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In this paper, we consider a new problem of portfolio optimization using stochastic information. In a setting where there is some uncertainty, we ask how to best select $k$ potential solutions, with the goal of optimizing the value of the best solution. More formally, given a combinatorial problem $\Pi$, a set of value functions $V$ over the solutions of $\Pi$, and a distribution $D$ over $V$, our goal is to select $k$ solutions of $\Pi$ that maximize or minimize the expected value of the {\em best} of those solutions. For a simple example, consider the classic knapsack problem: given a universe of elements each with unit weight and a positive value, the task is to select $r$ elements maximizing the total value. Now suppose that each element's weight comes from a (known) distribution. How should we select $k$ different solutions so that one of them is likely to yield a high value? In this work, we tackle this basic problem, and generalize it to the setting where the underlying set system forms a matroid. On the technical side, it is clear that the candidate solutions we select must be diverse and anti-correlated; however, it is not clear how to do so efficiently. Our main result is a polynomial-time algorithm that constructs a portfolio within a constant factor of the optimal., Comment: Accepted at ITCS 2025

Graph connectivity is a fundamental combinatorial optimization problem that arises in many practical applications, where usually a spanning subgraph of a network is used for its operation. However, in the real world, links may fail unexpectedly deeming the networks non-operational, while checking whether a link is damaged is costly and possibly erroneous. After an event that has damaged an arbitrary subset of the edges, the network operator must find a spanning tree of the network using non-damaged edges by making as few checks as possible. Motivated by such questions, we study the problem of finding a spanning tree in a network, when we only have access to noisy queries of the form "Does edge e exist?". We design efficient algorithms, even when edges fail adversarially, for all possible error regimes; 2-sided error (where any answer might be erroneous), false positives (where "no" answers are always correct) and false negatives (where "yes" answers are always correct). In the first two regimes we provide efficient algorithms and give matching lower bounds for general graphs. In the False Negative case we design efficient algorithms for large interesting families of graphs (e.g. bounded treewidth, sparse). Using the previous results, we provide tight algorithms for the practically useful family of planar graphs in all error regimes., Comment: 22 pages, 3 figures