Open algorithmic problems from a talk by Alon
The UCI mathematics department had a departmental colloquium today given by Noga Alon, titled “The Polynomial Method and its Algorithmic Aspects”. One part of his talk that I found very interesting was a collection of easy-to-state combinatorial problems where the existence of a solution can be proved, but there is no known efficient (polynomial time) algorithm for finding the solution:
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Let
be a prime number, and suppose that we are given two inputs: a sequence of elements of , of length , not necessarily distinct from each other, and a set , also consisting of elements of , distinct but not ordered. Can we pair them up by assigning an ordering to the elements of so that all of the sums are distinct? For instance, if all of the ’s are equal, then any ordering of the ’s will work. -
Let
be a bipartite graph that has a perfect matching. Assign “non-degrees” to the vertices on one side of the bipartition, numbers that should not be the degree of the vertex. Can we delete some subset of the vertices on the other side, so that in the remaining graph each vertex’s degree is different from its non-degree? For instance, if the non-degrees are all zero, then itself will work (it has nonzero degree because it has a perfect matching). If the non-degrees are all non-zero, then we can delete all of the vertices on the other side of the bipartition. -
Let
be a planar graph with degree three at every vertex, and suppose also that each edge of has a list of three colors available to it. Can we assign colors from these lists to the edges so that each vertex touches edges of three different colors? -
Let
and be proper colorings of a large -dimensional grid, using colors, and let and be subsets of the vertices of the grid that are far apart from each other (I am not sure how far this needs to be but it seems to be or maybe ). Can we find a proper coloring of the whole grid that blends between the two colorings, agreeing with each on ? This one comes from a new paper by Alon, Raimundo Briceño, Nishant Chandgotia, Alexander Magazinov, and Yinon Spinka, “Mixing properties of colourings of the lattice”, Combinatorics, Probability and Computing 2021. Fewer colors will not always work.
In all of these cases, the answer to the existence problem is yes, but we don’t know of an efficient algorithm for finding the structure that is supposed to exist. Instead, Alon proves the existence non-constructively, through a combinatorial Nullstellensatz encoding the principle that low-degree polynomials don’t have many roots. For one-variable polynomials of degree
The analogous principle Alon uses for higher numbers of variables is the following: Let
The hard parts about using this method to prove existence in combinatorial problems such as the ones above are finding the right polynomial and proving that it has a nonzero coefficient at the right monomial. The problem on reordering with distinct pairwise sums, for instance, comes from Alon’s paper “Additive Latin transversals”, Israel J. Math. 2000 and uses the polynomial
where each
The general problem of constructing a nonzero solution to an instance of the combinatorial Nullstellensatz whose polynomial is described by an arithmetic circuit is (according to Alon) as hard as inverting arbitrary one-way permutations, a standard cryptographic primitive. This suggests that a polynomial-time construction algorithm does not exist; if it did exist, it would break a lot of modern cryptography. On the other hand, proving that it does not exist would also prove