SIAM Journal on Computing

We prove that any weighted graph $G=(V,E,w)$ with $n$ points and $m$ edges has a spanning tree $T$ such that $\sum_{\{u,v\}\in E}\frac{d_T(u,v)}{w(u,v)}=O(m\log n\log\log n)$. Moreover, such a tree can be found in time $O(m\log n\log\log n)$. Our result is ...

In the Steiner point removal problem, we are given a weighted graph $G=(V,E)$ and a set of terminals $K\subset V$ of size $k$. The objective is to find a minor $M$ of $G$ with only the terminals as its vertex set, such that distances between the terminals ...

Approximate counting via correlation decay is the core algorithmic technique used in the sharp delineation of the computational phase transition that arises in the approximation of the partition function of antiferromagnetic 2-spin models. Previous analyses ...

Given a notion of pairwise similarity between objects, locality sensitive hashing (LSH) aims to construct a hash function family over the universe of objects such that the probability two objects hash to the same value is their similarity. LSH is a ...

The dispersion problem has been widely studied in computational geometry and facility location and is closely related to the packing problem. The goal is to locate $n$ points (e.g., facilities or persons) in a $k$-dimensional polytope, so that they are far ...

In the classic Minimum Bisection problem we are given as input an undirected graph $G$ and an integer $k$. The task is to determine whether there is a partition of $V(G)$ into two parts $A$ and $B$ such that $||A|-|B|| \leq 1$ and there are at most $k$ ...

This special section comprises eleven fully refereed papers whose extended abstracts were presented at the 57th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2016) in New Brunswick, New Jersey, October 9--11, 2016. The unrefereed ...

The most well-known and ubiquitous clustering problem encountered in nearly every branch of science is undoubtedly $k$-means: given a set of data points and a parameter $k$, select $k$ centers and partition the data points into $k$ clusters around these ...

It is a major open problem whether the $(\min,+)$-product of two $n\times n$ matrices has a truly subcubic (i.e., $O(n^{3-\varepsilon})$ for $\varepsilon>0$) time algorithm; in particular, since it is equivalent to the famous all-pairs-shortest-paths ...

A set $D$ of vertices of a graph $G$ is a dominating set if every vertex of $G$ is contained in $D$ or adjacent to some vertex of $D$. The number of vertices in a smallest dominating set of $G$ is denoted by $\gamma(G)$. We prove that, under the assumption ${...

We consider the problem of finding a low discrepancy coloring for sparse set systems where each element lies in at most $t$ sets. We give an efficient algorithm that finds a coloring with discrepancy $O((t \log n)^{1/2})$, matching the best known ...

Let $G$ be the special linear group ${SL}(2,q)$. We show that if $(a_1,\ldots,a_t)$ and $(b_1,\ldots,b_t)$ are sampled uniformly from large subsets $A$ and $B$ of $G^t$, then their interleaved product $a_1 b_1 a_2 b_2 \cdots a_t b_t$ is nearly uniform over $...

We study the hard-core (gas) model defined on independent sets of an input graph where the independent sets are weighted by a parameter (aka fugacity) $\lambda>0$. For constant $\Delta$, the previous work of Weitz [Proceedings of STOC, 2006, pp. 140--149] ...

We give the first polynomial-time approximation schemes (PTASs) for the following problems: (1) uniform facility location in edge-weighted planar graphs; (2) $k$-median and $k$-means in edge-weighted planar graphs; and (3) $k$-means in Euclidean space of ...

We show that every circuit with ${AND},{OR},{NOT}$, and ${MOD}_m$ gates, $m\in\mathbb{Z}^+$, of polynomial size and depth $d$ can be reduced to a depth-2, ${SYM}\circ{AND}$, circuit of size $2^{(\log n)^{O(d)}}$. This is an exponential size improvement ...

We prove that with high probability over the choice of a random graph $G$ from the Erdös--Rényi distribution $G(n,1/2)$, the $n^{O(d)}$-time degree $d$ sum-of-squares (SOS) semidefinite programming relaxation for the clique problem will give a value of at ...

We consider the hash function $h(x) = ((ax+b)mod p)mod n$ where $a,b$ are chosen uniformly at random from $\{0,1,\ldots,p-1\}$. We prove that when we use $h(x)$ in hashing with chaining to insert $n$ elements into a table of size $n$ the expected length of ...

We study high-dimensional distribution learning in an agnostic setting where an adversary is allowed to arbitrarily corrupt an $\varepsilon$-fraction of the samples. Such questions have a rich history spanning statistics, machine learning, and theoretical ...