ﻻ يوجد ملخص باللغة العربية
We consider the problem of designing sublinear time algorithms for estimating the cost of a minimum metric traveling salesman (TSP) tour. Specifically, given access to a $n times n$ distance matrix $D$ that specifies pairwise distances between $n$ points, the goal is to estimate the TSP cost by performing only sublinear (in the size of $D$) queries. For the closely related problem of estimating the weight of a metric minimum spanning tree (MST), it is known that for any $varepsilon > 0$, there exists an $tilde{O}(n/varepsilon^{O(1)})$ time algorithm that returns a $(1 + varepsilon)$-approximate estimate of the MST cost. This result immediately implies an $tilde{O}(n/varepsilon^{O(1)})$ time algorithm to estimate the TSP cost to within a $(2 + varepsilon)$ factor for any $varepsilon > 0$. However, no $o(n^2)$ time algorithms are known to approximate metric TSP to a factor that is strictly better than $2$. On the other hand, there were also no known barriers that rule out the existence of $(1 + varepsilon)$-approximate estimation algorithms for metric TSP with $tilde{O}(n)$ time for any fixed $varepsilon > 0$. In this paper, we make progress on both algorithms and lower bounds for estimating metric TSP cost. We also show that the problem of estimating metric TSP cost is closely connected to the problem of estimating the size of a maximum matching in a graph.
We consider the problem of testing graph cluster structure: given access to a graph $G=(V, E)$, can we quickly determine whether the graph can be partitioned into a few clusters with good inner conductance, or is far from any such graph? This is a ge
We present new lower bounds that show that a polynomial number of passes are necessary for solving some fundamental graph problems in the streaming model of computation. For instance, we show that any streaming algorithm that finds a weighted minimum
Robust optimization is a widely studied area in operations research, where the algorithm takes as input a range of values and outputs a single solution that performs well for the entire range. Specifically, a robust algorithm aims to minimize regret,
Given a metric $(V,d)$ and a $textsf{root} in V$, the classic $textsf{$k$-TSP}$ problem is to find a tour originating at the $textsf{root}$ of minimum length that visits at least $k$ nodes in $V$. In this work, motivated by applications where the inp
We prove that any two-pass graph streaming algorithm for the $s$-$t$ reachability problem in $n$-vertex directed graphs requires near-quadratic space of $n^{2-o(1)}$ bits. As a corollary, we also obtain near-quadratic space lower bounds for several o