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We construct an improved integrality gap instance for the Călinescu-Karloff-Rabani LP relaxation of the Multiway Cut problem. For \(k \geqslant 3\) terminals, our instance has an integrality ratio of \(6 / (5 + \frac{1}{k - 1}) - \varepsilon \), for every constant \(\varepsilon > 0\). For every \(k \geqslant 4\), this improves upon a long-standing lower bound of \(8 / (7 + \frac{1}{k - 1})\) by Freund and Karloff [7]. Due to the result by Manokaran et al. [9], our integrality gap also implies Unique Games hardness of approximating Multiway Cut of the same ratio.

An Improved Integrality Gap for the Călinescu-Karloff-Rabani Relaxation for Multiway Cut