We propose to perform a combined analysis of B → ππ and Bs → K + K − modes, in the framework of a global CKM fit. The method optimizes the constraining power of these decays and allows to derive constraints on NP contributions to penguin amplitudes or on the Bs mixing phase. We illustrate these capabilities with a simplified analysis using the recent measurements by the LHCb Collaboration, neglecting correlations with other SM observables.CP violation in B d,s decays plays a fundamental role in testing the consistency of the Cabibbo-Kobayashi-Maskawa (CKM) paradigm in the Standard Model (SM) and in probing virtual effects of heavy new particles. With the advent of the B-factories, the Gronau-London (GL) isospin analysis of B d → ππ decays [1] has been a precious source of information on the phase of the CKM matrix. Although the method allows a full determination of the weak phase and of the relevant hadronic parameters, it suffers from discrete ambiguities that limit its constraining power. It is however possible to reduce the impact of discrete ambiguities by adding information on hadronic parameters [2,3]. In particular, as noted in refs. [4][5][6], the hadronic parameters entering the B d → π + π − and the B s → K + K − decays are connected by U-spin, so that the experimental knowledge of B s → K + K − can definitely improve the extraction of the CKM phase with the GL analysis. Indeed, in ref.[3], the measurement of BR(B s → K + K − ) was used to obtain an upper bound on one of the hadronic parameters.After the pioneering studies performed at the TeVatron, very recently LHCb opened up the road to CP violation in B s → KK decays [7]. The present experimental information is summarized in Table I. At present, one has all the necessary information to use the U-spin strategy proposed by Fleischer (F) in refs. [4,5] to extract the CKM phase from a combined analysis of B d → π + π − and the B s → K + K − decays. However, as we will show explicitly below, this strategy alone suffers from a sizable dependence on the breaking of U-spin symmetry [8].Furthermore, in the B s system the measurement of any time-dependent CP asymmetry cannot be directly translated into a measurement of the angle β s = arg − V * tb Vts V * cb Vcs , even in the case of the so-called "gold-plated" b → ccs decays. This is due to the fact that the angle β s is small and correlated to the subdominant amplitude in b → ccs decays. Thus, measuring β s requires the determination of the subdominant decay amplitude. This is evident by noting that using CKM unitarity the b → s decay amplitude can be written asNaively dropping the doubly Cabibbo-suppressed term proportional to V ub V * us would lead to the conclusion that the CP asymmetry measures 2β s with the first choice or that the CP asymmetry should vanish in the second choice. Clearly, a full treatment of the decay amplitude, taking into account correlations between the various CKM terms, is necessary to give a meaningful interpretation to the CP asymmetry. This is at variance with the B d case, whe...