Computing A Near-Maximum Independent
Set in Linear Time by Reducing-Peeling
Lijun Chang, Wei Li, Wenjie Zhang
UNSW Sydney, Australia
[email protected]
Reducing-Peeling Framework
Problem Definition
Ø The Framework
Ø IndependentSet
Givenagraph𝐺 = (𝑉, 𝐸),avertexsubset𝐼 ⊆ 𝑉 isanindependentsetifforanytwo
vertices𝑢 and𝑣 in𝐼,thereisnoedgebetween𝑢 and𝑣 in𝐺.
Definition:(InexactReduction)
Givenagraph𝐺,we
remove/peelthevertexwith
thehighestdegreefrom𝐺.
Independent
Set
Ø Problem Statement
Givenagraph𝐺 = 𝑉, 𝐸 ,computeanindependentset𝐼of𝐺 whosesizeisthelargest
amongallindependentsetsof𝐺.
Maximum
Independent
Set
Ø Main Observations
• Realnetworksareusuallypower-lawgraphswithmanylow-degreevertices
• Reductionruleshavebeeneffectivelyusedforlow-degreevertices
• High-degreeverticesarelesslikelytobeinamaximumindependentset
Our Approaches
Ø BDOne & BDTwo
Degree-oneReduction
Existing Works
(a)𝛼 𝐺 = 𝛼 𝐺\ 𝑣
𝛼 𝐺 : 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑐𝑒𝑛𝑢𝑚𝑏𝑒𝑟𝑜𝑓𝐺
Ø NP-hard [Garey et al. Book’79] and APX-hard [J. Håstad. FOCS’96]
Degree-twoReductions
Ø Exact Algorithms
• branch-and-reduce paradigm [F.V.Fomin etal.J.ACM’09,T.Akiba etal.Theor.Comput.
(b)Isolation𝛼 𝐺 = 𝛼 𝐺\ 𝑣, 𝑤
(c)Folding𝛼 𝐺 = 𝛼 𝐺/ 𝑢, 𝑣, 𝑤
Sci.’16]
• Theoreticallyrunsin𝑂∗ 1.2201' timeandpracticallycomputestheexactsolutionfor
manysmallandmedium-sizedgraphs,butdoesnothandlelargegraphswell.
Ø Approximation Algorithms
• [U.Feige J.DiscreteMath’04,M.M.Halldórsson etal.Algorithmica’97,P.Berman.
Theor.Comput.Sys.’99]
• Approximationratiolargelydependsonn orΔ. Not practically useful
Ø Heuristic Algorithms
• Linear-timealgorithms
§ Greedy,anddynamicupdate
§ Efficient,butcanonlyfindsmallindependentsetsinpractice.
• Iterative randomized searching algorithms
§ ARW[D.V.Andrade.J.Heuristics’12],ReduMIS [S.Lamm.ALENEX’16], OnlineMIS [J.
Dahlum.SEA’16]
§ Can find large independent sets, but take long time
𝒅𝒆𝒈 𝒗𝟏 = 𝟏
+1
𝒅𝒆𝒈 𝒗𝟑 = 𝟐
Ø LinearTime
Lemma4.1:(Degree-twoPath
Reductions)Consideragraph 𝐺 =
𝑉, 𝐸 withminimumdegreetwo.
Foramaximaldegree-twopath
𝑃 = 𝑣N , 𝑣O , … , 𝑣Q ,let𝑣 ∉ 𝑃 and
𝑤 ∉ 𝑃 betheuniquevertices
connectedto𝑣N and𝑣Q ,
respectively.
Ø NearLinear
Overview of Our Approaches
Ø Computelarge independentsetforlargegraphsinatime-efficient andspace-effective
manner
• Subquadratic (orevenlinear)time
• 2m+O(n)space:m isthenumberofundirectededges.
Lemma5.1:(DominanceReduction)[F.V.Fomin etal.JACM’09] Vertexv dominatesvertexu
if 𝑣, 𝑢 ∈ 𝐸 andallneighbors ofv otherthanu arealsoconnectedtou (i.e.,𝑁 𝑣 \ 𝑢 ⊆
𝑁 𝑢 ).Ifv dominatesu,thenthereexistsamaximumindependentsetofG theexcludesu;
thus,wecanremoveu fromG,and𝛼 𝐺 = 𝛼 𝐺\ 𝑢 .
Lemma5.2:Vertexv dominatesitsneighbor u iff ∆ 𝑣, 𝑢 =
𝑑 𝑣 − 1,where∆ 𝑣, 𝑢 isthenumberoftriangles
containinguandv
Performance Studies
Accuracy
Boost ARW
Processing Time
Memory Usage