Steiner Point Removal with Distortion O(log⁡k)O(\log k)

In the Steiner point removal (SPR) 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 the distance between the terminals will be preserved up to a small multiplicative distortion. Kamma, Krauthgamer and Nguyen [KKN15] used a ball-growing algorithm with exponential distributions to show that the distortion is at most $O(\log^5 k)$. Cheung [Che17] improved the analysis of the same algorithm, bounding the distortion by $O(\log^2 k)$. We improve the analysis of this ball-growing algorithm even further, bounding the distortion by $O(\log k)$

Mimicking Networks and Succinct Representations of Terminal Cuts

Given a large edge-weighted network $G$ with $k$ terminal vertices, we wish to compress it and store, using little memory, the value of the minimum cut (or equivalently, maximum flow) between every bipartition of terminals. One appealing methodology to implement a compression of $G$ is to construct a \emph{mimicking network}: a small network $G'$ with the same $k$ terminals, in which the minimum cut value between every bipartition of terminals is the same as in $G$. This notion was introduced by Hagerup, Katajainen, Nishimura, and Ragde [JCSS '98], who proved that such $G'$ of size at most $2^{2^k}$ always exists. Obviously, by having access to the smaller network $G'$, certain computations involving cuts can be carried out much more efficiently. We provide several new bounds, which together narrow the previously known gap from doubly-exponential to only singly-exponential, both for planar and for general graphs. Our first and main result is that every $k$-terminal planar network admits a mimicking network $G'$ of size $O(k^2 2^{2k})$, which is moreover a minor of $G$. On the other hand, some planar networks $G$ require $|E(G')| \ge \Omega(k^2)$. For general networks, we show that certain bipartite graphs only admit mimicking networks of size $|V(G')| \geq 2^{\Omega(k)}$, and moreover, every data structure that stores the minimum cut value between all bipartitions of the terminals must use $2^{\Omega(k)}$ machine words

On Communication Complexity of Fixed Point Computation

Brouwer's fixed point theorem states that any continuous function from a compact convex space to itself has a fixed point. Roughgarden and Weinstein (FOCS 2016) initiated the study of fixed point computation in the two-player communication model, where each player gets a function from $[0,1]^n$ to $[0,1]^n$, and their goal is to find an approximate fixed point of the composition of the two functions. They left it as an open question to show a lower bound of $2^{\Omega(n)}$ for the (randomized) communication complexity of this problem, in the range of parameters which make it a total search problem. We answer this question affirmatively. Additionally, we introduce two natural fixed point problems in the two-player communication model. $\bullet$ Each player is given a function from $[0,1]^n$ to $[0,1]^{n/2}$, and their goal is to find an approximate fixed point of the concatenation of the functions. $\bullet$ Each player is given a function from $[0,1]^n$ to $[0,1]^{n}$, and their goal is to find an approximate fixed point of the interpolation of the functions. We show a randomized communication complexity lower bound of $2^{\Omega(n)}$ for these problems (for some constant approximation factor). Finally, we initiate the study of finding a panchromatic simplex in a Sperner-coloring of a triangulation (guaranteed by Sperner's lemma) in the two-player communication model: A triangulation $T$ of the $d$-simplex is publicly known and one player is given a set $S_A\subset T$ and a coloring function from $S_A$ to $\{0,\ldots ,d/2\}$, and the other player is given a set $S_B\subset T$ and a coloring function from $S_B$ to $\{d/2+1,\ldots ,d\}$, such that $S_A\dot\cup S_B=T$, and their goal is to find a panchromatic simplex. We show a randomized communication complexity lower bound of $|T|^{\Omega(1)}$ for the aforementioned problem as well (when $d$ is large)

Metric Extension Operators, Vertex Sparsifiers and Lipschitz Extendability

We study vertex cut and flow sparsifiers that were recently introduced by Moitra (2009), and Leighton and Moitra (2010). We improve and generalize their results. We give a new polynomialtime algorithm for constructing O(log k / log log k) cut and flow sparsifiers, matching the best existential upper bound on the quality of a sparsifier, and improving the previous algorithmic upper bound of O(log 2 k / log log k). We show that flow sparsifiers can be obtained from linear operators approximating minimum metric extensions. We introduce the notion of (linear) metric extension operators, prove that they exist, and give an exact polynomial-time algorithm for finding optimal operators. We then establish a direct connection between flow and cut sparsifiers and Lipschitz extendability of maps in Banach spaces, a notion studied in functional analysis since 1950s. Using this connection, we prove a lower bound of Ω ( √ log k / log log k) for flow sparsifiers and a lower bound of Ω ( 4 √ log k / log log k) for cut sparsifiers. We show that if a certain open question posed by Ball in 1992 has a positive answer, then there exist Õ( √ log k) cut sparsifiers. On the other hand, any lower bound on cut sparsifiers better than ˜ Ω ( √ log k) would imply a negative answer to this question.