In the Matroid Secretary Problem (MSP), the elements of the ground set of a Matroid are revealed on-line one by one, each together with its value. An algorithm for the Matroid Secretary Problem is Matroid-Unknown if, at every stage of its execution: (i) it only knows the elements that have been revealed so far and their values, and (ii) it has access to an oracle for testing whether or not any subset of the elements that have been revealed so far is an independent set. An algorithm is Known-Cardinality if, in addition to (i) and (ii), it also initially knows the cardinality of the ground set of the Matroid.We present here a Known-Cardinality and Order-Oblivious algorithm that, with constant probability, selects an independent set of elements, whose value is at least the optimal value divided by O(log log ρ), where ρ is the rank of the Matroid; that is, the algorithm has a competitive-ratio of O(log log ρ). The best previous results for a Known-Cardinality algorithm are a competitive-ratio of O(log ρ), by , and a competitive-ratio of O( √ log ρ), by Chakraborty and Lachish (2012). In many non-trivial cases the algorithm we present has a competitive-ratio that is better than the O(log log ρ). The cases in which it fails to do so are easily characterized. Understanding these cases may lead to improved algorithms for the problem or, conversely, to non-trivial lower bounds.is unknown until it is revealed. Immediately after each element is revealed, if the element together with the elements already selected does not form an independent set, then that element cannot be selected; however, if it does, then an irrevocable decision must be made whether or not to select the element. That is, if the element is selected, it will stay selected until the end of the process and likewise if it is not. The goal is to design an algorithm for this problem wit ha small competitiveratio, that is the ratio between the maximum sum of values of an independent set and the expected sum of values of the independent set returned by the algorithm.

The covering and boundedness problems for branching vector addition systems are shown complete for doubly-exponential time.

For a property P and a sub-property P , we say that P is Ppartially testable with q queries if there exists an algorithm that distinguishes, with high probability, inputs in P from inputs -far from P , using q queries. Some natural properties require many queries to test, but can be partitioned into a small number of subsets for which they are partially testable with very few queries, sometimes even a number independent of the input size.For properties over 0, 1, the notion of being thus partitionable ties in closely with Merlin-Arthur proofs of Proximity (MAPs) as defined independently in [14]; a partition into r partially-testable properties is the same as a Merlin-Arthur system where the proof consists of the identity of one of the r partially-testable properties, giving a 2-way translation to an O(log r) size proof.Our main result is that for some low complexity properties a partition as above cannot exist, and moreover that for each of our properties there does not exist even a single subproperty featuring both a large size and a query-efficient partial test, in particular improving the lower bound set in [14]. For this we use neither the traditional Yao-type arguments nor the more recent communication complexity method, but open up a new approach for proving lower bounds.First, we use entropy analysis, which allows us to apply our arguments directly to 2-sided tests, thus avoiding the cost of the conversion in [14] from 2-sided to 1-sided tests. Broadly speaking we use "distinguishing instances" of a supposed test to show that a uniformly random choice of a member of the sub-property has "low entropy areas", ultimately leading to it having a low total entropy and hence having a small base set.Additionally, to have our arguments apply to adaptive tests, we use a mechanism of "rearranging" the input bits (through a decision tree that adaptively reads the entire input) to expose the low entropy that would otherwise not be apparent.We also explore the possibility of a connection in the other direction, namely whether the existence of a good partition (or MAP) can lead to a relatively query-efficient standard property test. We provide some preliminary results concerning this question, including a simple lower bound on the possible trade-off.Our second major result is a positive trade-off result for the restricted framework of 1-sided proximity oblivious tests. This is achieved through the construction of a "universal tester" that works the same for all properties admitting the restricted test. Our tester is very related to the notion of sample-based testing (for a non-constant number of queries) as defined by Goldreich and Ron in [13]. In particular it partially resolves an open problem raised by [13].

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We show here that every non-adaptive property testing algorithm making a constant number of queries, over a fixed alphabet, can be converted to a sample-based (as per [Goldreich and Ron, 2015]) testing algorithm whose average number of queries is a fixed, smaller than 1, power of n. Since the query distribution of the sample-based algorithm is not dependent at all on the property, or the original algorithm, this has many implications in scenarios where there are many properties that need to be tested for concurrently, such as testing (relatively large) unions of properties, or converting a Merlin-Arthur Proximity proof (as per [Gur and Rothblum, 2013]) to a proper testing algorithm.The proof method involves preparing the original testing algorithm for a combinatorial analysis, which in turn involves a new result about the existence of combinatorial structures (essentially generalized sunflowers) that allow the sample-based tester to replace the original constant query complexity tester.