BE Certificate -2

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Agrawal and Vinay [FOCS 2008] showed how any polynomial size arithmetic circuit can be thought of as a depth four arithmetic circuit of subexponential size. The resulting circuit size in this simulation was more carefully analyzed by Koiran [TCS 2012] and subsequently by Tavenas [MFCS 2013]. We provide a simple proof of this chain of results. We then abstract the main ingredient to apply it to formulas and constant depth circuits, and show more structured depth reductions for them.In an apriori surprising result, Raz [STOC 2010] showed that for any $n$ and $d$, such that $\omega(1) \leq d \leq O(logn/loglogn)$, constructing explicit tensors $T: [n] \rightarrow F$ of high enough rank would imply superpolynomial lower bounds for arithmetic formulas over the field F. Using the additional structure we obtain from our proof of the depth reduction for arithmetic formulas, we give a new and arguably simpler proof of this connection. We also extend this result for homogeneous formulas to show that, in fact, the connection holds for any d such that $\omega(1) \leq d \leq n^{o(1)}$. Joint work with Mrinal Kumar, Ramprasad Saptharishi and V Vinay.

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We present a Logspace Approximation Scheme (LSAS), i.e. an approximation algorithm for maximum matching in planar graphs (not necessarily bipartite) that achieves an approximation ratio arbitrarily close to one, in Logspace This deviates from the well known Baker’s approach for approximation in planar graphs by avoiding the use of distance computation - which is not known to be in Logspace. Our algorithm actually works for any “recursively sparse” graph class which contains a linear size matching.The scheme is based on an LSAS in bounded degree graphs which are not known to be amenable to Baker’s method. We solve the bounded degree case by parallel augmentation of short augmenting paths. Finding a large number of such disjoint paths can, in turn, be reduced to finding a large independent set in a bounded degree graph. This is joint work with Samir Datta and Raghav Kulkarni.

Space-efficient Approximation Scheme for Maximum Matching in Sparse Graphs

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