计算复杂性
We introduce a `concrete complexity' model for studying algorithms for matching in bipartite graphs. The model is based on the "demand query" model used for combinatorial auctions. Most (but not all) known algorithms for bipartite matching…
We introduce a new decision problem, called Packed Interval Covering (PIC) and show that it is NP-complete.
Given a graph $G$ and a parameter $k$, the $k$-biclique problem asks whether $G$ contains a complete bipartite subgraph $K_{k,k}$. This is the most easily stated problem on graphs whose parameterized complexity is still unknown. We provide…
Robust mean estimation is the problem of estimating the mean $\mu \in \mathbb{R}^d$ of a $d$-dimensional distribution $D$ from a list of independent samples, an $\epsilon$-fraction of which have been arbitrarily corrupted by a malicious…
The $\epsilon$-approximate degree $deg_\epsilon(f)$ of a Boolean function $f$ is the least degree of a real-valued polynomial that approximates $f$ pointwise to error $\epsilon$. The approximate degree of $f$ is at least $k$ iff there…
We believe that economic design and computational complexity---while already important to each other---should become even more important to each other with each passing year. But for that to happen, experts in on the one hand such areas as…
We identify a new notion of pseudorandomness for randomness sources, which we call the average bias. Given a distribution $Z$ over $\{0,1\}^n$, its average bias is: $b_{\text{av}}(Z) =2^{-n} \sum_{c \in \{0,1\}^n} |\mathbb{E}_{z \sim…
We show that any DNNF circuit that expresses the set of linear orders over a set of $n$ candidates must be of size $2^{\Omega(n)}$. Moreover, we show that there exist DNNF circuits of size $2^{O(n)}$ expressing linear orders over $n$…
For any unsatisfiable CNF formula we give an exponential lower bound on the size of resolution refutations of a propositional statement that the formula has a resolution refutation. We describe three applications. (1) An open question in…
Counting problems in general and counting graph homomorphisms in particular have numerous applications in combinatorics, computer science, statistical physics, and elsewhere. One of the most well studied problems in this area is…
Finding paths in graphs is a fundamental graph-theoretic task. In this work, we we are concerned with finding a path with some constraints on its length and the number of vertices neighboring the path, that is, being outside of and incident…
This work examines the expected computational cost to determine an approximate global minimum of a class of cost functions characterized by the variance of coefficients. The cost function takes $N$-dimensional binary states as arguments and…
Even though every mathematician knows intuitively what it means to "simplify" a mathematical expression, there is still no universally accepted rigorous mathematical definition of "simplify". In this paper, we shall give a simple and…
Various applications of graphs, in particular applications related to finding shortest paths, naturally get inputs with real weights on the edges. However, for algorithmic or visualization reasons, inputs with integer weights would often be…
Despite persistent efforts, there is no known technique for obtaining unconditional super-linear lower bounds for the computational complexity of the problems in P. Vassilevska Williams and Williams introduce a fruitful approach to advance…
There has been significant recent progress on algorithms for approximating graph spanners, i.e., algorithms which approximate the best spanner for a given input graph. Essentially all of these algorithms use the same basic LP relaxation, so…
The $k$-ExactCover problem is a parameterized version of the ExactCover problem, in which we are given a universe $U$, a collection $S$ of subsets of $U$, and an integer $k$, and the task is to determine whether $U$ can be partitioned into…
Suppose a language $L$ can be decided by a bounded-error randomized algorithm that runs in space $S$ and time $n \cdot \text{poly}(S)$. We give a randomized algorithm for $L$ that still runs in space $O(S)$ and time $n \cdot \text{poly}(S)$…
We report a new limitation on the ability of physical systems to perform computation -- one that is based on generalizing the notion of memory, or storage space, available to the system to perform the computation. Roughly, we define memory…
We develop a notion of {\em inner rank} as a tool for obtaining lower bounds on the rank of matrix multiplication tensors. We use it to give a short proof that the border rank (and therefore rank) of the tensor associated with $n\times n$…