arXiv subjects

Zachary Friggstad

Publications and source records attributed to Zachary Friggstad.

29 records · Page 2Linked to original sources

arXiv2017

Approximation Schemes for Clustering with Outliers

Clustering problems are well-studied in a variety of fields such as data science, operations research, and computer science. Such problems include variants of centre location problems, $k$-median, and $k$-means to name a few. In some cases, not all data points need to be clustered; some may be discarded for various reasons. We study clustering problems with outliers. More specifically, we look at Uncapacitated Facility Location (UFL), $k$-Median, and $k$-Means. In UFL with outliers, we have to open some centres, discard up to $z$ points of $\cal X$ and assign every other point to the nearest open centre, minimizing the total assignment cost plus centre opening costs. In $k$-Median and $k$-Means, we have to open up to $k$ centres but there are no opening costs. In $k$-Means, the cost of assigning $j$ to $i$ is $δ^2(j,i)$. We present several results. Our main focus is on cases where $δ$ is a doubling metric or is the shortest path metrics of graphs from a minor-closed family of graphs. For uniform-cost UFL with outliers on such metrics we show that a multiswap simple local search heuristic yields a PTAS. With a bit more work, we extend this to bicriteria approximations for the $k$-Median and $k$-Means problems in the same metrics where, for any constant $ε> 0$, we can find a solution using $(1+ε)k$ centres whose cost is at most a $(1+ε)$-factor of the optimum and uses at most $z$ outliers. We also show that natural local search heuristics that do not violate the number of clusters and outliers for $k$-Median (or $k$-Means) will have unbounded gap even in Euclidean metrics. Furthermore, we show how our analysis can be extended to general metrics for $k$-Means with outliers to obtain a $(25+ε,1+ε)$ bicriteria.

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arXiv2017

Further Approximations for Demand Matching: Matroid Constraints and Minor-Closed Graphs

We pursue a study of the Generalized Demand Matching problem, a common generalization of the $b$-Matching and Knapsack problems. Here, we are given a graph with vertex capacities, edge profits, and asymmetric demands on the edges. The goal is to find a maximum-profit subset of edges so the demands of chosen edges do not violate vertex capacities. This problem is APX-hard and constant-factor approximations are known. Our results fall into two categories. First, using iterated relaxation and various filtering strategies, we show with an efficient rounding algorithm if an additional matroid structure $\mathcal M$ is given and we further only allow sets $F \subseteq E$ that are independent in $\mathcal M$, the natural LP relaxation has an integrality gap of at most $\frac{25}{3} \approx 8.333$. This can be improved in various special cases, for example we improve over the 15-approximation for the previously-studied Coupled Placement problem [Korupolu et al. 2014] by giving a $7$-approximation. Using similar techniques, we show the problem of computing a minimum-cost base in $\mathcal M$ satisfying vertex capacities admits a $(1,3)$-bicriteria approximation. This improves over the previous $(1,4)$-approximation in the special case that $\mathcal M$ is the graphic matroid over the given graph [Fukanaga and Nagamochi, 2009]. Second, we show Demand Matching admits a polynomial-time approximation scheme in graphs that exclude a fixed minor. If all demands are polynomially-bounded integers, this is somewhat easy using dynamic programming in bounded-treewidth graphs. Our main technical contribution is a sparsification lemma allowing us to scale the demands to be used in a more intricate dynamic programming algorithm, followed by randomized rounding to filter our scaled-demand solution to a feasible solution.

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arXiv2017

Local Search Yields a PTAS for k-Means in Doubling Metrics

The most well known and ubiquitous clustering problem encountered in nearly every branch of science is undoubtedly $k$-means: given a set of data points and a parameter $k$, select $k$ centres and partition the data points into $k$ clusters around these centres so that the sum of squares of distances of the points to their cluster centre is minimized. Typically these data points lie $\mathbb{R}^d$ for some $d\geq 2$. $k$-means and the first algorithms for it were introduced in the 1950's. Since then, hundreds of papers have studied this problem and many algorithms have been proposed for it. The most commonly used algorithm is known as Lloyd-Forgy, which is also referred to as "the" $k$-means algorithm, and various extensions of it often work very well in practice. However, they may produce solutions whose cost is arbitrarily large compared to the optimum solution. Kanungo et al. [2004] analyzed a simple local search heuristic to get a polynomial-time algorithm with approximation ratio $9+ε$ for any fixed $ε>0$ for $k$-means in Euclidean space. Finding an algorithm with a better approximation guarantee has remained one of the biggest open questions in this area, in particular whether one can get a true PTAS for fixed dimension Euclidean space. We settle this problem by showing that a simple local search algorithm provides a PTAS for $k$-means in $\mathbb{R}^d$ for any fixed $d$. More precisely, for any error parameter $ε>0$, the local search algorithm that considers swaps of up to $ρ=d^{O(d)}\cdotε^{-O(d/ε)}$ centres at a time finds a solution using exactly $k$ centres whose cost is at most a $(1+ε)$-factor greater than the optimum. Finally, we provide the first demonstration that local search yields a PTAS for the uncapacitated facility location problem and $k$-median with non-uniform opening costs in doubling metrics.

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arXiv2016

A Logarithmic Integrality Gap Bound for Directed Steiner Tree in Quasi-bipartite Graphs

We demonstrate that the integrality gap of the natural cut-based LP relaxation for the directed Steiner tree problem is $O(\log k)$ in quasi-bipartite graphs with $k$ terminals. Such instances can be seen to generalize set cover, so the integrality gap analysis is tight up to a constant factor. A novel aspect of our approach is that we use the primal-dual method; a technique that is rarely used in designing approximation algorithms for network design problems in directed graphs.

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arXiv2016

Tight Analysis of a Multiple-Swap Heuristic for Budgeted Red-Blue Median

Budgeted Red-Blue Median is a generalization of classic $k$-Median in that there are two sets of facilities, say $\mathcal{R}$ and $\mathcal{B}$, that can be used to serve clients located in some metric space. The goal is to open $k_r$ facilities in $\mathcal{R}$ and $k_b$ facilities in $\mathcal{B}$ for some given bounds $k_r, k_b$ and connect each client to their nearest open facility in a way that minimizes the total connection cost. We extend work by Hajiaghayi, Khandekar, and Kortsarz [2012] and show that a multiple-swap local search heuristic can be used to obtain a $(5+ε)$-approximation for Budgeted Red-Blue Median for any constant $ε> 0$. This is an improvement over their single swap analysis and beats the previous best approximation guarantee of 8 by Swamy [2014]. We also present a matching lower bound showing that for every $p \geq 1$, there are instances of Budgeted Red-Blue Median with local optimum solutions for the $p$-swap heuristic whose cost is $5 + Ω\left(\frac{1}{p}\right)$ times the optimum solution cost. Thus, our analysis is tight up to the lower order terms. In particular, for any $ε> 0$ we show the single-swap heuristic admits local optima whose cost can be as bad as $7-ε$ times the optimum solution cost.

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arXiv2015

An Improved Integrality Gap for Asymmetric TSP Paths

The Asymmetric Traveling Salesperson Path Problem (ATSPP) is one where, given an asymmetric metric space $(V,d)$ with specified vertices s and t, the goal is to find an s-t path of minimum length that passes through all the vertices in V. This problem is closely related to the Asymmetric TSP (ATSP), which seeks to find a tour (instead of an $s-t$ path) visiting all the nodes: for ATSP, a $ρ$-approximation guarantee implies an $O(ρ)$-approximation for ATSPP. However, no such connection is known for the integrality gaps of the linear programming relaxations for these problems: the current-best approximation algorithm for ATSPP is $O(\log n/\log\log n)$, whereas the best bound on the integrality gap of the natural LP relaxation (the subtour elimination LP) for ATSPP is $O(\log n)$. In this paper, we close this gap, and improve the current best bound on the integrality gap from $O(\log n)$ to $O(\log n/\log\log n)$. The resulting algorithm uses the structure of narrow $s$-$t$ cuts in the LP solution to construct a (random) tree spanning tree that can be cheaply augmented to contain an Eulerian $s$-$t$ walk. We also build on a result of Oveis Gharan and Saberi and show a strong form of Goddyn's conjecture about thin spanning trees implies the integrality gap of the subtour elimination LP relaxation for ATSPP is bounded by a constant. Finally, we give a simpler family of instances showing the integrality gap of this LP is at least 2.

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arXiv2013

Approximation Algorithms for Regret-Bounded Vehicle Routing and Applications to Distance-Constrained Vehicle Routing

We consider vehicle-routing problems (VRPs) that incorporate the notion of {\em regret} of a client, which is a measure of the waiting time of a client relative to its shortest-path distance from the depot. Formally, we consider both the additive and multiplicative versions of, what we call, the {\em regret-bounded vehicle routing problem} (RVRP). In these problems, we are given an undirected complete graph $G=(\{r\}\cup V,E)$ on $n$ nodes with a distinguished root (depot) node $r$, edge costs $\{c_{uv}\}$ that form a metric, and a regret bound $R$. Given a path $P$ rooted at $r$ and a node $v\in P$, let $c_P(v)$ be the distance from $r$ to $v$ along $P$. The goal is to find the fewest number of paths rooted at $r$ that cover all the nodes so that for every node $v$ covered by (say) path $P$: (i) its additive regret $c_P(v)-c_{rv}$, with respect to $P$ is at most $R$ in {\em additive-RVRP}; or (ii) its multiplicative regret, $c_P(c)/c_{rv}$, with respect to $P$ is at most $R$ in {\em multiplicative-RVRP}. Our main result is the {\em first} constant-factor approximation algorithm for additive-RVRP by devising rounding techniques for a natural {\em configuration-style LP}. This is a substantial improvement over the previous-best $O(\log n)$-approximation. Additive-RVRP turns out be a rather central vehicle-routing problem, whose study reveals insights into a variety of other regret-related problems as well as the classical {\em distance-constrained VRP} ({DVRP}). We obtain approximation ratios of $O\bigl(\log(\frac{R}{R-1})\bigr)$ for multiplicative-RVRP, and $O\bigl(\min\bigl\{\mathit{OPT},\frac{\log D}{\log\log D}\bigr\}\bigr)$ for DVRP with distance bound $D$ via reductions to additive-RVRP; the latter improves upon the previous-best approximation for DVRP.

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arXiv2013

Local-Search based Approximation Algorithms for Mobile Facility Location Problems

We consider the {\em mobile facility location} (\mfl) problem. We are given a set of facilities and clients located in a common metric space. The goal is to move each facility from its initial location to a destination and assign each client to the destination of some facility so as to minimize the sum of the movement-costs of the facilities and the client-assignment costs. This abstracts facility-location settings where one has the flexibility of moving facilities from their current locations to other destinations so as to serve clients more efficiently by reducing their assignment costs. We give the first {\em local-search based} approximation algorithm for this problem and achieve the best-known approximation guarantee. Our main result is $(3+ε)$-approximation for this problem for any constant $ε>0$ using local search. The previous best guarantee was an 8-approximation algorithm based on LP-rounding. Our guarantee {\em matches} the best-known approximation guarantee for the $k$-median problem. Since there is an approximation-preserving reduction from the $k$-median problem to \mfl, any improvement of our result would imply an analogous improvement for the $k$-median problem. Furthermore, {\em our analysis is tight} (up to $o(1)$ factors) since the tight example for the local-search based 3-approximation algorithm for $k$-median can be easily adapted to show that our local-search algorithm has a tight approximation ratio of 3. One of the chief novelties of the analysis is that in order to generate a suitable collection of local-search moves whose resulting inequalities yield the desired bound on the cost of a local-optimum, we define a tree-like structure that (loosely speaking) functions as a "recursion tree", using which we spawn off local-search moves by exploring this tree to a constant depth.

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arXiv2012

Approximating Minimum-Cost Connected T-Joins

We design and analyse approximation algorithms for the minimum-cost connected T-join problem: given an undirected graph G = (V;E) with nonnegative costs on the edges, and a subset of nodes T, find (if it exists) a spanning connected subgraph H of minimum cost such that every node in T has odd degree and every node not in T has even degree; H may have multiple copies of any edge of G. Two well-known special cases are the TSP (|T| = 0) and the s-t path TSP (|T| = 2). Recently, An, Kleinberg, and Shmoys [STOC 2012] improved on the long-standing 5/3-approximation guarantee for the latter problem and presented an algorithm based on LP rounding that achieves an approximation guarantee of (1+sqrt(5))/2 < 1.6181. We show that the methods of An et al. extend to the minimum-cost connected T-join problem. They presented a new proof for a 5/3-approximation guarantee for the s-t path TSP; their proof extends easily to the minimum-cost connected T-join problem. Next, we improve on the approximation guarantee of 5/3 by extending their LP-rounding algorithm to get an approximation guarantee of 13/8 = 1.625 for all |T| >= 4. Finally, we focus on the prize-collecting version of the problem, and present a primal-dual algorithm that is "Lagrangian multiplier preserving" and that achieves an approximation guarantee 3 - 2/(|T|-1) when |T| >= 4. Our primal-dual algorithm is a generalization of the known primal-dual 2-approximation for the prize-collecting s-t path TSP. Furthermore, we show that our analysis is tight by presenting instances with |T| >= 4 such that the cost of the solution found by the algorithm is exactly 3 - 2/(|T|-1) times the cost of the constructed dual solution.

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arXiv2011

Multiple Traveling Salesmen in Asymmetric Metrics

We consider some generalizations of the Asymmetric Traveling Salesman Path problem. Suppose we have an asymmetric metric G = (V,A) with two distinguished nodes s,t. We are also given a positive integer k. The goal is to find k paths of minimum total cost from s to t whose union spans all nodes. We call this the k-Person Asymmetric Traveling Salesmen Path problem (k-ATSPP). Our main result for k-ATSPP is a bicriteria approximation that, for some parameter b >= 1 we may choose, finds between k and k + k/b paths of total length O(b log |V|) times the optimum value of an LP relaxation based on the Held-Karp relaxation for the Traveling Salesman problem. On one extreme this is an O(log |V|)-approximation that uses up to 2k paths and on the other it is an O(k log |V|)-approximation that uses exactly k paths. Next, we consider the case where we have k pairs of nodes (s_1,t_1), ..., (s_k,t_k). The goal is to find an s_i-t_i path for every pair such that each node of G lies on at least one of these paths. Simple approximation algorithms are presented for the special cases where the metric is symmetric or where s_i = t_i for each i. We also show that the problem can be approximated within a factor O(log n) when k=2. On the other hand, we demonstrate that the general problem cannot be approximated within any bounded ratio unless P = NP.

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arXiv2010

Asymmetric Traveling Salesman Path and Directed Latency Problems

We study integrality gaps and approximability of two closely related problems on directed graphs. Given a set V of n nodes in an underlying asymmetric metric and two specified nodes s and t, both problems ask to find an s-t path visiting all other nodes. In the asymmetric traveling salesman path problem (ATSPP), the objective is to minimize the total cost of this path. In the directed latency problem, the objective is to minimize the sum of distances on this path from s to each node. Both of these problems are NP-hard. The best known approximation algorithms for ATSPP had ratio O(log n) until the very recent result that improves it to O(log n/ log log n). However, only a bound of O(sqrt(n)) for the integrality gap of its linear programming relaxation has been known. For directed latency, the best previously known approximation algorithm has a guarantee of O(n^(1/2+eps)), for any constant eps > 0. We present a new algorithm for the ATSPP problem that has an approximation ratio of O(log n), but whose analysis also bounds the integrality gap of the standard LP relaxation of ATSPP by the same factor. This solves an open problem posed by Chekuri and Pal [2007]. We then pursue a deeper study of this linear program and its variations, which leads to an algorithm for the k-person ATSPP (where k s-t paths of minimum total length are sought) and an O(log n)-approximation for the directed latency problem.

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