Exact Algorithms for Finding Well-Connected 2-Clubs in Sparse - - PowerPoint PPT Presentation

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 102 103 104 105 106 20 40 60 5 Graph size (n) Time (minutes) 102 103 104 105 106 107 20 40 60 5 Graph size (m) Time (minutes)

Komusiewicz, Nichterlein, Niedermeier, Picker Well-Connected 2-Club 1.1 2-Club 3 / 16

Analysis: 2-Club size

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101 102 103 2-club size

2-club size

1 2 3 4 2-club size − ∆

2-club size 2-club size−∆

Almost optimal algorithm: Return a maximum degree vertex with its neighbors

Komusiewicz, Nichterlein, Niedermeier, Picker Well-Connected 2-Club 1.1 2-Club 4 / 16

2-Club extensions

Definition ([Veremyev, Boginski; Eur J Oper Res, 2012]) t-robust 2-club G: Any pair of vertices is connected by t internally vertex-disjoint paths of length at most two. 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. ⇐ ⇒ any pair of nonadjacent vertices has t + 1 common neighbors. Definition ([Pattillo, Youssef, Butenko; Eur J Oper Res, 2013]) t-connected 2-club G: G is a 2-club and t-vertex-connected. u1 v1 u2 v2 u3 v3 Example: A K3,3 is a

◮ 1-robust 2-club ◮ 2-hereditary 2-club ◮ 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(2ℓnm) ℓ . . . number of vertices not in a solution Note: no 2(1−ε)ℓnO(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

coAuthorsCiteseer coAuthorsDBLP citationCiteseer coPapersDBLP coPapersCiteseer

101 102 103 104 105 106 107

n m maximum degree average degree maximum 2-neighborhood average 2-neighborhood h2-index

Advantage: Turing kernelization allows to store data for each pair

• f vertices (e. g. number of

common neighbors)

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 10−1 100 101 102 103 10−2 10−1 100 101 102 103 103.56 103.56 Our Implementation Their Implementation

×2 ×5 ×25 HKN CHLS

2-Club implementations:

◮ HKN (Java) [Hartung, Komusiewicz, N.; Journal of Graph Algorithms and Applications, 2015] ◮ CHLS (c++) [Chang, Hung, Lin, Su; Computing, 2013.]

Komusiewicz, Nichterlein, Niedermeier, Picker Well-Connected 2-Club

• 3. Experimental Results

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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 10−1 100 101 102 103 10−2 10−1 100 101 102 103 103.56 103.56 Our Implementation Their Implementation

×2 ×5 ×25 BB ILP

Implementations: 2-connected 2-Club

◮ BB - Branch & Bound (c++) ◮ ILP (c++ & Gurobi) [Yezerska, Pajouh, Butenko; European Journal of Operational Research, 2017.]

Komusiewicz, Nichterlein, Niedermeier, Picker Well-Connected 2-Club

• 3. Experimental Results

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Experiments III

100 101 102 103 102 103 104 t Running time

(t − 1)-hereditary t-robust t-connected

101 102 103 500 1,000 t 2-club size Graph: coPapersCiteseer

Komusiewicz, Nichterlein, Niedermeier, Picker Well-Connected 2-Club

• 3. Experimental Results

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Experiments IV

100 101 102 103 101 102 103 t Running time

(t − 1)-hereditary t-robust t-connected

100 101 102 103 500 1,000 1,500 t 2-club size Graph: coAuthorsCiteseer

Komusiewicz, Nichterlein, Niedermeier, Picker Well-Connected 2-Club

• 3. Experimental Results

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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

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