not always find all 2-cliques, so we compute all 2-cyclesand remove them if they are somewhere included in a clique. We create a partial order onall vertices adjacent to directed edges, denoted as E . The constraints can be interpreted aso(s) < o(t) ∨ x(s) ∨ x(t): edges must be part of a DAG over the order or either vertex chosenfor the solution. We furthermore compute a hint of short cycles K as in Section 3.1. Thevertices chosen can directly be interpreted as a result for DFVS.3.3 Underlying Vertex CoverWe take the undirected part of the graph and compute a minimum vertex cover using [7]. Ifno cycles are remaining, we return the solution. Otherwise, we apply the hitting set gadgetreduction rules (Rules 13 and 14) and solve the resulting instance again. At this point, wecould turn towards iterative solving as in Section 3.1 but did not implement this as the aboveapproach was already sufficient on all of the test instances.IPEC 202230:4 PACE Solver Description: GraPA-JAVAAlgorithm 2 Linear ordered ILP formulation.foreach vi ∈ V (G) addvariable x(vi) = xi ∈ X with constraint xi ∈ [0, 1]variable o(vi) = yi ∈ Y with constraint yi ∈ [0, n]foreach clique C ∈ C add constraint∑vi∈C x(vi) ≥ |C| − 1foreach edge (s, t) ∈ E add constraint o(t) − o(s) + x(t) · (n + 1) + x(s) · (n + 1) ≥ 1foreach cycle K ∈ K add constraint∑vi∈K x(vi) ≥ 1minimize∑xi∈X xiReferences1 Benjamin Bergougnoux, Eduard Eiben, Robert Ganian, Sebastian Ordyniak, and M. S.Ramanujan. Towards a Polynomial Kernel for Directed Feedback Vertex Set. In Kim G.Larsen, Hans L. Bodlaender, and Jean-Francois Raskin, editors, 42nd International Symposiumon Mathematical Foundations of Computer Science (MFCS 2017), volume 83 of LeibnizInternational Proceedings in Informatics (LIPIcs), pages 36:1–36:15, Dagstuhl, Germany, 2017.Schloss Dagstuhl – Leibniz-Zentrum fuer Informatik. doi:10.4230/LIPIcs.MFCS.2017.36.2 Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, MarcinPilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2016.doi:10.1007/978-3-319-21275-3.3 David Eppstein, Maarten Löffler, and Darren Strash. Listing all maximal cliques in large sparsereal-world graphs. ACM J. Exp. Algorithmics, 18, November 2013. doi:10.1145/2543629.4 Michael R. Fellows, Lars Jaffke, Aliz Izabella Király, Frances A. Rosamond, and MathiasWeller. What is known about vertex cover kernelization? In Adventures Between LowerBounds and Higher Altitudes, pages 330–356, 2018. doi:10.1007/978-3-319-98355-4_19.5 Lisa K. Fleischer, Bruce Hendrickson, and Ali Pınar. On identifying strongly connectedcomponents in parallel. In José Rolim, editor, Parallel and Distributed Processing, pages 505–511, Berlin, Heidelberg, 2000. Springer Berlin Heidelberg. doi:10.1007/3-540-45591-4_68.6 Rudolf Fleischer, Xi Wu, and Liwei Yuan. Experimental study of fpt algorithms for thedirected feedback vertex set problem. In Amos Fiat and Peter Sanders, editors, Algorithms– ESA 2009, pages 611–622, Berlin, Heidelberg, 2009. Springer Berlin Heidelberg. doi:10.1007/978-3-642-04128-0_55.7 Demian Hespe, Sebastian Lamm, Christian Schulz, and Darren Strash. WeGotYouCovered:The winning solver from the pace 2019 implementation challenge, vertex cover track. ArXiv,abs/1908.06795, 2019. arXiv:1908.06795.

PACE Solver Description: GraPA-JAVA

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PACE Solver Description: GraPA-JAVA

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