${{\bf Ger}}^+_{\infty}$-algebra is a blend of two ubiquitous algebraic structures: $A_{\infty}$-algebra and homotopy Gerstenhaber algebra. The notion of $A_{\infty}$-algebra was invented by J. Stasheff [@Stasheff]. The notion of homotopy Gerstenhaber algebra has its origin in works [@Ger], [@Ezra], [@braces] of M. Gerstenhaber, E. Getzler, and T. Kadeishvili. This latter notion was developed and formalized in works [@BF], [@GerVor], [@GJ], [@KS], [@M-Smith], [@Dima-Proof], [@Sasha]. In this paper we describe interesting relations of the operad ${{\bf Ger}}^+_{\infty}$ to Kontsevich’s formality theorem [@K], to the operad ${{\mathcal{O C}}}_{\infty}$ governing open-closed homotopy algebras [@OCHAStasheff], and to Voronov’s Swiss Cheese operad [@Sasha-SC].

We define the DG operad ${{\bf Ger}}^+_{\infty}$ as the cobar construction of a cooperad ${{\bf S}}$ which is closely related to the cohomology cooperad $H^{{{\bullet}}}({{\rm SC}})$ of Voronov’s Swiss Cheese operad ${{\rm SC}}$. We show that algebras over ${{\bf Ger}}^+_{\infty}$ are pairs $(V,A)$ with the following data:

-   A ${{\bf Ger}}_{\infty}$-structure on $V$,

-   An $A_{\infty}$-structure on $A$, and

-   an $L_{\infty}$-morphism from $V$ to the Hochschild cochain complex ${C^{\bullet}}(A,A)$ of $A$.

Using this description we show that for every $A_{\infty}$-algebra $A$ Tamarkin’s ${{\bf Ger}}_{\infty}$-structure on the Hochschild cochain complex ${C^{\bullet}}(A,A)$ extends naturally to a ${{\bf Ger}}^+_{\infty}$-structure on the pair $({C^{\bullet}}(A,A), A)$.

We 