Delaunay Edge Flips in Dense Surface Triangulations

Delaunay flip is an elegant, simple tool to convert a triangulation of a point set to its Delaunay triangulation. The technique has been researched extensively for full dimensional triangulations of point sets. However, an important case of triangulations which are not full dimensional is surface triangulations in three dimensions. In this paper we address the question of converting a surface triangulation to a subcomplex of the Delaunay triangulation with edge flips. We show that the surface triangulations which closely approximate a smooth surface with uniform density can be transformed to a Delaunay triangulation with a simple edge flip algorithm. The condition on uniformity becomes less stringent with increasing density of the triangulation. If the condition is dropped completely, the flip algorithm still terminates although the output surface triangulation becomes "almost Delaunay" instead of exactly Delaunay.Comment: This paper is prelude to "Maintaining Deforming Surface Meshes" by Cheng-Dey in SODA 200

Restricted Max-Min Allocation: Approximation and Integrality Gap

Asadpour, Feige, and Saberi proved that the integrality gap of the configuration LP for the restricted max-min allocation problem is at most 4. However, their proof does not give a polynomial-time approximation algorithm. A lot of efforts have been devoted to designing an efficient algorithm whose approximation ratio can match this upper bound for the integrality gap. In ICALP 2018, we present a (6 + delta)-approximation algorithm where delta can be any positive constant, and there is still a gap of roughly 2. In this paper, we narrow the gap significantly by proposing a (4+delta)-approximation algorithm where delta can be any positive constant. The approximation ratio is with respect to the optimal value of the configuration LP, and the running time is poly(m,n)* n^{poly(1/(delta))} where n is the number of players and m is the number of resources. We also improve the upper bound for the integrality gap of the configuration LP to 3 + 21/26 =~ 3.808

Solving Fr\'echet Distance Problems by Algebraic Geometric Methods

We study several polygonal curve problems under the Fr\'{e}chet distance via algebraic geometric methods. Let $\mathbb{X}_m^d$ and $\mathbb{X}_k^d$ be the spaces of all polygonal curves of $m$ and $k$ vertices in $\mathbb{R}^d$, respectively. We assume that $k \leq m$. Let $\mathcal{R}^d_{k,m}$ be the set of ranges in $\mathbb{X}_m^d$ for all possible metric balls of polygonal curves in $\mathbb{X}_k^d$ under the Fr\'{e}chet distance. We prove a nearly optimal bound of $O(dk\log (km))$ on the VC dimension of the range space $(\mathbb{X}_m^d,\mathcal{R}_{k,m}^d)$, improving on the previous $O(d^2k^2\log(dkm))$ upper bound and approaching the current $\Omega(dk\log k)$ lower bound. Our upper bound also holds for the weak Fr\'{e}chet distance. We also obtain exact solutions that are hitherto unknown for curve simplification, range searching, nearest neighbor search, and distance oracle.Comment: To appear at SODA24, correct some reference

Curve Simplification and Clustering under Fr\'echet Distance

We present new approximation results on curve simplification and clustering under Fr\'echet distance. Let $T = \{\tau_i : i \in [n] \}$ be polygonal curves in $R^d$ of $m$ vertices each. Let $l$ be any integer from $[m]$. We study a generalized curve simplification problem: given error bounds $\delta_i > 0$ for $i \in [n]$, find a curve $\sigma$ of at most $l$ vertices such that $d_F(\sigma,\tau_i) \le \delta_i$ for $i \in [n]$. We present an algorithm that returns a null output or a curve $\sigma$ of at most $l$ vertices such that $d_F(\sigma,\tau_i) \le \delta_i + \epsilon\delta_{\max}$ for $i \in [n]$, where $\delta_{\max} = \max_{i \in [n]} \delta_i$. If the output is null, there is no curve of at most $l$ vertices within a Fr\'echet distance of $\delta_i$ from $\tau_i$ for $i \in [n]$. The running time is $\tilde{O}\bigl(n^{O(l)} m^{O(l^2)} (dl/\epsilon)^{O(dl)}\bigr)$. This algorithm yields the first polynomial-time bicriteria approximation scheme to simplify a curve $\tau$ to another curve $\sigma$, where the vertices of $\sigma$ can be anywhere in $R^d$, so that $d_F(\sigma,\tau) \le (1+\epsilon)\delta$ and $|\sigma| \le (1+\alpha) \min\{|c| : d_F(c,\tau) \le \delta\}$ for any given $\delta > 0$ and any fixed $\alpha, \epsilon \in (0,1)$. The running time is $\tilde{O}\bigl(m^{O(1/\alpha)} (d/(\alpha\epsilon))^{O(d/\alpha)}\bigr)$. By combining our technique with some previous results in the literature, we obtain an approximation algorithm for $(k,l)$-median clustering. Given $T$, it computes a set $\Sigma$ of $k$ curves, each of $l$ vertices, such that $\sum_{i \in [n]} \min_{\sigma \in \Sigma} d_F(\sigma,\tau_i)$ is within a factor $1+\epsilon$ of the optimum with probability at least $1-\mu$ for any given $\mu, \epsilon \in (0,1)$. The running time is $\tilde{O}\bigl(n m^{O(kl^2)} \mu^{-O(kl)} (dkl/\epsilon)^{O((dkl/\epsilon)\log(1/\mu))}\bigr)$.Comment: 28 pages; Corrected some wrong descriptions concerning related wor

Restricted Max-Min Fair Allocation

The restricted max-min fair allocation problem seeks an allocation of resources to players that maximizes the minimum total value obtained by any player. It is NP-hard to approximate the problem to a ratio less than 2. Comparing the current best algorithm for estimating the optimal value with the current best for constructing an allocation, there is quite a gap between the ratios that can be achieved in polynomial time: 4+delta for estimation and 6 + 2 sqrt{10} + delta ~~ 12.325 + delta for construction, where delta is an arbitrarily small constant greater than 0. We propose an algorithm that constructs an allocation with value within a factor 6 + delta from the optimum for any constant delta > 0. The running time is polynomial in the input size for any constant delta chosen