12 research outputs found

    Tight Bounds for Gomory-Hu-like Cut Counting

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    By a classical result of Gomory and Hu (1961), in every edge-weighted graph G=(V,E,w)G=(V,E,w), the minimum stst-cut values, when ranging over all s,tVs,t\in V, take at most V1|V|-1 distinct values. That is, these (V2)\binom{|V|}{2} instances exhibit redundancy factor Ω(V)\Omega(|V|). They further showed how to construct from GG a tree (V,E,w)(V,E',w') that stores all minimum stst-cut values. Motivated by this result, we obtain tight bounds for the redundancy factor of several generalizations of the minimum stst-cut problem. 1. Group-Cut: Consider the minimum (A,B)(A,B)-cut, ranging over all subsets A,BVA,B\subseteq V of given sizes A=α|A|=\alpha and B=β|B|=\beta. The redundancy factor is Ωα,β(V)\Omega_{\alpha,\beta}(|V|). 2. Multiway-Cut: Consider the minimum cut separating every two vertices of SVS\subseteq V, ranging over all subsets of a given size S=k|S|=k. The redundancy factor is Ωk(V)\Omega_{k}(|V|). 3. Multicut: Consider the minimum cut separating every demand-pair in DV×VD\subseteq V\times V, ranging over collections of D=k|D|=k demand pairs. The redundancy factor is Ωk(Vk)\Omega_{k}(|V|^k). This result is a bit surprising, as the redundancy factor is much larger than in the first two problems. A natural application of these bounds is to construct small data structures that stores all relevant cut values, like the Gomory-Hu tree. We initiate this direction by giving some upper and lower bounds.Comment: This version contains additional references to previous work (which have some overlap with our results), see Bibliographic Update 1.

    Lower Bounds for Multiplication via Network Coding

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    Multiplication is one of the most fundamental computational problems, yet its true complexity remains elusive. The best known upper bound, very recently proved by Harvey and van der Hoeven (2019), shows that two n-bit numbers can be multiplied via a boolean circuit of size O(n lg n). In this work, we prove that if a central conjecture in the area of network coding is true, then any constant degree boolean circuit for multiplication must have size Omega(n lg n), thus almost completely settling the complexity of multiplication circuits. We additionally revisit classic conjectures in circuit complexity, due to Valiant, and show that the network coding conjecture also implies one of Valiant\u27s conjectures