Exact Algorithms for Finding Well-Connected 2-Clubs in Sparse Real-World Graphs: Theory and Experiments Andr´ e Nichterlein Algorithmics and Computational Complexity, Faculty IV, TU Berlin Shonan Meeting 144, March 5th Based on joint work with Christian Komusiewicz, Rolf Niedermeier, and Marten Picker appearing in European Journal of Operational Research, 2019. Komusiewicz, Nichterlein, Niedermeier, Picker Well-Connected 2-Club 1 / 16
2-Club Problem Input: An undirected graph G = ( V, E ) . Task: Find the maximum size 2-club ( = diameter-two subgraph) in G . ◮ proposed as clique relaxation in social network analysis [Mokken; Quality and Quantity, 1979] ◮ NP-hard [Balasundaram, Butenko, Trukhanov; Journal of Combinatorial Optimization, 2005] ◮ NP-hard to approximate within a factor | V | 1 / 2 − ǫ [Asahiro, Doi, Miyano, Samizo, Shimizu; Algorithmica, 2018] Komusiewicz, Nichterlein, Niedermeier, Picker Well-Connected 2-Club 1.1 2-Club 2 / 16
2-Club is hard? Not in practice! Existing implementation: [Hartung, Komusiewicz, N.; Journal of Graph Algorithms and Applications, 2015] Data set: Graphs from clustering and coauthor category of the 10th DIMACS challenge Implementation: Written in Java Machine: CPU 3.60 GHz (Xeon); 64 GB main memory 60 60 Time (minutes) Time (minutes) 40 40 20 20 5 5 0 0 10 2 10 3 10 4 10 5 10 6 10 2 10 3 10 4 10 5 10 6 10 7 Graph size ( n ) Graph size ( m ) Komusiewicz, Nichterlein, Niedermeier, Picker Well-Connected 2-Club 1.1 2-Club 3 / 16
Analysis: 2-Club size Return a maximum degree vertex with its neighbors � Almost optimal algorithm: Komusiewicz, Nichterlein, Niedermeier, Picker 2-club size 10 1 10 2 10 3 karate dolphins adjnoun polbooks celegans-metabolic football netscience jazz email Well-Connected 2-Club power uk add20 add32 2-club size − ∆ 2-club size 2-club size data hep-th PGPgiantcompo polblogs whitaker3 coAuthorsCiteseer crack coAuthorsDBLP cs4 citationCiteseer 1.1 2-Club graph-thres-01 coPapersDBLP coPapersCiteseer 1 2 3 4 2-club size − ∆ 4 / 16
2-Club extensions Definition ( [Veremyev, Boginski; Eur J Oper Res, 2012] ) v 1 u 2 t -robust 2-club G : Any pair of vertices is connected by t internally vertex-disjoint paths of length at most two. u 1 v 2 Definition ( [Pattillo, Youssef, Butenko; Eur J Oper Res, 2013] ) t -hereditary 2-club G : G − U is a 2-club for all U ⊂ V ( G ) where | U | ≤ t . v 3 u 3 ⇐ ⇒ any pair of nonadjacent vertices has t + 1 common neighbors. Example: A K 3 , 3 is a Definition ( [Pattillo, Youssef, Butenko; Eur J Oper Res, 2013] ) ◮ 1-robust 2-club t -connected 2-club G : ◮ 2-hereditary 2-club G is a 2-club and t -vertex-connected. ◮ 3-connected 2-club Komusiewicz, Nichterlein, Niedermeier, Picker Well-Connected 2-Club 1.2 Well-Connected 2-Clubs 5 / 16
Our results Goal: transfer algorithmic work (theoretical & practical) from 2-Club to t -robust / t -hereditary / t -connected 2-Club Results: ◮ “unifying view” on all three considered models ◮ FPT algorithms ◮ competitive implementation Komusiewicz, Nichterlein, Niedermeier, Picker Well-Connected 2-Club 1.3 Our results 6 / 16
Simple search tree Example: Find largest 2 -hereditary 2-club (deleting any 2 vertices yields a 2-club ) Observation: At most one red vertex in a solution. Generic Search tree: FindSolution( G ) 1. If G is a solution then return G 2. u, v ← two “ incompatible ” vertices 3. Return max { FindSolution ( G − v ), FindSolution ( G − u ) } � running time O (2 ℓ nm ) ℓ . . . number of vertices not in a solution Note: no 2 (1 − ε ) ℓ n O (1) algorithm for any ε > 0 , unless SETH fails [Hartung, Komusiewicz, N.; Journal of Graph Algorithms and Applications, 2015] Komusiewicz, Nichterlein, Niedermeier, Picker Well-Connected 2-Club 2.1 Unifying view 7 / 16
Compatible vertices — unifying view Definition Two vertices v and w in a graph are called compatible ◮ for t -robust 2-clubs if they are adjacent and have at least t − 1 common neighbors, or if they have at least t common neighbors, ◮ for t -hereditary 2-clubs if they are adjacent or if they have at least t + 1 common neighbors, ◮ for t -connected 2-clubs if they are at distance at most two and are connected by at least t internally vertex-disjoint paths. Komusiewicz, Nichterlein, Niedermeier, Picker Well-Connected 2-Club 2.1 Unifying view 8 / 16
Turing Kernelization 1. 2. 3. 4. 5. 6. 1. sol ← ∅ 2. foreach v ∈ V do 3. T ← all vertices at distance ≥ 2 from v 4. S ← largest solution in T that contains v 5. if S is larger than sol then sol ← S 6. delete v 7. return sol Komusiewicz, Nichterlein, Niedermeier, Picker Well-Connected 2-Club 2.2 Turing Kernelization & Data reduction 9 / 16
Turing kernelization — practical effect n 10 7 m maximum degree 10 6 average degree maximum 2-neighborhood 10 5 average 2-neighborhood 10 4 h 2 -index 10 3 Advantage: Turing kernelization allows 10 2 to store data for each pair of vertices (e. g. number of 10 1 common neighbors) coPapersDBLP coPapersCiteseer coAuthorsCiteseer coAuthorsDBLP citationCiteseer Komusiewicz, Nichterlein, Niedermeier, Picker Well-Connected 2-Club 2.2 Turing Kernelization & Data reduction 10 / 16
Data reduction & lower bounds Reduction Rule Remove vertices whose degree is too low. Incompatibility graph: Two vertices are adjacent in the the incompatibility graph iff they are not compatible . input graph incompatibility graph Observation: The size of a maximum independent set in the incompatibility graph is an upper bound on the solution size in the input graph. � upper bound worse than best previously found solution ⇒ discard current Turing kernel Komusiewicz, Nichterlein, Niedermeier, Picker Well-Connected 2-Club 2.2 Turing Kernelization & Data reduction 11 / 16
Experiments I Data set: Graphs from clustering and coauthor category of the 10th DIMACS challenge Implementation: Written in Java Machine: CPU 3.60 GHz (Xeon); 64 GB main memory 10 3 . 56 10 3 Their Implementation 2-Club implementations: 10 2 ◮ HKN (Java) 10 1 [Hartung, Komusiewicz, N.; Journal of × 2 10 0 Graph Algorithms and Applications, × 5 2015] × 25 10 − 1 ◮ CHLS (c++) HKN CHLS [Chang, Hung, Lin, Su; Computing, 10 − 2 2013.] 10 − 1 10 0 10 1 10 2 10 3 10 3 . 56 Our Implementation Komusiewicz, Nichterlein, Niedermeier, Picker Well-Connected 2-Club 3. Experimental Results 12 / 16
Experiments II Data set: Graphs from clustering and coauthor category of the 10th DIMACS challenge Implementation: Written in Java Machine: CPU 3.60 GHz (Xeon); 64 GB main memory 10 3 . 56 10 3 Their Implementation 10 2 Implementations: 10 1 2-connected 2-Club × 2 ◮ BB - Branch & Bound (c++) 10 0 × 5 ◮ ILP (c++ & Gurobi) × 25 10 − 1 BB [Yezerska, Pajouh, Butenko; European ILP 10 − 2 Journal of Operational Research, 2017.] 10 − 1 10 0 10 1 10 2 10 3 10 3 . 56 Our Implementation Komusiewicz, Nichterlein, Niedermeier, Picker Well-Connected 2-Club 3. Experimental Results 13 / 16
Experiments III ( t − 1) -hereditary t -robust t -connected 10 4 1 , 000 Running time 2-club size 10 3 500 10 2 0 10 0 10 1 10 2 10 3 10 1 10 2 10 3 t t Graph: coPapersCiteseer Komusiewicz, Nichterlein, Niedermeier, Picker Well-Connected 2-Club 3. Experimental Results 14 / 16
Experiments IV ( t − 1) -hereditary t -robust t -connected 1 , 500 10 3 Running time 1 , 000 2-club size 10 2 500 10 1 0 10 0 10 1 10 2 10 3 10 0 10 1 10 2 10 3 t t Graph: coAuthorsCiteseer Komusiewicz, Nichterlein, Niedermeier, Picker Well-Connected 2-Club 3. Experimental Results 15 / 16
Summary & Outlook Key results: ◮ Unifying approach for several 2-club variants. ◮ Efficient implementation (= data reduction + Turing kernelization + search tree). Work in progress: γ -relative robust 2-club S : 0 < γ ≤ 1 : Any pair of vertices connected by at least γ · | S | paths of length at most two. Example: γ = 0 . 5 input graph incompatibility graph input graph incompatibility graph Open Question: Is t -robust / t -hereditary / t -connected 2-club fixed parameter tractable with respect to the solution size? Thank you! Komusiewicz, Nichterlein, Niedermeier, Picker Well-Connected 2-Club 4. Conclusion 16 / 16
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