Tight local approximation results for max-min linear programs

In a bipartite max-min LP, we are given a bipartite graph \myG = (V \cup I \cup K, E), where each agent $v \in V$ is adjacent to exactly one constraint $i \in I$ and exactly one objective $k \in K$. Each agent $v$ controls a variable $x_v$. For each $i \in I$ we have a nonnegative linear constraint on the variables of adjacent agents. For each $k \in K$ we have a nonnegative linear objective function of the variables of adjacent agents. The task is to maximise the minimum of the objective functions. We study local algorithms where each agent $v$ must choose $x_v$ based on input within its constant-radius neighbourhood in \myG. We show that for every $\epsilon>0$ there exists a local algorithm achieving the approximation ratio ${\Delta_I (1 - 1/\Delta_K)} + \epsilon$. We also show that this result is the best possible -- no local algorithm can achieve the approximation ratio ${\Delta_I (1 - 1/\Delta_K)}$. Here $\Delta_I$ is the maximum degree of a vertex $i \in I$, and $\Delta_K$ is the maximum degree of a vertex $k \in K$. As a methodological contribution, we introduce the technique of graph unfolding for the design of local approximation algorithms.Comment: 16 page

Asymmetric Hashing for Fast Ranking via Neural Network Measures

Fast item ranking is an important task in recommender systems. In previous works, graph-based Approximate Nearest Neighbor (ANN) approaches have demonstrated good performance on item ranking tasks with generic searching/matching measures (including complex measures such as neural network measures). However, since these ANN approaches must go through the neural measures several times during ranking, the computation is not practical if the neural measure is a large network. On the other hand, fast item ranking using existing hashing-based approaches, such as Locality Sensitive Hashing (LSH), only works with a limited set of measures. Previous learning-to-hash approaches are also not suitable to solve the fast item ranking problem since they can take a significant amount of time and computation to train the hash functions. Hashing approaches, however, are attractive because they provide a principle and efficient way to retrieve candidate items. In this paper, we propose a simple and effective learning-to-hash approach for the fast item ranking problem that can be used for any type of measure, including neural network measures. Specifically, we solve this problem with an asymmetric hashing framework based on discrete inner product fitting. We learn a pair of related hash functions that map heterogeneous objects (e.g., users and items) into a common discrete space where the inner product of their binary codes reveals their true similarity defined via the original searching measure. The fast ranking problem is reduced to an ANN search via this asymmetric hashing scheme. Then, we propose a sampling strategy to efficiently select relevant and contrastive samples to train the hashing model. We empirically validate the proposed method against the existing state-of-the-art fast item ranking methods in several combinations of non-linear searching functions and prominent datasets

How Fast Can We Play Tetris Greedily With Rectangular Pieces?

Consider a variant of Tetris played on a board of width $w$ and infinite height, where the pieces are axis-aligned rectangles of arbitrary integer dimensions, the pieces can only be moved before letting them drop, and a row does not disappear once it is full. Suppose we want to follow a greedy strategy: let each rectangle fall where it will end up the lowest given the current state of the board. To do so, we want a data structure which can always suggest a greedy move. In other words, we want a data structure which maintains a set of $O(n)$ rectangles, supports queries which return where to drop the rectangle, and updates which insert a rectangle dropped at a certain position and return the height of the highest point in the updated set of rectangles. We show via a reduction to the Multiphase problem [P\u{a}tra\c{s}cu, 2010] that on a board of width $w=\Theta(n)$, if the OMv conjecture [Henzinger et al., 2015] is true, then both operations cannot be supported in time $O(n^{1/2-\epsilon})$ simultaneously. The reduction also implies polynomial bounds from the 3-SUM conjecture and the APSP conjecture. On the other hand, we show that there is a data structure supporting both operations in $O(n^{1/2}\log^{3/2}n)$ time on boards of width $n^{O(1)}$, matching the lower bound up to a $n^{o(1)}$ factor.Comment: Correction of typos and other minor correction

On strong Menger-connectivity of star graphs

AbstractMotivated by parallel routing in networks with faults, we study the following graph theoretical problem. Let G be a graph of minimum vertex degree d. We say that G is strongly Menger-connected if for any copy Gf of G with at most d−2 nodes removed, every pair of nodes u and v in Gf are connected by min{degf(u),degf(v)} node-disjoint paths in Gf, where degf(u) and degf(v) are the degrees of the nodes u and v in Gf, respectively. We show that the star graphs, which are a recently proposed attractive alternative to the widely used hypercubes as network models, are strongly Menger-connected. An algorithm of optimal running time is developed that constructs the maximum number of node-disjoint paths of nearly optimal length in star graphs with faults

Robust and Listening-Efficient Contention Resolution

This paper shows how to achieve contention resolution on a shared communication channel using only a small number of channel accesses -- both for listening and sending -- and the resulting algorithm is resistant to adversarial noise. The shared channel operates over a sequence of synchronized time slots, and in any slot agents may attempt to broadcast a packet. An agent's broadcast succeeds if no other agent broadcasts during that slot. If two or more agents broadcast in the same slot, then the broadcasts collide and both broadcasts fail. An agent listening on the channel during a slot receives ternary feedback, learning whether that slot had silence, a successful broadcast, or a collision. Agents are (adversarially) injected into the system over time. The goal is to coordinate the agents so that each is able to successfully broadcast its packet. A contention-resolution protocol is measured both in terms of its throughput and the number of slots during which an agent broadcasts or listens. Most prior work assumes that listening is free and only tries to minimize the number of broadcasts. This paper answers two foundational questions. First, is constant throughput achievable when using polylogarithmic channel accesses per agent, both for listening and broadcasting? Second, is constant throughput still achievable when an adversary jams some slots by broadcasting noise in them? Specifically, for $N$ packets arriving over time and $J$ jammed slots, we give an algorithm that with high probability in $N+J$ guarantees $\Theta(1)$ throughput and achieves on average $O(\texttt{polylog}(N+J))$ channel accesses against an adaptive adversary. We also have per-agent high-probability guarantees on the number of channel accesses -- either $O(\texttt{polylog}(N+J))$ or $O((J+1) \texttt{polylog}(N))$, depending on how quickly the adversary can react to what is being broadcast