Optimal Bounds for the k -cut Problem

In the k -cut problem, we want to find the lowest-weight set of edges whose deletion breaks a given (multi)graph into k connected components. Algorithms of Karger and Stein can solve this in roughly O ( n 2k ) time. However, lower bounds from conjectures about the k -clique problem imply that Ω ( n (1- o (1)) k ) time is likely needed. Recent results of Gupta, Lee, and Li have given new algorithms for general k -cut in n 1.98k + O(1) time, as well as specialized algorithms with better performance for certain classes of graphs (e.g., for small integer edge weights). In this work, we resolve the problem for general graphs. We show that the Contraction Algorithm of Karger outputs any fixed k -cut of weight α λ k with probability Ω k ( n - α k ), where λ k denotes the minimum k -cut weight. This also gives an extremal bound of O k ( n k ) on the number of minimum k -cuts and an algorithm to compute λ k with roughly n k polylog( n ) runtime. Both are tight up to lower-order factors, with the algorithmic lower bound assuming hardness of max-weight k -clique. The first main ingredient in our result is an extremal bound on the number of cuts of weight less than 2 λ k / k , using the Sunflower lemma. The second ingredient is a fine-grained analysis of how the graph shrinks—and how the average degree evolves—in the Karger process.

Sunflower Theorems in Monotone Circuit Complexity

Alexander Razborov (1985) developed the approximation method to obtain lower bounds on the size of monotone circuits deciding if a graph contains a clique. Given a "small" circuit, this technique consists in finding a monotone Boolean function which approximates the circuit in a distribution of interest, but makes computation errors in that same distribution. To prove that such a function is indeed a good approximation, Razborov used the sunflower lemma of Erd\H{o}s and Rado (1960). This technique was improved by Alon and Boppana (1987) to show lower bounds for a larger class of monotone computational problems. In that same work, the authors also improved the result of Razborov for the clique problem, using a relaxed variant of sunflowers. More recently, Rossman (2010) developed another variant of sunflowers, now called "robust sunflowers", to obtain lower bounds for the clique problem in random graphs. In the following years, the concept of robust sunflowers found applications in many areas of computational complexity, such as DNF sparsification, randomness extractors and lifting theorems. Even more recent was the breakthrough result of Alweiss, Lovett, Wu and Zhang (2020), which improved Rossman's bound on the size of hypergraphs without robust sunflowers. This result was employed to obtain a significant progress on the sunflower conjecture. In this work, we will show how the recent progress in sunflower theorems can be applied to improve monotone circuit lower bounds. In particular, we will show the best monotone circuit lower bound obtained up to now, breaking a 20-year old record of Harnik and Raz (2000). We will also improve the lower bound of Alon and Boppana for the clique function in a slightly more restricted range of clique sizes. Our exposition is self-contained. These results were obtained in a collaboration with Benjamin Rossman and Mrinal Kumar.

Sunflowers in Lattices

A Sunflower is a subset $S$ of a lattice, with the property that the meet of any two elements in $S$ coincides with the meet of all of $S$. The Sunflower Lemma of Erdös and Rado asserts that a set of size at least $1 + k!(t-1)^k$ of elements of rank $k$ in a Boolean Lattice contains a sunflower of size $t$. We develop counterparts of the Sunflower Lemma for distributive lattices, graphic matroids, and matroids representable over a fixed finite field. We also show that there is no counterpart for arbitrary matroids.