Algorithms solving the Matching Cut problem

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• 作者：Dieter Kratsch^a ; ^{kratsch@univ-metz.fr" class="auth_mail" title="E-mail the corresponding author} ; Van Bang Le^b ; ^{van-bang.le@uni-rostock.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Matching cut ; Graph algorithm
• 刊名：Theoretical Computer Science
• 出版年：2016
• 出版时间：4 January 2016
• 年：2016
• 卷：609
• 期：part_P2
• 页码：328-335
• 全文大小：336 K

文摘

In a graph, a matching cut is an edge cut that is a matching. class="smallcaps">Matching Cut is the problem of deciding whether or not a given graph has a matching cut, which is known to be NP-complete. This paper provides a first branching algorithm solving class="smallcaps">Matching Cut in time class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304397515008993&_mathId=si1.gif&_user=111111111&_pii=S0304397515008993&_rdoc=1&_issn=03043975&md5=e89aa36093d1fbea42ca3020aeee3d8d" title="Click to view the MathML source">O^⁎(2^n/2)=O^⁎(1.4143ⁿ)class="mathContainer hidden">class="mathCode">$O^{⁎}(2^{n/2})=O^{⁎}({1.4143}^n)$ for an n-vertex input graph, and shows that class="smallcaps">Matching Cut parameterized by the vertex cover number class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304397515008993&_mathId=si2.gif&_user=111111111&_pii=S0304397515008993&_rdoc=1&_issn=03043975&md5=1b9ac3ea7b27b403b16c39ccdfb2aa60" title="Click to view the MathML source">τ(G)class="mathContainer hidden">class="mathCode">$\tau (G)$ can be solved by a single-exponential algorithm in time class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0304397515008993&_mathId=si225.gif&_user=111111111&_pii=S0304397515008993&_rdoc=1&_issn=03043975&md5=b18ea95918222d4e8e2a9cc890345c9b" title="Click to view the MathML source">2^τ(G)O(n²)class="mathContainer hidden">class="mathCode">$2^{\tau (G)}O(n^2)$. Moreover, the paper also gives a polynomially solvable case for class="smallcaps">Matching Cut which covers previous known results on graphs of maximum degree three, line graphs, and claw-free graphs.