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Zachary Friggstad

Publications and source records attributed to Zachary Friggstad.

At least 19 recordsLinked to original sources
arXiv2025

Budget-constrained Active Learning to Effectively De-censor Survival Data

Standard supervised learners attempt to learn a model from a labeled dataset. Given a small set of labeled instances, and a pool of unlabeled instances, a budgeted learner can use its given budget to pay to acquire the labels of some unlabeled instances, which it can then use to produce a model. Here, we explore budgeted learning in the context of survival datasets, which include (right) censored instances, where we know only a lower bound on an instance's time-to-event. Here, that learner can pay to (partially) label a censored instance -- e.g., to acquire the actual time for an instance [perhaps go from (3 yr, censored) to (7.2 yr, uncensored)], or other variants [e.g., learn about one more year, so go from (3 yr, censored) to either (4 yr, censored) or perhaps (3.2 yr, uncensored)]. This serves as a model of real world data collection, where follow-up with censored patients does not always lead to uncensoring, and how much information is given to the learner model during data collection is a function of the budget and the nature of the data itself. We provide both experimental and theoretical results for how to apply state-of-the-art budgeted learning algorithms to survival data and the respective limitations that exist in doing so. Our approach provides bounds and time complexity asymptotically equivalent to the standard active learning method BatchBALD. Moreover, empirical analysis on several survival tasks show that our model performs better than other potential approaches on several benchmarks.

cs.LG↗
arXiv2025

Approximating Multiple-Depot Capacitated Vehicle Routing via LP Rounding

In Capacitated Vehicle Routing with Multiple Depots (CVRP-MD) we are given a set of client locations $C$ and a set of depots $R$ located in a metric space with costs $c(i,j)$ between $u,v \in C \cup R$. Additionally, we are given a capacity bound $k$. The goal is to find a collection of tours of minimum total cost such that each tour starts and ends at some depot $r \in R$ and includes at most $k$ clients and such that each client lies on at least one tour. Our main result is a $3.9365$-approximation based on rounding a new LP relaxation for CVRP-MD.

cs.DS↗
arXiv2025

A Polynomial-Time Approximation for Pairwise Fair $k$-Median Clustering

In this work, we study pairwise fair clustering with $\ell \ge 2$ groups, where for every cluster $C$ and every group $i \in [\ell]$, the number of points in $C$ from group $i$ must be at most $t$ times the number of points in $C$ from any other group $j \in [\ell]$, for a given integer $t$. To the best of our knowledge, only bi-criteria approximation and exponential-time algorithms follow for this problem from the prior work on fair clustering problems when $\ell > 2$. In our work, focusing on the $\ell > 2$ case, we design the first polynomial-time $O(k^2\cdot \ell \cdot t)$-approximation for this problem with $k$-median cost that does not violate the fairness constraints. We complement our algorithmic result by providing hardness of approximation results, which show that our problem even when $\ell=2$ is almost as hard as the popular uniform capacitated $k$-median, for which no polynomial-time algorithm with an approximation factor of $o(\log k)$ is known.

cs.DS↗
arXiv2024

Approximation Algorithms for Clustering with Minimum Sum of Radii, Diameters, and Squared Radii

In this paper, we present an improved approximation algorithm for three related problems. In the Minimum Sum of Radii clustering problem (MSR), we aim to select $k$ balls in a metric space to cover all points while minimizing the sum of the radii. In the Minimum Sum of Diameters clustering problem (MSD), we are to pick $k$ clusters to cover all the points such that sum of diameters of all the clusters is minimized. At last, in the Minimum Sum of Squared Radii problem (MSSR), the goal is to choose $k$ balls, similar to MSR. However in MSSR, the goal is to minimize the sum of squares of radii of the balls. We present a 3.389-approximation for MSR and a 6.546-approximation for MSD, improving over respective 3.504 and 7.008 developed by Charikar and Panigrahy (2001). In particular, our guarantee for MSD is better than twice our guarantee for MSR. In the case of MSSR, the best known approximation guarantee is $4\cdot(540)^{2}$ based on the work of Bhowmick, Inamdar, and Varadarajan in their general analysis of the $t$-Metric Multicover Problem. At last with our analysis, we get a 11.078-approximation algorithm for Minimum Sum of Squared Radii.

cs.DS↗
arXiv2024

Approximating Traveling Salesman Problems Using a Bridge Lemma

We give improved approximations for two metric Traveling Salesman Problem (TSP) variants. In Ordered TSP (OTSP) we are given a linear ordering on a subset of nodes $o_1, \ldots, o_k$. The TSP solution must have that $o_{i+1}$ is visited at some point after $o_i$ for each $1 \leq i < k$. This is the special case of Precedence-Constrained TSP ($PTSP$) in which the precedence constraints are given by a single chain on a subset of nodes. In $k$-Person TSP Path (k-TSPP), we are given pairs of nodes $(s_1,t_1), \ldots, (s_k,t_k)$. The goal is to find an $s_i$-$t_i$ path with minimum total cost such that every node is visited by at least one path. We obtain a $3/2 + e^{-1} < 1.878$ approximation for OTSP, the first improvement over a trivial $\alpha+1$ approximation where $\alpha$ is the current best TSP approximation. We also obtain a $1 + 2 \cdot e^{-1/2} < 2.214$ approximation for k-TSPP, the first improvement over a trivial $3$-approximation. These algorithms both use an adaptation of the Bridge Lemma that was initially used to obtain improved Steiner Tree approximations [Byrka et al., 2013]. Roughly speaking, our variant states that the cost of a cheapest forest rooted at a given set of terminal nodes will decrease by a substantial amount if we randomly sample a set of non-terminal nodes to also become terminals such provided each non-terminal has a constant probability of being sampled. We believe this view of the Bridge Lemma will find further use for improved vehicle routing approximations beyond this paper.

cs.DS↗
arXiv2024

A QPTAS for Facility Location on Unit Disk graphs

We study the classic \textsc{(Uncapacitated) Facility Location} problem on Unit Disk Graphs (UDGs). For a given point set $P$ in the plane, the unit disk graph UDG(P) on $P$ has vertex set $P$ and an edge between two distinct points $p, q \in P$ if and only if their Euclidean distance $|pq|$ is at most 1. The weight of the edge $pq$ is equal to their distance $|pq|$. An instance of \fl on UDG(P) consists of a set $C\subseteq P$ of clients and a set $F\subseteq P$ of facilities, each having an opening cost $f_i$. The goal is to pick a subset $F'\subseteq F$ to open while minimizing $\sum_{i\in F'} f_i + \sum_{v\in C} d(v,F')$, where $d(v,F')$ is the distance of $v$ to nearest facility in $F'$ through UDG(P). In this paper, we present the first Quasi-Polynomial Time Approximation Schemes (QPTAS) for the problem. While approximation schemes are well-established for facility location problems on sparse geometric graphs (such as planar graphs), there is a lack of such results for dense graphs. Specifically, prior to this study, to the best of our knowledge, there was no approximation scheme for any facility location problem on UDGs in the general setting.

cs.DS↗
arXiv2024

Exact Algorithms and Lower Bounds for Stable Instances of Euclidean k-Means

We investigate the complexity of solving stable or perturbation-resilient instances of $k$-Means and $k$-Median clustering in fixed dimension Euclidean metrics (more generally doubling metrics). The notion of stable (perturbation resilient) instances was introduced by Bilu and Linial [2010] and Awasthi et al. [2012]. In our context we say a $k$-Means instance is $α$-stable if there is a unique OPT which remains optimum if distances are (non-uniformly) stretched by a factor of at most $α$. Stable clustering instances have been studied to explain why heuristics such as Lloyd's algorithm perform well in practice. In this work we show that for any fixed $ε>0$, $(1+ε)$-stable instances of $k$-Means in doubling metrics can be solved in polynomial time. More precisely we show a natural multiswap local search algorithm finds OPT for $(1+ε)$-stable instances of $k$-Means and $k$-Median in a polynomial number of iterations. We complement this result by showing that under a new PCP theorem, this is essentially tight: that when the dimension d is part of the input, there is a fixed $ε_0>0$ s.t. there is not even a PTAS for $(1+ε_0)$-stable $k$-Means in $R^d$ unless NP=RP. To do this, we consider a robust property of CSPs; call an instance stable if there is a unique optimum solution $x^*$ and for any other solution $x'$, the number of unsatisfied clauses is proportional to the Hamming distance between $x^*$ and $x'$. Dinur et al. have already shown stable QSAT is hard to approximate for some constant Q, our hypothesis is simply that stable QSAT with bounded variable occurrence is also hard. Given this hypothesis we consider "stability-preserving" reductions to prove our hardness for stable k-Means. Such reductions seem to be more fragile than standard L-reductions and may be of further use to demonstrate other stable optimization problems are hard.

cs.DS↗
arXiv2023

An $O(\log k)$-Approximation for Directed Steiner Tree in Planar Graphs

We present an $O(\log k)$-approximation for both the edge-weighted and node-weighted versions of \DST in planar graphs where $k$ is the number of terminals. We extend our approach to \MDST (in general graphs \MDST and \DST are easily seen to be equivalent but in planar graphs this is not the case necessarily) in which we get an $O(R+\log k)$-approximation for planar graphs for where $R$ is the number of roots.

cs.DS↗
arXiv2023

On Narrative Information and the Distillation of Stories

The act of telling stories is a fundamental part of what it means to be human. This work introduces the concept of narrative information, which we define to be the overlap in information space between a story and the items that compose the story. Using contrastive learning methods, we show how modern artificial neural networks can be leveraged to distill stories and extract a representation of the narrative information. We then demonstrate how evolutionary algorithms can leverage this to extract a set of narrative templates and how these templates -- in tandem with a novel curve-fitting algorithm we introduce -- can reorder music albums to automatically induce stories in them. In the process of doing so, we give strong statistical evidence that these narrative information templates are present in existing albums. While we experiment only with music albums here, the premises of our work extend to any form of (largely) independent media.

cs.CL↗
arXiv2022

A Constant-Factor Approximation for Quasi-bipartite Directed Steiner Tree on Minor-Free Graphs

We give the first constant-factor approximation algorithm for quasi-bipartite instances of Directed Steiner Tree on graphs that exclude fixed minors. In particular, for $K_r$-minor-free graphs our approximation guarantee is $O(r\cdot\sqrt{\log r})$ and, further, for planar graphs our approximation guarantee is 20. Our algorithm uses the primal-dual scheme. We employ a more involved method of determining when to buy an edge while raising dual variables since, as we show, the natural primal-dual scheme fails to raise enough dual value to pay for the purchased solution. As a consequence, we also demonstrate integrality gap upper bounds on the standard cut-based linear programming relaxation for the Directed Steiner Tree instances we consider.

cs.DS↗
arXiv2021

Parameterized Approximation Algorithms for $k$-Center Clustering and Variants

$k$-center is one of the most popular clustering models. While it admits a simple 2-approximation in polynomial time in general metrics, the Euclidean version is NP-hard to approximate within a factor of 1.93, even in the plane, if one insists the dependence on $k$ in the running time be polynomial. Without this restriction, a classic algorithm yields a $2^{O((k\log k)/ε)}dn$-time $(1+ε)$-approximation for Euclidean $k$-center, where $d$ is the dimension. We give a faster algorithm for small dimensions: roughly speaking an $O^*(2^{O((1/ε)^{O(d)} \cdot k^{1-1/d} \cdot \log k)})$-time $(1+ε)$-approximation. In particular, the running time is roughly $O^*(2^{O((1/ε)^{O(1)}\sqrt{k}\log k)})$ in the plane. We complement our algorithmic result with a matching hardness lower bound. We also consider a well-studied generalization of $k$-center, called Non-uniform $k$-center (NUkC), where we allow different radii clusters. NUkC is NP-hard to approximate within any factor, even in the Euclidean case. We design a $2^{O(k\log k)}n^2$ time $3$-approximation for NUkC in general metrics, and a $2^{O((k\log k)/ε)}dn$ time $(1+ε)$-approximation for Euclidean NUkC. The latter time bound matches the bound for $k$-center.

cs.DS↗
arXiv2021

Improved Approximations for CVRP with Unsplittable Demands

In this paper, we present improved approximation algorithms for the (unsplittable) Capacitated Vehicle Routing Problem (CVRP) in general metrics. In CVRP, introduced by Dantzig and Ramser (1959), we are given a set of points (clients) $V$ together with a depot $r$ in a metric space, with each $v\in V$ having a demand $d_v>0$, and a vehicle of bounded capacity $Q$. The goal is to find a minimum cost collection of tours for the vehicle, each starting and ending at the depot, such that each client is visited at least once and the total demands of the clients in each tour is at most $Q$. In the unsplittable variant we study, the demand of a node must be served entirely by one tour. We present two approximation algorithms for unsplittable CVRP: a combinatorial $(α+1.75)$-approximation, where $α$ is the approximation factor for the Traveling Salesman Problem, and an approximation algorithm based on LP rounding with approximation guarantee $α+\ln(2) + δ\approx 3.194 + δ$ in $n^{O(1/δ)}$ time. Both approximations can further be improved by a small amount when combined with recent work by Blauth, Traub, and Vygen (2021), who obtained an $(α+ 2\cdot (1 -ε))$-approximation for unsplittable CVRP for some constant $ε$ depending on $α$ ($ε> 1/3000$ for $α= 1.5$).

cs.DS↗
arXiv2021

Combinatorial Algorithms for Rooted Prize-Collecting Walks and Applications to Orienteering and Minimum-Latency Problems

We consider the rooted prize-collecting walks (PCW) problem, wherein we seek a collection $C$ of rooted walks having minimum prize-collecting cost, which is the (total cost of walks in $C$) + (total node-reward of nodes not visited by any walk in $C$). This problem arises naturally as the Lagrangian relaxation of both orienteering, where we seek a length-bounded walk of maximum reward, and the $\ell$-stroll problem, where we seek a minimum-length walk covering at least $\ell$ nodes. Our main contribution is to devise a simple, combinatorial algorithm for the PCW problem in directed graphs that returns a rooted tree whose prize-collecting cost is at most the optimum value of the prize-collecting walks problem. We utilize our algorithm to develop combinatorial approximation algorithms for two fundamental vehicle-routing problems (VRPs): (1) orienteering; and (2) $k$-minimum-latency problem ($k$-MLP), wherein we seek to cover all nodes using $k$ paths starting at a prescribed root node, so as to minimize the sum of the node visiting times. Our combinatorial algorithm allows us to sidestep the part where we solve a preflow-based LP in the LP-rounding algorithms of Friggstand and Swamy (2017) for orienteering, and in the state-of-the-art $7.183$-approximation algorithm for $k$-MP in Post and Swamy (2015). Consequently, we obtain combinatorial implementations of these algorithms with substantially improved running times compared with the current-best approximation factors. We report computational results for our resulting (combinatorial implementations of) orienteering algorithms, which show that the algorithms perform quite well in practice, both in terms of the quality of the solution they return, as also the upper bound they yield on the orienteering optimum (which is obtained by leveraging the workings of our PCW algorithm).

cs.DS↗
arXiv2021

Automatic Embedding of Stories Into Collections of Independent Media

We look at how machine learning techniques that derive properties of items in a collection of independent media can be used to automatically embed stories into such collections. To do so, we use models that extract the tempo of songs to make a music playlist follow a narrative arc. Our work specifies an open-source tool that uses pre-trained neural network models to extract the global tempo of a set of raw audio files and applies these measures to create a narrative-following playlist. This tool is available at https://github.com/dylanashley/playlist-story-builder/releases/tag/v1.0.0

cs.CL↗
arXiv2020

A Constant-Factor Approximation for Directed Latency in Quasi-Polynomial Time

We give the first constant-factor approximation for the Directed Latency problem in quasi-polynomial time. Here, the goal is to visit all nodes in an asymmetric metric with a single vehicle starting at a depot $r$ to minimize the average time a node waits to be visited by the vehicle. The approximation guarantee is an improvement over the polynomial-time $O(\log n)$-approximation [Friggstad, Salavatipour, Svitkina, 2013] and no better quasi-polynomial time approximation algorithm was known. To obtain this, we must extend a recent result showing the integrality gap of the Asymmetric TSP-Path LP relaxation is bounded by a constant [Köhne, Traub, and Vygen, 2019], which itself builds on the breakthrough result that the integrality gap for standard Asymmetric TSP is also a constant [Svensson, Tarnawsi, and Vegh, 2018]. We show the standard Asymmetric TSP-Path integrality gap is bounded by a constant even if the cut requirements of the LP relaxation are relaxed from $x(δ^{in}(S)) \geq 1$ to $x(δ^{in}(S)) \geq ρ$ for some constant $1/2 < ρ\leq 1$. We also give a better approximation guarantee in the special case of Directed Latency in regret metrics where the goal is to find a path $P$ minimize the average time a node $v$ waits in excess of $c_{rv}$, i.e. $\frac{1}{|V|} \cdot \sum_{v \in V} (c_v(P)-c_{rv})$.

cs.DS↗
arXiv2019

Graph Pricing with Limited Supply

We study approximation algorithms for graph pricing with vertex capacities yet without the traditional envy-free constraint. Specifically, we have a set of items $V$ and a set of customers $X$ where each customer $i \in X$ has a budget $b_i$ and is interested in a bundle of items $S_i \subseteq V$ with $|S_i| \leq 2$. However, there is a limited supply of each item: we only have $μ_v$ copies of item $v$ to sell for each $v \in V$. We should assign prices $p(v)$ to each $v \in V$ and chose a subset $Y \subseteq X$ of customers so that each $i \in Y$ can afford their bundle ($p(S_i) \leq b_i$) and at most $μ_v$ chosen customers have item $v$ in their bundle for each item $v \in V$. Each customer $i \in Y$ pays $p(S_i)$ for the bundle they purchased: our goal is to do this in a way that maximizes revenue. Such pricing problems have been studied from the perspective of envy-freeness where we also must ensure that $p(S_i) \geq b_i$ for each $i \notin Y$. However, the version where we simply allocate items to customers after setting prices and do not worry about the envy-free condition has received less attention. Our main result is an 8-approximation for the capacitated case via local search and a 7.8096-approximation in simple graphs with uniform vertex capacities. The latter is obtained by combing a more involved analysis of a multi-swap local search algorithm for constant capacities and an LP-rounding algorithm for larger capacities. If all capacities are bounded by a constant $C$, we further show a multi-swap local search algorithm yields an $\left(4 \cdot \frac{2C-1}{C} + ε\right)$-approximation. We also give a $(4+ε)$-approximation in simple graphs through LP rounding when all capacities are very large as a function of $ε$.

cs.DS↗
arXiv2017

Compact, Provably-Good LPs for Orienteering and Regret-Bounded Vehicle Routing

We develop polynomial-size LP-relaxations for {\em orienteering} and the {\em regret-bounded vehicle routing problem} (\rvrp) and devise suitable LP-rounding algorithms that lead to various new insights and approximation results for these problems. In orienteering, the goal is to find a maximum-reward $r$-rooted path, possibly ending at a specified node, of length at most some given budget $B$. In \rvrp, the goal is to find the minimum number of $r$-rooted paths of {\em regret} at most a given bound $R$ that cover all nodes, where the regret of an $r$-$v$ path is its length $-$ $c_{rv}$. For {\em rooted orienteering}, we introduce a natural bidirected LP-relaxation and obtain a simple $3$-approximation algorithm via LP-rounding. This is the {\em first LP-based} guarantee for this problem. We also show that {\em point-to-point} (\ptp) {\em orienteering} can be reduced to a regret-version of rooted orienteering at the expense of a factor-2 loss in approximation. For \rvrp, we propose two compact LPs that lead to significant improvements, in both approximation ratio and running time, over the approach in~\cite{FriggstadS14}. One of these is a natural modification of the LP for rooted orienteering; the other is an unconventional formulation that is motivated by certain structural properties of an \rvrp-solution, which leads to a $15$-approximation algorithm for \rvrp.

cs.DS↗