Computing A Near-Maximum Independent Set in Linear Time by Reducing-Peeling
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1 Computing A Near-Maximum Independent Set in Linear Time by Reducing-Peeling Computer Science and Engineering Lijun Chang University of New South Wales, Australia Lijun.Chang@unsw.edu.au Joint work with Wei Li, Wenjie Zhang
2 Outline q Introduction q Existing Works q Our Reducing-Peeling Framework q Our Approaches q Experimental Studies q Conclusion
3 Introduction Independent Set Given a graph! = ($, &), a vertex subset ( $ is an independent set if for any two vertices * and + in (, there is no edge between * and + in!. Maximum Independent Set An independent set ( of! is a maximum independent set if its size is the largest among all independent sets of!.
4 Introduction Independent Set Given a graph! = ($, &), a vertex subset ( $ is an independent set if for any two vertices * and + in (, there is no edge between * and + in!. Maximum Independent Set An independent set ( of! is a maximum independent set if its size is the largest among all independent sets of!.sets of!. Independent Set
5 Introduction Independent Set Given a graph! = ($, &), a vertex subset ( $ is an independent set if for any two vertices * and + in (, there is no edge between * and + in!. Maximum Independent Set An independent set ( of! is a maximum independent set if its size is the largest among all independent sets of!. Maximum Independent Set
6 Introduction Applications v Build index for shortest path/distance queries [Cheng et al. SIGMOD 12, Fu et al. VLDB 13] v Refine the result of matching two graphs [Zhu et al. VLDB J 13] v Social network coverage [Puthal et al. BigData 15]; vertex cover Hardness v NP-hard to compute a maximum independent set [Garey et al. Book 79] v Hard to approximate NP-hard to approximate within a factor of -./0 for any 0 < 3 < 1 [J. Håstad. FOCS 96]
7 Outline q Introduction q Existing Works q Our Reducing-Peeling Framework q Our Approaches q Experimental Studies q Conclusion
8 Existing Works Exact algorithms -- branch-and-reduce paradigm v [F. V. Fomin et al.j.acm 09] Theoretically runs in time v [T. Akiba et al. Theor. Comput. Sci. 16] Practically computes the exact solution for many small and medium-sized graphs Approximation algorithms v [U. Feige J. Discrete Math 04, M. M. Halldórsson et al. Algorithmica 97, P. Berman. Theor.Comput. Sys. 99] Approximation ratio largely depends on n or Δ Not practically useful
9 Existing Works Heuristic algorithms for large graphs v Linear-time algorithms Greedy, dynamic update Efficient, but can only find small independent sets in practice v Iterative randomized searching Local search algorithm: ARW [D. V. Andrade. J.Heuristics 12] Evolutionary algorithm: ReduMIS [S. Lamm. ALENEX 16] Local search + simple reduction rules: OnlineMIS [J. Dahlum. SEA 16] Can find large independent sets, but take long time Our goal: find large independent sets in a time-efficient and space-effective manner
10 Outline q Introduction q Existing Works q Our Reducing-Peeling Framework q Our Approaches q Experimental Studies q Conclusion
11 Three Observations Utilized in Our Framework v Observation I: Real networks are usually power-law graphs with many low-degree vertices ;< =>? C v Observation-II: Reduction rules have been effectively used for lowdegree vertices v Observation-III: High-degree vertices are less likely to be in a maximum independent set
12 Three Observations Utilized in Our Framework v Observation I: Real networks are usually power-law graphs with many low-degree vertices v Observation-II: Reduction rules have been effectively used for low-degree vertices Degree-one Reduction (a) D! = D!\ + D! : J-KLML-KL-NL -*OPLQ RS! Degree-two Reductions (b) Isolation D! = D!\ +, F (c) Folding D! = D!/ *, +, F + 1 v Observation-III: High-degree vertices are less likely to be in a maximum independent set
13 Three Observations Utilized in Our Framework v Observation I: Real networks are usually power-law graphs with many low-degree vertices v Observation-II: Reduction rules have been effectively used for lowdegree vertices v Observation-III: High-degree vertices are less likely to be in a maximum independent set Ø If a high-degree vertex is added into the independent set, then all its neighbors, which are of a large quantity, are ruled out from the independent set [J. Dahlum et al SEA 16] Ø Removing/peeling high-degree vertices can further sparsify the graph [Y. Lim et al TKDE 14]
14 The Reducing-Peeling Framework Definition 3.1: (Inexact Reduction) Given a graph!, we remove/peel the vertex with the highest degree from!. v Phase 1: Reducing Ø While a reduction rule can be applied on a vertex * then Apply the exact reduction rule on * v Phase 2: Peeling Ø Apply the inexact reduction rule to temporarily remove a highdegree vertex v Repeat the above two phases until there is no edge in the graph v Post-process: Iteratively add a temporarily removed vertex to the solution if the independence requirement is not violated
15 Outline q Introduction q Existing Works q Our Reducing-Peeling Framework q Our Approaches q Experimental Studies q Conclusion
16 Overview of Our Approaches v Compute large independent set for large graphs in a time-efficient and space-effective manner Subquadratic (or even linear) time. 2m + O(n) space: m is the number of undirected edges. A graph is stored in 2m + n + O(1) space by the adjacency array (aka, Compressed Sparse Row) graph representation A graph with one billion edges takes slightly more than 8GB memory
17 An Efficient Baseline Algorithm v BDOne Step 1: While $ T. or $ WX If $ T. then DegreeOne-Reduction Else Inexact-Reduction =>? Y B = B Step 2: Recover temporarily removed vertices
18 An Efficient Baseline Algorithm v BDOne =>? Y B = B Step 1: While $ T. or $ WX If $ T. then DegreeOne-Reduction Else Inexact-Reduction Step 2: Recover temporarily removed vertices Y Z [\ ][^_ ^_> _[?_>\^ =>?<>>
19 An Efficient Baseline Algorithm v BDOne =>? Y B = B Step 1: While $ T. or $ WX If $ T. then DegreeOne-Reduction Else Inexact-Reduction Step 2: Recover temporarily removed vertices Y Z [\ ][^_ ^_> _[?_>\^ =>?<>>
20 An Efficient Baseline Algorithm v BDOne =>? Y B = B Step 1: While $ T. or $ WX If $ T. then DegreeOne-Reduction Else Inexact-Reduction Step 2: Recover temporarily removed vertices Y Z [\ ][^_ ^_> _[?_>\^ =>?<>>
21 An Efficient Baseline Algorithm v BDOne =>? Y B = B Step 1: While $ T. or $ WX If $ T. then DegreeOne-Reduction Else Inexact-Reduction Step 2: Recover temporarily removed vertices Y Z [\ ][^ [?_>\^ =>?<>> Complexity Analysis Time: 5 O Space: 2O + 5 -
22 An Effective Baseline Algorithm v BDTwo Step 1: While $ T. or $ TX or $ W` If $ T. then DegreeOne-Reduction Else if $ TX then DegreeTwo-Reduction Else Inexact-Reduction Step 2: Recover temporarily removed vertices =>? Y B = B
23 An Effective Baseline Algorithm v BDTwo =>? Y B = B Step 1: While $ T. or $ TX or $ W` If $ T. then DegreeOne-Reduction Else if $ TX then DegreeTwo-Reduction Else Inexact-Reduction Step 2: Recover temporarily removed vertices =>? Y a = b
24 An Effective Baseline Algorithm v BDTwo Step 1: =>? Y B = B While $ T. or $ TX or $ W` If $ T. then DegreeOne-Reduction Else if $ TX then DegreeTwo-Reduction Else Inexact-Reduction Step 2: Recover temporarily removed vertices =>? Y a = b Complexity Analysis Time: 5 - O and g O + -hri- Space: 6O + 5 -
25 An Effective Linear-Time Algorithm v LinearTime Lemma 4.1: (Degree-two Path Reductions) Consider a graph! = $, & with minimum degree two. For a maximal degree-two path k = +., + X,, + m, let + k and F k be the unique vertices connected to +. and + m, respectively. Case 1: + = F D! = D!\ +
26 An Effective Linear-Time Algorithm v LinearTime Lemma 4.1: (Degree-two Path Reductions) Consider a graph! = $, & with minimum degree two. For a maximal degree-two path k = +., + X,, + m, let + k and F k be the unique vertices connected to +. and + m, respectively. Case 2: k is odd and +, F & D! = D!\ +, F
27 An Effective Linear-Time Algorithm v LinearTime Lemma 4.1: (Degree-two Path Reductions) Consider a graph! = $, & with minimum degree two. For a maximal degree-two path k = +., + X,, + m, let + k and F k be the unique vertices connected to +. and + m, respectively. Case 3: k is odd and +, F & D! = D!\ + X,, + m +., F + k 1 2
28 An Effective Linear-Time Algorithm v LinearTime Lemma 4.1: (Degree-two Path Reductions) Consider a graph! = $, & with minimum degree two. For a maximal degree-two path k = +., + X,, + m, let + k and F k be the unique vertices connected to +. and + m, respectively. Case 4: k is even and +, F & D! = D!\ +.,, + m + k 2
29 An Effective Linear-Time Algorithm v LinearTime Lemma 4.1: (Degree-two Path Reductions) Consider a graph! = $, & with minimum degree two. For a maximal degree-two path k = +., + X,, + m, let + k and F k be the unique vertices connected to +. and + m, respectively. Case 5: k is even and +, F & D! = D!\ +.,, + m +, F + k 2
30 An Effective Linear-Time Algorithm v LinearTime Step 1: While $ T. or $ TX or $ W` If $ T. then DegreeOne-Reduction Else if $ TX then DegreeTwoPath-Reduction Else Inexact-Reduction Step 2: Recover temporarily removed vertices
31 An Effective Linear-Time Algorithm v LinearTime Step 1: While $ T. or $ TX or $ W` If $ T. then DegreeOne-Reduction Else if $ TX then DegreeTwoPath-Reduction Else Inexact-Reduction Step 2: Recover temporarily removed vertices
32 An Effective Linear-Time Algorithm v LinearTime Step 1: While $ T. or $ TX or $ W` If $ T. then DegreeOne-Reduction Else if $ TX then DegreeTwoPath-Reduction Else Inexact-Reduction Step 2: Recover temporarily removed vertices
33 An Effective Linear-Time Algorithm v LinearTime Step 1: While $ T. or $ TX or $ W` If $ T. then DegreeOne-Reduction Else if $ TX then DegreeTwoPath-Reduction Else Inexact-Reduction Step 2: Recover temporarily removed vertices
34 An Effective Linear-Time Algorithm v LinearTime Step 1: While $ T. or $ TX or $ W` If $ T. then DegreeOne-Reduction Else if $ TX then DegreeTwoPath-Reduction Else Inexact-Reduction Step 2: Recover temporarily removed vertices
35 An Effective Linear-Time Algorithm v LinearTime Step 1: While $ T. or $ TX or $ W` If $ T. then DegreeOne-Reduction Else if $ TX then DegreeTwoPath-Reduction Else Inexact-Reduction Step 2: Recover temporarily removed vertices Complexity Analysis Time: 5 O Space: 2O + 5 -
36 A Near-Linear-Time Algorithm v NearLinear Lemma 5.1: (Dominance Reduction) [F. V. Fomin et al. JACM 09] Vertex v dominates vertex u if +, * & and all neighbors of v other than u are also connected to u (i.e., s + \ * s * ). If v dominates u, then there exists a maximum independent set of G the excludes u; thus, we can remove u from G, and D! = D!\ *. Lemma 5.2: Vertex v dominates its neighbor u iff +, * = K + 1, where +, * is the number of triangles containing u and v
37 A Near-Linear-Time Algorithm v NearLinear Step1: Maintain the set u of candidate dominated vertices, and also maintain +, * for every edge +, * Step 2: While $ TX or u or $ W` If $ TX then DegreeTwoPath-Reduction Else if u then dominance reduction Else Inexact-Reduction Step 2: Recover temporarily removed vertices Complexity Analysis Time: 5 O Δ (Δ is the maximum degree in!) Space: 4O in worst case and 2O in practice
38 Extensions of Our Algorithms v Accelerate ARW v Compute Upper Bound of D!
39 Outline q Introduction q Existing Works q Our Reducing-Peeling Framework q Our Approaches q Experimental Studies q Conclusion
40 Experimental Settings v Datasets v Environments Ø All algorithms are implemented in C++ Ø All experiments are conducted on a machine with an Intel(R) Xeon(R) 3.4GHz CPU and 16GB main memory running Linux
41 Accuracy v Gap to the maximum independent set size
42 Processing Time (a) Compared with Existing Techniques (b) Compare Our Techniques
43 Memory Usage (a) Compared with Existing Techniques (b) Compare Our Techniques
44 Boost ARW ARW-NL, ARW-LT: ARW boosted by NearLinear and LinearTime, respectively. Convergence plots of local search algorithms
45 Outline q Introduction q Existing Works q Our Reducing-Peeling Framework q Our Approaches q Experimental Studies q Conclusion
46 Conclusion p A new Reducing-Peeling framework p Time-efficient and space-effective techniques to implement the reducing-peeling framework p Find large independent sets efficiently for large real-world graphs with billions of edges
47
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