We present randomized algorithms that solve Subset Sum and Knapsack instances with n items in O * (2 0.86n ) time, where the O * (·) notation suppresses factors polynomial in the input size, and polynomial space, assuming random read-only access to exponentially many random bits. These results can be extended to solve Binary Linear Programming on n variables with few constraints in a similar running time. We also show that for any constant k ≥ 2, random instances of k-Sum can be solved using O(n k−0.5 polylog(n)) time and O(log n) space, without the assumption of random access to random bits.Underlying these results is an algorithm that determines whether two given lists of length n with integers bounded by a polynomial in n share a common value. Assuming random read-only access to random bits, we show that this problem can be solved using O(log n) space significantly faster than the trivial O(n 2 ) time algorithm if no value occurs too often in the same list.
We consider distribution-based objectives for Markov Decision Processes (MDP). This class of objectives gives rise to an interesting trade-off between full and partial information. As in full observation, the strategy in the MDP can depend on the state of the system, but similar to partial information, the strategy needs to account for all the states at the same time.In this paper, we focus on two safety problems that arise naturally in this context, namely, existential and universal safety. Given an MDP A and a closed and convex polytope H of probability distributions over the states of A, the existential safety problem asks whether there exists some distribution ∆ in H and a strategy of A, such that starting from ∆ and repeatedly applying this strategy keeps the distribution forever in H . The universal safety problem asks whether for all distributions in H , there exists such a strategy of A which keeps the distribution forever in H . We prove that both problems are decidable, with tight complexity bounds: we show that existential safety is PTIME-complete, while universal safety is co-NP-complete.Further, we compare these results with existential and universal safety problems for Rabin's probabilistic finite-state automata (PFA), the subclass of Partially Observable MDPs which have zero observation. Compared to MDPs, strategies of PFAs are not statedependent. In sharp contrast to the PTIME result, we show that existential safety for PFAs is undecidable, with H having closed and open boundaries. On the other hand, it turns out that the universal safety for PFAs is decidable in EXPTIME, with a co-NP lower bound. Finally, we show that an alternate representation of the input polytope allows us to improve the complexity of universal safety for MDPs and PFAs.
We study the problem of almost-everywhere reliable message transmission ; a key component in designing efficient and secure Multi-party Computation (MPC) protocols for sparsely connected networks. The goal is to design low-degree networks which allow a large fraction of honest nodes to communicate reliably even when a small constant fraction of nodes experience byzantine corruption and deviate arbitrarily from the assigned protocol. In this paper, we achieve a -degree network with a polylogarithmic work complexity protocol, thereby improving over the state-of-the-art result of Chandran et al. (ICALP 2010) who required a polylogarithmic-degree network and had a linear work complexity. In addition, we also achieve: A work efficient version of Dwork et al.’s (STOC 1986) butterfly network. An improvement upon the state of the art protocol of Ben-or and Ron (Information Processing Letters 1996) in the randomized corruption model—both in work-efficiency and in resilience.
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