arXiv subjects

Ramin Mousavi

Publications and source records attributed to Ramin Mousavi.

5 recordsLinked to original sources

arXiv2026

A Constant-Factor Approximation for Directed Latency

In the Directed Latency problem, we are given an asymmetric metric space $(V \cup \{s\},c)$ on a set $V$ of clients and a depot $s$. We are looking for a path $P$ starting in $s$ that visits all clients and minimizes the sum of the clients' waiting times (also known as latency) before being visited on the path. In contrast to the symmetric version of this problem (also known as the Deliveryperson problem and the Repairperson problem in the literature), there are significant gaps in our understanding of Directed Latency. The best approximation factor has remained at $O(\log |V|)$, as shown by [Friggstad, Salavatipour, and Svitkina, '13], for more than a decade. Only recently, [Friggstad and Swamy, '22] presented a constant-factor approximation but in quasi-polynomial time. Both results follow similar ideas: they consider buckets with geometrically increasing distances, build a path on each bucket, and then stitch together all these paths to get a feasible solution. [Friggstad and Swamy, '22] showed that by guessing a vertex from each bucket and augmenting a standard LP relaxation with these guesses, one can reduce the stitching cost. Unfortunately, the number of buckets is logarithmic in the number of vertices, so the running time of their algorithm is quasi-polynomial. In this paper, we present the first constant-factor approximation for Directed Latency in polynomial time by introducing a completely new way of bucketing, which helps us strengthen a standard LP relaxation with less aggressive guessing. Although the resulting LP is no longer a relaxation of Directed Latency, it still admits a good solution. We present a rounding algorithm for fractional solutions of our LP, crucially exploiting the way we restricted the feasibility region of the LP formulation.

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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.

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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.

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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.

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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 $α= 1.5$).

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