arXiv subjects

Ohad Trabelsi

Publications and source records attributed to Ohad Trabelsi.

17 recordsLinked to original sources

arXiv2026

A Note on Approximability of Densest At-Least-k-Subgraph

We study the Densest At-Least-$k$-Subgraph (DAL$k$S) problem, in which we are given an undirected graph $G$ and an integer $k$, and the goal is to find a subgraph of $G$ with at least $k$ vertices with maximum density. The best-known algorithm, independently discovered by Khuller and Saha (2009) and by Andersen (2007), yields a 2-approximation for DAL$k$S in polynomial time. In this note, we provide a (simple) reduction from Densest $k$-Subgraph (D$k$S) to Densest At-Least-$k$-Subgraph, which shows that, if D$k$S is hard to approximate to within any constant factor, then DAL$k$S is hard to approximate to within $(3/2 - \varepsilon)$ factor for every $\varepsilon > 0$. This holds in both the normal (non-parameterized) and the parameterized (by $k$) settings. We then generalize the reduction to provide a tight $(2 - \varepsilon)$ factor hardness of approximating Densest At-Least-$k$-Subgraph, albeit under a stronger hypothesis which roughly states that Densest $k$-Subgraph is hard to approximate to within $k^{1 - δ}$ factor for any constant $δ> 0$. Once again, this extends naturally to the parameterized setting. Previously, $(2 - \varepsilon)$ factor inapproximability for DAL$k$S was only known under the Small Set Expansion Hypothesis (Bergner, 2013; Manurangsi, 2017), which does not apply to the parameterized version of the problem. Furthermore, we show that the exact version of DAL$k$S is W[1]-hard (parameterized by $k$).

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arXiv2025

Faster Algorithms for Global Minimum Vertex-Cut in Directed Graphs

We study the directed global minimum vertex-cut problem: given a directed vertex-weighted graph $G$, compute a vertex-cut $(L,S,R)$ in $G$ of minimum value, which is defined to be the total weight of all vertices in $S$. The problem, together with its edge-based variant, is one of the most basic in graph theory and algorithms, and has been studied extensively. The fastest currently known algorithm for directed global minimum vertex-cut (Henzinger, Rao and Gabow, FOCS 1996 and J. Algorithms 2000) has running time $\tilde{O}(mn)$, where $m$ and $n$ denote the number of edges and vertices in the input graph, respectively. A long line of work over the past decades led to faster algorithms for other main versions of the problem, including the undirected edge-based setting (Karger, STOC 1996 and J. ACM 2000), directed edge-based setting (Cen et al., FOCS 2021), and undirected vertex-based setting (Chuzhoy and Trabelsi, STOC 2025). However, for the vertex-based version in directed graphs, the 29 year-old $\tilde{O}(mn)$-time algorithm of Henzinger, Rao and Gabow remains the state of the art to this day, in all edge-density regimes. In this paper we break the $Θ(mn)$ running time barrier for the first time, by providing a randomized algorithm for directed global minimum vertex-cut, with running time $O\left(mn^{0.976}\cdot\operatorname{polylog} W\right)$ where $W$ is the ratio of largest to smallest vertex weight. Additionally, we provide a randomized $O\left(\min\left\{m^{1+o(1)}\cdot k,n^{2+o(1)}\right\}\right)$-time algorithm for the unweighted version of directed global minimum vertex-cut, where $k$ is the value of the optimal solution. The best previous algorithm for the problem achieved running time $\tilde O\left(\min\left\{k^2 \cdot m, mn^{11/12+o(1)}, n^{2+o(1)}\right\}\right)$ (Forster et al., SODA 2020, Li et al., STOC 2021).

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arXiv2025

Breaking the O(mn)-Time Barrier for Vertex-Weighted Global Minimum Cut

We consider the Global Minimum Vertex-Cut problem: given an undirected vertex-weighted graph $G$, compute a minimum-weight subset of its vertices whose removal disconnects $G$. The problem is closely related to Global Minimum Edge-Cut, where the weights are on the graph edges instead of vertices, and the goal is to compute a minimum-weight subset of edges whose removal disconnects the graph. Global Minimum Cut is one of the most basic and extensively studied problems in combinatorial optimization and graph theory. While an almost-linear time algorithm was known for the edge version of the problem for awhile (Karger, STOC 1996 and J. ACM 2000), the fastest previous algorithm for the vertex version (Henzinger, Rao and Gabow, FOCS 1996 and J. Algorithms 2000) achieves a running time of $\tilde{O}(mn)$, where $m$ and $n$ denote the number of edges and vertices in the input graph, respectively. For the special case of unit vertex weights, this bound was broken only recently (Li {et al.}, STOC 2021); their result, combined with the recent breakthrough almost-linear time algorithm for Maximum $s$-$t$ Flow (Chen {et al.}, FOCS 2022, van den Brand {et al.}, FOCS 2023), yields an almost-linear time algorithm for Global Minimum Vertex-Cut with unit vertex weights. In this paper we break the $28$ years old bound of Henzinger {et al.} for the general weighted Global Minimum Vertex-Cut, by providing a randomized algorithm for the problem with running time $O(\min\{mn^{0.99+o(1)},m^{1.5+o(1)}\})$.

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arXiv2024

(Almost) Ruling Out SETH Lower Bounds for All-Pairs Max-Flow

The All-Pairs Max-Flow problem has gained significant popularity in the last two decades, and many results are known regarding its fine-grained complexity. Despite this, wide gaps remain in our understanding of the time complexity for several basic variants of the problem. In this paper, we aim to bridge these gaps by providing algorithms, conditional lower bounds, and non-reducibility results. Notably, we show that for most problem settings, deterministic reductions based on the Strong Exponential Time Hypothesis (SETH) cannot rule out $O(n^{4-\varepsilon})$ time algorithms for some small $\varepsilon>0$, under a hypothesis called NSETH. To obtain our result for the setting of undirected graphs with unit node-capacities, we design a new randomized Las Vegas $O(m^{2+o(1)})$ time combinatorial algorithm. This is our main technical result, improving over the recent $O(m^{11/5+o(1)})$ time Monte Carlo algorithm [Huang et al., STOC 2023] and matching their $m^{2-o(1)}$ lower bound (up to subpolynomial factors), thus essentially settling the time complexity for this setting of the problem.

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arXiv2023

Bridge Girth: A Unifying Notion in Network Design

A classic 1993 paper by Althőfer et al. proved a tight reduction from spanners, emulators, and distance oracles to the extremal function $γ$ of high-girth graphs. This paper initiated a large body of work in network design, in which problems are attacked by reduction to $γ$ or the analogous extremal function for other girth concepts. In this paper, we introduce and study a new girth concept that we call the bridge girth of path systems, and we show that it can be used to significantly expand and improve this web of connections between girth problems and network design. We prove two kinds of results: 1) We write the maximum possible size of an $n$-node, $p$-path system with bridge girth $>k$ as $β(n, p, k)$, and we write a certain variant for "ordered" path systems as $β^*(n, p, k)$. We identify several arguments in the literature that implicitly show upper or lower bounds on $β, β^*$, and we provide some polynomially improvements to these bounds. In particular, we construct a tight lower bound for $β(n, p, 2)$, and we polynomially improve the upper bounds for $β(n, p, 4)$ and $β^*(n, p, \infty)$. 2) We show that many state-of-the-art results in network design can be recovered or improved via black-box reductions to $β$ or $β^*$. Examples include bounds for distance/reachability preservers, exact hopsets, shortcut sets, the flow-cut gaps for directed multicut and sparsest cut, an integrality gap for directed Steiner forest. We believe that the concept of bridge girth can lead to a stronger and more organized map of the research area. Towards this, we leave many open problems, related to both bridge girth reductions and extremal bounds on the size of path systems with high bridge girth.

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arXiv2022

Conditional Lower Bounds for All-Pairs Max-Flow

We provide evidence that computing the maximum flow value between every pair of nodes in a directed graph on $n$ nodes, $m$ edges,and capacities in the range $[1..n]$, which we call the All-Pairs Max-Flow problem, cannot be solved in time that is significantly faster (i.e., by a polynomial factor) than $O(n^3)$ even for sparse graphs. Since a single maximum $st$-flow can be solved in time $\tilde{O}(m\sqrt{n})$ [Lee and Sidford, FOCS 2014], we conclude that the all-pairs version might require time equivalent to $\tildeΩ(n^{3/2})$ computations of maximum $st$-flow,which strongly separates the directed case from the undirected one. Moreover, if maximum $st$-flow can be solved in time $\tilde{O}(m)$,then the runtime of $\tildeΩ(n^2)$ computations is needed. The latter settles a conjecture of Lacki, Nussbaum, Sankowski, and Wulf-Nilsen [FOCS 2012] negatively. Specifically, we show that in sparse graphs $G=(V,E,w)$, if one can compute the maximum $st$-flow from every $s$ in an input set of sources $S\subseteq V$ to every $t$ in an input set of sinks $T\subseteq V$ in time $O((|S| |T| m)^{1-ε})$,for some $|S|$, $|T|$, and a constant $ε>0$,then MAX-CNF-SAT with $n'$ variables and $m'$ clauses can be solved in time ${m'}^{O(1)}2^{(1-δ)n'}$ for a constant $δ(ε)>0$,a problem for which not even $2^{n'}/poly(n')$ algorithms are known. Such runtime for MAX-CNF-SAT would in particular refute the Strong Exponential Time Hypothesis (SETH). Hence, we improve the lower bound of Abboud, Vassilevska-Williams, and Yu [STOC 2015], who showed that for every fixed $ε>0$ and $|S|=|T|=O(\sqrt{n})$, if the above problem can be solved in time $O(n^{3/2-ε})$, then some incomparable conjecture is false. Furthermore, a larger lower bound than ours implies strictly super-linear time for maximum $st$-flow problem, which would be an amazing breakthrough.

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arXiv2022

Breaking the Cubic Barrier for All-Pairs Max-Flow: Gomory-Hu Tree in Nearly Quadratic Time

In 1961, Gomory and Hu showed that the All-Pairs Max-Flow problem of computing the max-flow between all $n\choose 2$ pairs of vertices in an undirected graph can be solved using only $n-1$ calls to any (single-pair) max-flow algorithm. Even assuming a linear-time max-flow algorithm, this yields a running time of $O(mn)$, which is $O(n^3)$ when $m = Θ(n^2)$. While subsequent work has improved this bound for various special graph classes, no subcubic-time algorithm has been obtained in the last 60 years for general graphs. We break this longstanding barrier by giving an $\tilde{O}(n^{2})$-time algorithm on general, weighted graphs. Combined with a popular complexity assumption, we establish a counter-intuitive separation: all-pairs max-flows are strictly easier to compute than all-pairs shortest-paths. Our algorithm produces a cut-equivalent tree, known as the Gomory-Hu tree, from which the max-flow value for any pair can be retrieved in near-constant time. For unweighted graphs, we refine our techniques further to produce a Gomory-Hu tree in the time of a poly-logarithmic number of calls to any max-flow algorithm. This shows an equivalence between the all-pairs and single-pair max-flow problems, and is optimal up to poly-logarithmic factors. Using the recently announced $m^{1+o(1)}$-time max-flow algorithm (Chen et al., March 2022), our Gomory-Hu tree algorithm for unweighted graphs also runs in $m^{1+o(1)}$-time.

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arXiv2021

Friendly Cut Sparsifiers and Faster Gomory-Hu Trees

We devise new cut sparsifiers that are related to the classical sparsification of Nagamochi and Ibaraki [Algorithmica, 1992], which is an algorithm that, given an unweighted graph $G$ on $n$ nodes and a parameter $k$, computes a subgraph with $O(nk)$ edges that preserves all cuts of value up to $k$. We put forward the notion of a friendly cut sparsifier, which is a minor of $G$ that preserves all friendly cuts of value up to $k$, where a cut in $G$ is called friendly if every node has more edges connecting it to its own side of the cut than to the other side. We present an algorithm that, given a simple graph $G$, computes in almost-linear time a friendly cut sparsifier with $\tilde{O}(n \sqrt{k})$ edges. Using similar techniques, we also show how, given in addition a terminal set $T$, one can compute in almost-linear time a terminal sparsifier, which preserves the minimum $st$-cut between every pair of terminals, with $\tilde{O}(n \sqrt{k} + |T| k)$ edges. Plugging these sparsifiers into the recent $n^{2+o(1)}$-time algorithms for constructing a Gomory-Hu tree of simple graphs, along with a relatively simple procedure for handling the unfriendly minimum cuts, we improve the running time for moderately dense graphs (e.g., with $m=n^{1.75}$ edges). In particular, assuming a linear-time Max-Flow algorithm, the new state-of-the-art for Gomory-Hu tree is the minimum between our $(m+n^{1.75})^{1+o(1)}$ and the known $m n^{1/2+o(1)}$. We further investigate the limits of this approach and the possibility of better sparsification. Under the hypothesis that an $\tilde{O}(n)$-edge sparsifier that preserves all friendly minimum $st$-cuts can be computed efficiently, our upper bound improves to $\tilde{O}(m+n^{1.5})$ which is the best possible without breaking the cubic barrier for constructing Gomory-Hu trees in non-simple graphs.

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arXiv2021

APMF < APSP? Gomory-Hu Tree for Unweighted Graphs in Almost-Quadratic Time

We design an $n^{2+o(1)}$-time algorithm that constructs a cut-equivalent (Gomory-Hu) tree of a simple graph on $n$ nodes. This bound is almost-optimal in terms of $n$, and it improves on the recent $\tilde{O}(n^{2.5})$ bound by the authors (STOC 2021), which was the first to break the cubic barrier. Consequently, the All-Pairs Maximum-Flow (APMF) problem has time complexity $n^{2+o(1)}$, and for the first time in history, this problem can be solved faster than All-Pairs Shortest Paths (APSP). We further observe that an almost-linear time algorithm (in terms of the number of edges $m$) is not possible without first obtaining a subcubic algorithm for multigraphs. Finally, we derandomize our algorithm, obtaining the first subcubic deterministic algorithm for Gomory-Hu Tree in simple graphs, showing that randomness is not necessary for beating the $n-1$ times max-flow bound from 1961. The upper bound is $\tilde{O}(n^{2\frac{2}{3}})$ and it would improve to $n^{2+o(1)}$ if there is a deterministic single-pair maximum-flow algorithm that is almost-linear. The key novelty is in using a ``dynamic pivot'' technique instead of the randomized pivot selection that was central in recent works.

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arXiv2021

Subcubic Algorithms for Gomory-Hu Tree in Unweighted Graphs

Every undirected graph $G$ has a (weighted) cut-equivalent tree $T$, commonly named after Gomory and Hu who discovered it in 1961. Both $T$ and $G$ have the same node set, and for every node pair $s,t$, the minimum $(s,t)$-cut in $T$ is also an exact minimum $(s,t)$-cut in $G$. We give the first subcubic-time algorithm that constructs such a tree for a simple graph $G$ (unweighted with no parallel edges). Its time complexity is $\tilde{O}(n^{2.5})$, for $n=|V(G)|$; previously, only $\tilde{O}(n^3)$ was known, except for restricted cases like sparse graphs. Consequently, we obtain the first algorithm for All-Pairs Max-Flow in simple graphs that breaks the cubic-time barrier. Gomory and Hu compute this tree using $n-1$ queries to (single-pair) Max-Flow; the new algorithm can be viewed as a fine-grained reduction to $\tilde{O}(\sqrt{n})$ Max-Flow computations on $n$-node graphs.

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arXiv2020

Cut-Equivalent Trees are Optimal for Min-Cut Queries

Min-Cut queries are fundamental: Preprocess an undirected edge-weighted graph, to quickly report a minimum-weight cut that separates a query pair of nodes $s,t$. The best data structure known for this problem simply builds a cut-equivalent tree, discovered 60 years ago by Gomory and Hu, who also showed how to construct it using $n-1$ minimum $st$-cut computations. Using state-of-the-art algorithms for minimum $st$-cut (Lee and Sidford, FOCS 2014) arXiv:1312.6713, one can construct the tree in time $\tilde{O}(mn^{3/2})$, which is also the preprocessing time of the data structure. (Throughout, we focus on polynomially-bounded edge weights, noting that faster algorithms are known for small/unit edge weights.) Our main result shows the following equivalence: Cut-equivalent trees can be constructed in near-linear time if and only if there is a data structure for Min-Cut queries with near-linear preprocessing time and polylogarithmic (amortized) query time, and even if the queries are restricted to a fixed source. That is, equivalent trees are an essentially optimal solution for Min-Cut queries. This equivalence holds even for every minor-closed family of graphs, such as bounded-treewidth graphs, for which a two-decade old data structure (Arikati et al., J.~Algorithms 1998) implies the first near-linear time construction of cut-equivalent trees. Moreover, unlike all previous techniques for constructing cut-equivalent trees, ours is robust to relying on approximation algorithms. In particular, using the almost-linear time algorithm for $(1+ε)$-approximate minimum $st$-cut (Kelner et al., SODA 2014), we can construct a $(1+ε)$-approximate flow-equivalent tree (which is a slightly weaker notion) in time $n^{2+o(1)}$. This leads to the first $(1+ε)$-approximation for All-Pairs Max-Flow that runs in time $n^{2+o(1)}$, and matches the output size almost-optimally.

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arXiv2019

New Algorithms and Lower Bounds for All-Pairs Max-Flow in Undirected Graphs

We investigate the time-complexity of the All-Pairs Max-Flow problem: Given a graph with $n$ nodes and $m$ edges, compute for all pairs of nodes the maximum-flow value between them. If Max-Flow (the version with a given source-sink pair $s,t$) can be solved in time $T(m)$, then an $O(n^2) \cdot T(m)$ is a trivial upper bound. But can we do better? For directed graphs, recent results in fine-grained complexity suggest that this time bound is essentially optimal. In contrast, for undirected graphs with edge capacities, a seminal algorithm of Gomory and Hu (1961) runs in much faster time $O(n)\cdot T(m)$. Under the plausible assumption that Max-Flow can be solved in near-linear time $m^{1+o(1)}$, this half-century old algorithm yields an $nm^{1+o(1)}$ bound. Several other algorithms have been designed through the years, including $\tilde{O}(mn)$ time for unit-capacity edges (unconditionally), but none of them break the $O(mn)$ barrier. Meanwhile, no super-linear lower bound was shown for undirected graphs. We design the first hardness reductions for All-Pairs Max-Flow in undirected graphs, giving an essentially optimal lower bound for the $\textit{node-capacities}$ setting. For edge capacities, our efforts to prove similar lower bounds have failed, but we have discovered a surprising new algorithm that breaks the $O(mn)$ barrier for graphs with unit-capacity edges! Assuming $T(m)=m^{1+o(1)}$, our algorithm runs in time $m^{3/2 +o(1)}$ and outputs a cut-equivalent tree (similarly to the Gomory-Hu algorithm). Even with current Max-Flow algorithms we improve state-of-the-art as long as $m=O(n^{5/3-\varepsilon})$. Finally, we explain the lack of lower bounds by proving a $\textit{non-reducibility}$ result. This result is based on a new quasi-linear time $\tilde{O}(m)$ $\textit{non-deterministic}$ algorithm for constructing a cut-equivalent tree and may be of independent interest.

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arXiv2019

Faster Algorithms for All-Pairs Bounded Min-Cuts

The All-Pairs Min-Cut problem (aka All-Pairs Max-Flow) asks to compute a minimum $s$-$t$ cut (or just its value) for all pairs of vertices $s,t$. We study this problem in directed graphs with unit edge/vertex capacities (corresponding to edge/vertex connectivity). Our focus is on the $k$-bounded case, where the algorithm has to find all pairs with min-cut value less than $k$, and report only those. The most basic case $k=1$ is the Transitive Closure (TC) problem, which can be solved in graphs with $n$ vertices and $m$ edges in time $O(mn)$ combinatorially, and in time $O(n^ω)$ where $ω<2.38$ is the matrix-multiplication exponent. These time bounds are conjectured to be optimal. We present new algorithms and conditional lower bounds that advance the frontier for larger $k$, as follows: (i) A randomized algorithm for vertex capacities that runs in time $O((nk)^ω)$. (ii) Two deterministic algorithms for edge capacities (which is more general) that work in DAGs and further reports a minimum cut for each pair. The first algorithm is combinatorial (does not involve matrix multiplication) and runs in time $O(2^{O(k^2)}\cdot mn)$. The second algorithm can be faster on dense DAGs and runs in time $O((k\log n)^{4^k+o(k)} n^ω)$. (iii) The first super-cubic lower bound of $n^{ω-1-o(1)} k^2$ time under the $4$-Clique conjecture, which holds even in the simplest case of DAGs with unit vertex capacities. It improves on the previous (SETH-based) lower bounds even in the unbounded setting $k=n$. For combinatorial algorithms, our reduction implies an $n^{2-o(1)} k^2$ conditional lower bound. Thus, we identify new settings where the complexity of the problem is (conditionally) higher than that of TC.

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arXiv2019

Nearly Optimal Time Bounds for kPath in Hypergraphs

We give almost tight conditional lower bounds on the running time of the kHyperPath problem. Given an $r$-uniform hypergraph for some integer $r$, kHyperPath seeks a tight path of length $k$. That is, a sequence of $k$ nodes such that every consecutive $r$ of them constitute a hyperedge in the graph. This problem is a natural generalization of the extensively-studied kPath problem in graphs. We show that solving kHyperPath in time $O^*(2^{(1-γ)k})$ where $γ>0$ is independent of $r$ is probably impossible. Specifically, it implies that Set Cover on $n$ elements can be solved in time $O^*(2^{(1 - δ)n})$ for some $δ>0$. The only known lower bound for the kPath problem is $2^{Ω(k)} poly(n)$ where $n$ is the number of nodes assuming the Exponential Time Hypothesis (ETH), and finding any conditional lower bound with an explicit constant in the exponent has been an important open problem. We complement our lower bound with an almost tight upper bound. Formally, for every integer $r\geq 3$ we give algorithms that solve kHyperPath and kHyperCycle on $r$-uniform hypergraphs with $n$ nodes and $m$ edges in time $2^k m \cdot poly(n)$ and $2^k m^2 poly(n)$ respectively, and that is even for the directed version of these problems. To the best of our knowledge, this is the first algorithm for kHyperPath. The fastest algorithms known for kPath run in time $2^k poly(n)$ for directed graphs (Williams, 2009), and in time $1.66^k poly(n)$ for undirected graphs (Björklund \etal, 2014).

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arXiv2019

The Set Cover Conjecture and Subgraph Isomorphism with a Tree Pattern

In the Set Cover problem, the input is a ground set of $n$ elements and a collection of $m$ sets, and the goal is to find the smallest sub-collection of sets whose union is the entire ground set. The fastest algorithm known runs in time $O(mn2^n)$ [Fomin et al., WG 2004], and the Set Cover Conjecture (SeCoCo) [Cygan et al., TALG 2016] asserts that for every fixed $\varepsilon>0$, no algorithm can solve Set Cover in time $2^{(1-\varepsilon)n}poly(m)$, even if set sizes are bounded by $Δ=Δ(\varepsilon)$. We show strong connections between this problem and kTree, a special case of Subgraph Isomorphism where the input is an $n$-node graph $G$ and a $k$-node tree $T$, and the goal is to determine whether $G$ has a subgraph isomorphic to $T$. First, we propose a weaker conjecture Log-SeCoCo, that allows input sets of size $Δ=O(1/\varepsilon \cdot\log n)$, and show that an algorithm breaking Log-SeCoCo would imply a faster algorithm than the currently known $2^n poly(n)$-time algorithm [Koutis and Williams, TALG 2016] for Directed nTree, which is kTree with $k=n$ and arbitrary directions to the edges of $G$ and $T$. This would also improve the running time for Directed Hamiltonicity, for which no algorithm significantly faster than $2^n poly(n)$ is known despite extensive research. Second, we prove that if Set Cover cannot be solved significantly faster than $2^npoly(m)$ (an assumption even weaker than Log-SeCoCo), then kTree cannot be computed significantly faster than $2^kpoly(n)$, the running time of the Koutis and Williams' algorithm. Applying the same techniques to the p-Partial Cover problem, a parameterized version of Set Cover that requires covering at least $p$ elements, we obtain a new algorithm with running time $(2+\varepsilon)^p (m+n)^{O(1/\varepsilon)}$ for arbitrary $\varepsilon>0$, which improves previous work and is nearly optimal assuming say Log-SeCoCo.

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arXiv2018

Relaxed Voronoi: a Simple Framework for Terminal-Clustering Problems

We reprove three known algorithmic bounds for terminal-clustering problems, using a single framework that leads to simpler proofs. In this genre of problems, the input is a metric space $(X,d)$ (possibly arising from a graph) and a subset of terminals $K\subset X$, and the goal is to partition the points $X$ such that each part, called a cluster, contains exactly one terminal (possibly with connectivity requirements) so as to minimize some objective. The three bounds we reprove are for Steiner Point Removal on trees [Gupta, SODA 2001], for Metric $0$-Extension in bounded doubling dimension [Lee and Naor, unpublished 2003], and for Connected Metric $0$-Extension [Englert et al., SICOMP 2014]. A natural approach is to cluster each point with its closest terminal, which would partition $X$ into so-called Voronoi cells, but this approach can fail miserably due to its stringent cluster boundaries. A now-standard fix, which we call the Relaxed-Voronoi framework, is to use enlarged Voronoi cells, but to obtain disjoint clusters, the cells are computed greedily according to some order. This method, first proposed by Calinescu, Karloff and Rabani [SICOMP 2004], was employed successfully to provide state-of-the-art results for terminal-clustering problems on general metrics. However, for restricted families of metrics, e.g., trees and doubling metrics, only more complicated, ad-hoc algorithms are known. Our main contribution is to demonstrate that the Relaxed-Voronoi algorithm is applicable to restricted metrics, and actually leads to relatively simple algorithms and analyses.

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arXiv2018

Conditional Lower Bound for Subgraph Isomorphism with a Tree Pattern

The kTree problem is a special case of Subgraph Isomorphism where the pattern graph is a tree, that is, the input is an $n$-node graph $G$ and a $k$-node tree $T$, and the goal is to determine whether $G$ has a subgraph isomorphic to $T$. We provide evidence that this problem cannot be computed significantly faster than $2^{k} \textsf{poly}(n)$, which matches the fastest algorithm known for this problem by Koutis and Williams [ICALP 2009 and TALG 2016]. Specifically, we show that if kTree can be solved in time $(2-\varepsilon)^k \textsf{poly}(n)$ for some constant $\varepsilon>0$, then Set Cover with $n'$ elements and $m'$ sets can be solved in time $(2-δ)^{n'} \textsf{poly}(m')$ for a constant $δ(\varepsilon) > 0$, which would refute the Set Cover Conjecture by Cygan et al. [CCC 2012 and TALG 2016]. Our techniques yield a new algorithm for the p-Partial Cover problem, a parameterized version of Set Cover that requires covering at least $p$ elements (rather than all elements). Its running time is $(2+\varepsilon)^p (m')^{O(1/\varepsilon)}$ for any fixed $\varepsilon>0$, which improves the previous $2.597^p \textsf{poly}(m')$-time algorithm by Zehavi [ESA 2015]. Our running time is nearly optimal, as a $(2-\varepsilon')^p \textsf{poly}(m')$-time algorithm would refute the Set Cover Conjecture.

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