Related papers: Deterministic Monotone Min-Plus Product and Convol…
Min-plus product of two $n\times n$ matrices is a fundamental problem in algorithm research. It is known to be equivalent to APSP, and in general it has no truly subcubic algorithms. In this paper, we focus on the min-plus product on a…
In this paper, we show that the time complexity of monotone min-plus product of two $n\times n$ matrices is $\tilde{O}(n^{(3+\omega)/2})=\tilde{O}(n^{2.687})$, where $\omega < 2.373$ is the fast matrix multiplication exponent [Alman and…
Recently, Pagh presented a randomized approximation algorithm for the multiplication of real-valued matrices building upon work for detecting the most frequent items in data streams. We continue this line of research and present new {\em…
In a recent breakthrough paper, Chi et al. (STOC'22) introduce an $\tilde{O}(n^{\frac{3 + \omega}{2}})$ time algorithm to compute Monotone Min-Plus Product between two square matrices of dimensions $n \times n$ and entries bounded by…
Min-plus matrix multiplication is used in many problems operating on distances in graphs or solvable by dynamic programming. Assuming the APSP hypothesis, there is no subcubic-time algorithm for the min-plus product of two general $n\times…
The goal of this paper is to get truly subcubic algorithms for Min-Plus product for less structured inputs than what was previously known, and to apply them to versions of All-Pairs Shortest Paths (APSP) and other problems. The results are…
The most studied linear algebraic operation, matrix multiplication, has surprisingly fast $O(n^\omega)$ time algorithms for $\omega<2.373$. On the other hand, the $(\min,+)$ matrix product which is at the heart of many fundamental graph…
One of the most basic graph problems, All-Pairs Shortest Paths (APSP) is known to be solvable in $n^{3-o(1)}$ time, and it is widely open whether it has an $O(n^{3-\epsilon})$ time algorithm for $\epsilon > 0$. To better understand APSP,…
We revisit the problem of finding optimal strategies for deterministic Markov Decision Processes (DMDPs), and a closely related problem of testing feasibility of systems of $m$ linear inequalities on $n$ real variables with at most two…
We study dynamic $(1+\epsilon)$-approximation algorithms for the all-pairs shortest paths problem in unweighted undirected $n$-node $m$-edge graphs under edge deletions. The fastest algorithm for this problem is a randomized algorithm with…
We consider the classical Minimum Balanced Cut problem: given a graph $G$, compute a partition of its vertices into two subsets of roughly equal volume, while minimizing the number of edges connecting the subsets. We present the first {\em…
We revisit the classic 0-1-Knapsack problem, in which we are given $n$ items with their weights and profits as well as a weight budget $W$, and the goal is to find a subset of items of total weight at most $W$ that maximizes the total…
Motivated by studying the power of randomness, certifying algorithms and barriers for fine-grained reductions, we investigate the question whether the multiplication of two $n\times n$ matrices can be performed in near-optimal…
Symmetric submodular maximization is an important class of combinatorial optimization problems, including MAX-CUT on graphs and hyper-graphs. The state-of-the-art algorithm for the problem over general constraints has an approximation ratio…
We present a pseudopolynomial-time algorithm for the Knapsack problem that has running time $\widetilde{O}(n + t\sqrt{p_{\max}})$, where $n$ is the number of items, $t$ is the knapsack capacity, and $p_{\max}$ is the maximum item profit.…
We give a deterministic algorithm for finding the minimum (weight) cut of an undirected graph on $n$ vertices and $m$ edges using $\text{polylog}(n)$ calls to any maximum flow subroutine. Using the current best deterministic maximum flow…
We study the time complexity of computing the $(\min,+)$ matrix product of two $n\times n$ integer matrices in terms of $n$ and the number of monotone subsequences the rows of the first matrix and the columns of the second matrix can be…
Interior point algorithms for solving linear programs have been studied extensively for a long time [e.g. Karmarkar 1984; Lee, Sidford FOCS'14; Cohen, Lee, Song STOC'19]. For linear programs of the form $\min_{Ax=b, x \ge 0} c^\top x$ with…
In this work, we study the problem of finding the maximum value of a non-negative submodular function subject to a limit on the number of items selected, a ubiquitous problem that appears in many applications, such as data summarization and…
In the $\{-1,0,1\}$-APSP problem the goal is to compute all-pairs shortest paths (APSP) on a directed graph whose edge weights are all from $\{-1,0,1\}$. In the (min,max)-product problem the input is two $n\times n$ matrices $A$ and $B$,…