Prophet inequalities for rewards maximization are fundamental results from optimal stopping theory with several applications to mechanism design and online optimization. We study the cost minimization counterpart of the classical prophet inequality due to Krengel, Sucheston, and Garling [KS77], where one is facing a sequence of costs X 1 , X 2 , . . . , X n in an online manner and must "stop" at some point and take the last cost seen. 1 Given that the X i 's are independent random variables drawn from known distributions, the goal is to devise a stopping strategy S (online algorithm) that minimizes the expected cost. The best cost possible is E [min i X i ] (offline optimum), achievable only by a prophet who can see the realizations of all X i 's upfront. We say that strategy S is an α-approximation to the prophet (α ≥ 1) ifWe first observe that if the X i 's are not identically distributed, then no strategy can achieve a bounded approximation, no matter if the arrival order is adversarial or random, even when restricted to n = 2 and distributions with support size at most two. 2 This leads us to consider the case where the X i 's are independent and identically distributed (I.I.D.). For the I.I.D. case, we give a complete characterization of the optimal stopping strategy. We show that it achieves a (distribution-dependent) constant-factor approximation to the prophet's cost for almost all distributions and that this constant is tight. In particular, for distributions for which the integral of the hazard rate 3 is a polynomial H(x) = k i=1 a i x di , where d 1 < • • • < d k , the approximation factor is λ(d 1 ), a decreasing function of d 1 , and is the best possible for H(x) = x d1 . Furthermore, when the hazard rate is monotonically increasing (i.e. the distribution is MHR), we show that this constant is at most 2, and this again is the best possible for the MHR distributions.For the classical prophet inequality for reward maximization, single-threshold strategies have been powerful enough to achieve the best possible approximation factor. Motivated by this, we analyze single-threshold strategies for the cost prophet inequality problem. We design a threshold that achieves a O (polylog n)-factor approximation, where the exponent in the logarithmic factor is a distribution-dependent constant, and we show a matching lower bound.We believe that our results may be of independent interest for analyzing approximately optimal (posted price-style) mechanisms for procuring items.

In recent years, Contention Resolution Schemes (CRSs), introduced by Chekuri, Vondrák, and Zenklusen, have emerged as a general framework for obtaining feasible solutions to combinatorial optimization problems with constraints. The idea is to first solve a continuous relaxation and then round the fractional solution. When one does not have any control on the order of rounding, Online Contention Resolution Schemes (OCRSs) can be used instead, and have been successfully applied in settings such as prophet inequalities and stochastic probing. Intuitively, a greedy OCRS has to decide which elements to include in the integral solution before the online process starts.In this work, we give simple 1/e -selectable greedy OCRSs for rank-1 matroids, partition matroids and transversal matroids. We also show that our greedy OCRSs are optimal, even for the simple single-item case.

We study the problem of finding fair and efficient allocations of a set of indivisible items to a set of agents, where each item may be a good (positively valued) for some agents and a bad (negatively valued) for others, i.e., a mixed manna. As fairness notions, we consider arguably the strongest possible relaxations of envy-freeness and proportionality, namely envy-free up to any item (EFX and EFX 0 ), and proportional up to the maximin good or any bad (PropMX and PropMX 0 ). Our efficiency notion is Pareto-optimality (PO).We study two types of instances: (i) Separable, where the item set can be partitioned into goods and bads, and (ii) Restricted mixed goods (RMG), where for each item j, every agent has either a non-positive value for j, or values j at the same v j > 0. We obtain polynomial-time algorithms for the following:• Separable instances: PropMX 0 allocation.• RMG instances: Let pure bads be the set of items that everyone values negatively.-PropMX allocation for general pure bads.-EFX+PropMX allocation for identically-ordered pure bads.-EFX+PropMX+PO allocation for identical pure bads.Finally, if the RMG instances are further restricted to binary mixed goods where all the v j 's are the same, we strengthen the results to guarantee EFX 0 and PropMX 0 respectively.