Given a graph G=(V,E)\documentclass[12pt]{minimal}
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\begin{document}$$G=(V,E)$$\end{document} with a label set L={ℓ1,ℓ2,…,ℓq}\documentclass[12pt]{minimal}
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\begin{document}$$L = \{\ell _1, \ell _2, \ldots , \ell _q \}$$\end{document}, in which each edge has a label from L, and a source s∈V\documentclass[12pt]{minimal}
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\begin{document}$$s \in V$$\end{document} together with a sink t∈V\documentclass[12pt]{minimal}
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\begin{document}$$t \in V$$\end{document}, the Minimum Labels-tCut problem asks to pick a set L′⊆L\documentclass[12pt]{minimal}
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\begin{document}$$L' \subseteq L$$\end{document} of labels with minimized cardinality, such that the removal of all edges with labels in L′\documentclass[12pt]{minimal}
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\begin{document}$$L'$$\end{document} from G disconnects s and t. Let n=|V|\documentclass[12pt]{minimal}
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\begin{document}$$n = |V|$$\end{document} and m=|E|\documentclass[12pt]{minimal}
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\begin{document}$$m = |E|$$\end{document}. The previous best known approximation ratio for this problem in literature is O(m1/2)\documentclass[12pt]{minimal}
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\begin{document}$$O(m^{1/2})$$\end{document}. We present two simple and purely combinatorial approximation algorithms for the problem with ratios O(n2/3/OPT1/3)\documentclass[12pt]{minimal}
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\begin{document}$$O(n^{2/3}/\text {OPT}^{1/3})$$\end{document} and O(m1/2/OPT1/2)\documentclass[12pt]{minimal}
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\begin{document}$$O(m^{1/2} / \text {OPT}^{1/2})$$\end{document}, where OPT\documentclass[12pt]{minimal}
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\begin{document}$$\text {OPT}$$\end{document} is the optimal value of the input instance. The former result gives the first approximation ratio which is sublinear in n for the problem, and in particular applies to the instances with dense graphs (e.g., m=Θ(n2)\documentclass[12pt]{minimal}
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\begin{document}$$m = \varTheta (n^2)$$\end{document}). The latter result improves the previous ratio O(m1/2)\documentclass[12pt]{minimal}
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\begin{document}$$O(m^{1/2})$$\end{document}, as we can always assume that OPT\documentclass[12pt]{minimal}
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\begin{document}$$\text {OPT}$$\end{document} is a super-constant.
