arXiv subjects

Guy Kortsarz

Publications and source records attributed to Guy Kortsarz.

At least 19 recordsLinked to original sources

arXiv2026

A $4/3$ ratio approximation algorithm for the Tree Augmentation Problem by deferred local-ratio and climbing

The \emph{Tree Augmentation Problem (TAP)} is given a tree $T=(V,E_T)$ and additional set of {\em links} $E$ on $V\times V$, find $F \subseteq E$ such that $T \cup F$ is $2$-edge-connected, and $|F|$ is minimum. The problem is APX-hard \cite{r} even in if links are only between leaves \cite{r}. The best known approximation ratio for TAP is $1.393$, due to Traub and Zenklusen~\cite{tr1} J.~ACM,~2025 using the {\em relative greedy} technique \cite{zel}. \noindent We introduce a new technique called the {\em deferred local ratio technique}. In this technique, the disjointness of the local-ratio primal-dual type does not hold. The technique applies Set Cover problem under certain conditions (see Section \ref{lr}). We use it provide a We use it to provide a $4/3$ approximation algorithm for TAP. It is possible this technique will find future applications. The running time is The running time is $O(m\cdot\sqrt{n})$ time \cite{vaz}, \cite{vaz1}. Faster than \cite{tr1} \cite{LS} and LP based algorithms as we do not enumeratestructures of size $exp(Θ(f(1/ε)\cdot \log n)).$ Nor do we scale and round. \noindent \cite{ed} has an implementation \cite{kol} that is extensively used in the industry.

cs.CC↗

arXiv2026

The Telephone $k$-Multicast Problem

We consider minimum time multicasting problems in directed and undirected graphs: given a root node and a subset of $t$ terminal nodes, multicasting seeks to find the minimum number of rounds within which all terminals can be informed with a message originating at the root. In each round, the telephone model we study allows the information to move via a matching from the informed nodes to the uninformed nodes. Since minimum time multicasting in digraphs is poorly understood compared to the undirected variant, we study an intermediate problem in undirected graphs that specifies a target $k < t$, and requires that only $k$ of the terminals be informed in the minimum number of rounds. For this problem, we improve the implications of the previous results and obtain a multiplicative approximation factor of $\tilde{O}(t^{1/3})$. For the directed version, we obtain an additive $\tilde{O}(k^{1/2})$ approximation algorithm (with a polylogarithmic multiplicative factor). Our algorithms are based on reductions to the related problems of finding $k$-trees of minimum poise (sum of maximum degree and diameter) and applying a combination of greedy network decomposition techniques and set covering under partition matroid constraints. We also study the problem of bounded degree Directed Steiner Tree, for which we obtain improved polylogarithmic approximations for the special case of bounded treewidth graphs. This extends prior work on the Group Steiner Tree problem.

cs.DS↗

arXiv2024

Approximations and Hardness of Packing Partially Ordered Items

Motivated by applications in production planning and storage allocation in hierarchical databases, we initiate the study of covering partially ordered items (CPO). Given a capacity $k \in \mathbb{Z}^+$, and a directed graph $G=(V,E)$ where each vertex has a size in $\{0,1, \ldots,k\}$, we seek a collection of subsets of vertices $S_1, \ldots, S_m$ that cover all the vertices, such that for any $1 \leq j \leq m$, the total size of vertices in $S_j$ is bounded by $k$, and there are no edges from $V \setminus S_j$ to $S_j$. The objective is to minimize the number of subsets $m$. CPO is closely related to the rule caching problem (RCP) that is of wide interest in the networking area. The input for RCP is a directed graph $G=(V,E)$, a profit function $p:V \rightarrow \mathbb{Z}_{0}^+$, and $k \in \mathbb{Z}^+$. The output is a subset $S \subseteq V$ of maximum profit such that $|S| \leq k$ and there are no edges from $V \setminus S$ to $S$. Our main result is a $2$-approximation algorithm for CPO on out-trees, complemented by an asymptotic $1.5$-hardness of approximation result. We also give a two-way reduction between RCP and the densest $k$-subhypergraph problem, surprisingly showing that the problems are equivalent w.r.t. polynomial-time approximation within any factor $ρ\geq 1$. This implies that RCP cannot be approximated within factor $|V|^{1-\eps}$ for any fixed $\eps>0$, under standard complexity assumptions. Prior to this work, RCP was just known to be strongly NP-hard. We further show that there is no EPTAS for the special case of RCP where the profits are uniform, assuming Gap-ETH. Since this variant admits a PTAS, we essentially resolve the complexity status of this problem.

cs.DS↗

arXiv2023

Improved Approximations for Relative Survivable Network Design

One of the most important and well-studied settings for network design is edge-connectivity requirements. This encompasses uniform demands such as the Minimum $k$-Edge-Connected Spanning Subgraph problem as well as nonuniform demands such as the Survivable Network Design problem (SND). In a recent paper by [Dinitz, Koranteng, Kortsarz APPROX '22] , the authors observed that a weakness of these formulations is that it does not enable one to consider fault-tolerance in graphs that have just a few small cuts. To remedy this, they introduced new variants of these problems under the notion "relative" fault-tolerance. Informally, this requires not that two nodes are connected if there are a bounded number of faults (as in the classical setting), but that two nodes are connected if there are a bounded number of faults and the two nodes are connected in the underlying graph post-faults. The problem is already highly non-trivial even for the case of a single demand. Due to difficulties introduced by this new notion of fault-tolerance, the results in [Dinitz, Koranteng, Kortsarz APPROX '22] are quite limited. For the Relative Survivable Network Design problem (RSND), when the demands are not uniform they give a nontrivial result only when there is a single demand with a connectivity requirement of $3$: a non-optimal $27/4$-approximation. We strengthen this result in two significant ways: We give a $2$-approximation for RSND where all requirements are at most $3$, and a $2^{O(k^2)}$-approximation for RSND with a single demand of arbitrary value $k$. To achieve these results, we first use the "cactus representation'' of minimum cuts to give a lossless reduction to normal SND. Second, we extend the techniques of [Dinitz, Koranteng, Kortsarz APPROX '22] to prove a generalized and more complex version of their structure theorem, which we then use to design a recursive approximation algorithm.

cs.DS↗

arXiv2023

Degrees and Network Design: New Problems and Approximations

While much of network design focuses mostly on cost (number or weight of edges), node degrees have also played an important role. They have traditionally either appeared as an objective, to minimize the maximum degree (e.g., the Minimum Degree Spanning Tree problem), or as constraints which might be violated to give bicriteria approximations (e.g., the Minimum Cost Degree Bounded Spanning Tree problem). We extend the study of degrees in network design in two ways. First, we introduce and study a new variant of the Survivable Network Design Problem where in addition to the traditional objective of minimizing the cost of the chosen edges, we add a constraint that the $\ell_p$-norm of the node degree vector is bounded by an input parameter. This interpolates between the classical settings of maximum degree (the $\ell_{\infty}$-norm) and the number of edges (the $\ell_1$-degree), and has natural applications in distributed systems and VLSI design. We give a constant bicriteria approximation in both measures using convex programming. Second, we provide a polylogrithmic bicriteria approximation for the Degree Bounded Group Steiner problem on bounded treewidth graphs, solving an open problem from [Kortsarz and Nutov, Discret. Appl. Math. 2022] and [Guo et al., Algorithmica 2022].

cs.DS↗

arXiv2022

Relative Survivable Network Design

One of the most important and well-studied settings for network design is edge-connectivity requirements. This encompasses uniform demands such as the Minimum $k$-Edge-Connected Spanning Subgraph problem ($k$-ECSS), as well as nonuniform demands such as the Survivable Network Design problem. A weakness of these formulations, though, is that we are not able to ask for fault-tolerance larger than the connectivity. We introduce and study new variants of these problems under a notion of relative fault-tolerance. Informally, we require not that two nodes are connected if there are a bounded number of faults (as in the classical setting), but that two nodes are connected if there are a bounded number of faults and the two nodes are connected in the underlying graph post-faults. That is, the subgraph we build must "behave" identically to the underlying graph with respect to connectivity after bounded faults. We define and introduce these problems, and provide the first approximation algorithms: a $(1+4/k)$-approximation for the unweighted relative version of $k$-ECSS, a $2$-approximation for the weighted relative version of $k$-ECSS, and a $27/4$-approximation for the special case of Relative Survivable Network Design with only a single demand with a connectivity requirement of $3$. To obtain these results, we introduce a number of technical ideas that may of independent interest. First, we give a generalization of Jain's iterative rounding analysis that works even when the cut-requirement function is not weakly supermodular, but instead satisfies a weaker definition we introduce and term local weak supermodularity. Second, we prove a structure theorem and design an approximation algorithm utilizing a new decomposition based on important separators, which are structures commonly used in fixed-parameter algorithms that have not commonly been used in approximation algorithms.

cs.DS↗

arXiv2020

Tight Bounds on Subexponential Time Approximation of Set Cover and Related Problems

We show that Set Cover on instances with $N$ elements cannot be approximated within $(1-γ)\ln N$-factor in time exp($N^{γ-δ})$, for any $0 < γ< 1$ and any $δ> 0$, assuming the Exponential Time Hypothesis. This essentially matches the best upper bound known by Cygan et al.\ (IPL, 2009) of $(1-γ)\ln N$-factor in time $exp(O(N^γ))$. The lower bound is obtained by extracting a standalone reduction from Label Cover to Set Cover from the work of Moshkovitz (Theory of Computing, 2015), and applying it to a different PCP theorem than done there. We also obtain a tighter lower bound when conditioning on the Projection Games Conjecture. We also treat three problems (Directed Steiner Tree, Submodular Cover, and Connected Polymatroid) that strictly generalize Set Cover. We give a $(1-γ)\ln N$-approximation algorithm for these problems that runs in $exp(\tilde{O}(N^γ))$ time, for any $1/2 \le γ< 1$.

cs.DS↗

arXiv2020

On Approximating Degree-Bounded Network Design Problems

Directed Steiner Tree (DST) is a central problem in combinatorial optimization and theoretical computer science: Given a directed graph $G=(V, E)$ with edge costs $c \in \mathbb{R}_{\geq 0}^E$, a root $r \in V$ and $k$ terminals $K\subseteq V$, we need to output the minimum-cost arborescence in $G$ that contains an $r$\textrightarrow $t$ path for every $t \in K$. Recently, Grandoni, Laekhanukit and Li, and independently Ghuge and Nagarajan, gave quasi-polynomial time $O(\log^2k/\log \log k)$-approximation algorithms for the problem, which are tight under popular complexity assumptions. In this paper, we consider the more general Degree-Bounded Directed Steiner Tree (DB-DST) problem, where we are additionally given a degree bound $d_v$ on each vertex $v \in V$, and we require that every vertex $v$ in the output tree has at most $d_v$ children. We give a quasi-polynomial time $(O(\log n \log k), O(\log^2 n))$-bicriteria approximation: The algorithm produces a solution with cost at most $O(\log n\log k)$ times the cost of the optimum solution that violates the degree constraints by at most a factor of $O(\log^2n)$. This is the first non-trivial result for the problem. While our cost-guarantee is nearly optimal, the degree violation factor of $O(\log^2n)$ is an $O(\log n)$-factor away from the approximation lower bound of $Ω(\log n)$ from the set-cover hardness. The hardness result holds even on the special case of the {\em Degree-Bounded Group Steiner Tree} problem on trees (DB-GST-T). With the hope of closing the gap, we study the question of whether the degree violation factor can be made tight for this special case. We answer the question in the affirmative by giving an $(O(\log n\log k), O(\log n))$-bicriteria approximation algorithm for DB-GST-T.

cs.DS↗

arXiv2019

Bounded Degree Group Steiner Tree Problems

We study two problems that seek a subtree $T$ of a graph $G=(V,E)$ such that $T$ satisfies a certain property and has minimal maximum degree. - In the Min-Degree Group Steiner Tree problem we are given a collection ${\cal S}$ of groups (subsets of $V$) and $T$ should contain a node from every group. - In the Min-Degree Steiner $k$-Tree problem we are given a set $R$ of terminals and an integer $k$, and $T$ should contain at least $k$ terminals. We show that if the former problem admits approximation ratio $ρ$ then the later problem admits approximation ratio $ρ\cdot O(\log k)$. For bounded treewidth graphs, we obtain approximation ratio $O(\log^3 n)$ for Min-Degree Group Steiner Tree. In the more general Bounded Degree Group Steiner Tree problem we are also given edge costs and degree bounds $\{b(v):v \in V\}$, and $T$ should obey the degree constraints $deg_T(v) \leq b(v)$ for all $v \in V$. We give a bicriteria $(O(\log N \log |{\cal S}|),O(\log^2 n))$-approximation algorithm for this problem on tree inputs, where $N$ is the size of the largest group, generalizing the approximation of Garg, Konjevod, and Ravi for the case without degree bounds.

cs.DS↗

arXiv2018

On subexponential running times for approximating directed Steiner tree and related problems

This paper concerns proving almost tight (super-polynomial) running times, for achieving desired approximation ratios for various problems. To illustrate, the question we study, let us consider the Set-Cover problem with n elements and m sets. Now we specify our goal to approximate Set-Cover to a factor of (1-d)ln n, for a given parameter 0 = 2^{n^{c d}}, for some constant 0 = exp((1+o(1)){log^{d-c}n}), for any c>0, unless the ETH is false. Our result follows by analyzing the work of Halperin and Krauthgamer [STOC, 2003]. The same lower and upper bounds hold for CST.

cs.DS↗

arXiv2018

Spanning Trees With Edge Conflicts and Wireless Connectivity

We introduce the problem of finding a spanning tree along with a partition of the tree edges into fewest number of feasible sets, where constraints on the edges define feasibility. The motivation comes from wireless networking, where we seek to model the irregularities seen in actual wireless environments. Not all node pairs may be able to communicate, even if geographically close --- thus, the available pairs are modeled with a link graph $\mathcal{L}=(V,E)$. Also, signal attenuation need not follow a nice geometric formulas --- hence, interference is modeled by a conflict (hyper)graph $\mathcal{C}=(E,F)$ on the links. The objective is to maximize the efficiency of the communication, or equivalently minimizing the length of a schedule of the tree edges in the form of a coloring. We find that in spite of all this generality, the problem can be approximated linearly in terms of a versatile parameter, the inductive independence of the interference graph. Specifically, we give a simple algorithm that attains a $O(ρ\log n)$-approximation, where $n$ is the number of nodes and $ρ$ is the inductive independence, and show that near-linear dependence on $ρ$ is also necessary. We also treat an extension to Steiner trees, modeling multicasting, and obtain a comparable result. Our results suggest that several canonical assumptions of geometry, regularity and "niceness" in wireless settings can sometimes be relaxed without a significant hit in algorithm performance.

cs.NI↗

arXiv2018

Improved approximation algorithm for the Dense-3-Subhypergraph Problem

The study of Dense-$3$-Subhypergraph problem was initiated in Chlamt{á}c et al. [Approx'16]. The input is a universe $U$ and collection ${\cal S}$ of subsets of $U$, each of size $3$, and a number $k$. The goal is to choose a set $W$ of $k$ elements from the universe, and maximize the number of sets, $S\in {\cal S}$ so that $S\subseteq W$. The members in $U$ are called {\em vertices} and the sets of ${\cal S}$ are called the {\em hyperedges}. This is the simplest extension into hyperedges of the case of sets of size $2$ which is the well known Dense $k$-subgraph problem. The best known ratio for the Dense-$3$-Subhypergraph is $O(n^{0.69783..})$ by Chlamt{á}c et al. We improve this ratio to $n^{0.61802..}$. More importantly, we give a new algorithm that approximates Dense-$3$-Subhypergraph within a ratio of $\tilde O(n/k)$, which improves the ratio of $O(n^2/k^2)$ of Chlamt{á}c et al. We prove that under the {\em log density conjecture} (see Bhaskara et al. [STOC'10]) the ratio cannot be better than $Ω(\sqrt{n})$ and demonstrate some cases in which this optimum can be attained.

cs.DS↗

arXiv2017

From Gap-ETH to FPT-Inapproximability: Clique, Dominating Set, and More

We consider questions that arise from the intersection between the areas of polynomial-time approximation algorithms, subexponential-time algorithms, and fixed-parameter tractable algorithms. The questions, which have been asked several times (e.g., [Marx08, FGMS12, DF13]), are whether there is a non-trivial FPT-approximation algorithm for the Maximum Clique (Clique) and Minimum Dominating Set (DomSet) problems parameterized by the size of the optimal solution. In particular, letting $\text{OPT}$ be the optimum and $N$ be the size of the input, is there an algorithm that runs in $t(\text{OPT})\text{poly}(N)$ time and outputs a solution of size $f(\text{OPT})$, for any functions $t$ and $f$ that are independent of $N$ (for Clique, we want $f(\text{OPT})=ω(1)$)? In this paper, we show that both Clique and DomSet admit no non-trivial FPT-approximation algorithm, i.e., there is no $o(\text{OPT})$-FPT-approximation algorithm for Clique and no $f(\text{OPT})$-FPT-approximation algorithm for DomSet, for any function $f$ (e.g., this holds even if $f$ is the Ackermann function). In fact, our results imply something even stronger: The best way to solve Clique and DomSet, even approximately, is to essentially enumerate all possibilities. Our results hold under the Gap Exponential Time Hypothesis (Gap-ETH) [Dinur16, MR16], which states that no $2^{o(n)}$-time algorithm can distinguish between a satisfiable 3SAT formula and one which is not even $(1 - ε)$-satisfiable for some constant $ε> 0$. Besides Clique and DomSet, we also rule out non-trivial FPT-approximation for Maximum Balanced Biclique, Maximum Subgraphs with Hereditary Properties, and Maximum Induced Matching in bipartite graphs. Additionally, we rule out $k^{o(1)}$-FPT-approximation algorithm for Densest $k$-Subgraph although this ratio does not yet match the trivial $O(k)$-approximation algorithm.

cs.CC↗

arXiv2016

Approximating Spanners and Directed Steiner Forest: Upper and Lower Bounds

It was recently found that there are very close connections between the existence of additive spanners (subgraphs where all distances are preserved up to an additive stretch), distance preservers (subgraphs in which demand pairs have their distance preserved exactly), and pairwise spanners (subgraphs in which demand pairs have their distance preserved up to a multiplicative or additive stretch) [Abboud-Godwin SODA '16, Godwin-Williams SODA '16]. We study these problems from an optimization point of view, where rather than studying the existence of extremal instances we are given an instance and are asked to find the sparsest possible spanner/preserver. We give an $O(n^{3/5 + ε})$-approximation for distance preservers and pairwise spanners (for arbitrary constant $ε> 0$). This is the first nontrivial upper bound for either problem, both of which are known to be as hard to approximate as Label Cover. We also prove Label Cover hardness for approximating additive spanners, even for the cases of additive 1 stretch (where one might expect a polylogarithmic approximation, since the related multiplicative 2-spanner problem admits an $O(\log n)$-approximation) and additive polylogarithmic stretch (where the related multiplicative spanner problem has an $O(1)$-approximation). Interestingly, the techniques we use in our approximation algorithm extend beyond distance-based problem to pure connectivity network design problems. In particular, our techniques allow us to give an $O(n^{3/5 + ε})$-approximation for the Directed Steiner Forest problem (for arbitrary constant $ε> 0$) when all edges have uniform costs, improving the previous best $O(n^{2/3 + ε})$-approximation due to Berman et al.~[ICALP '11] (which holds for general edge costs).

cs.DS↗

arXiv2016

The Densest k-Subhypergraph Problem

The Densest $k$-Subgraph (D$k$S) problem, and its corresponding minimization problem Smallest $p$-Edge Subgraph (S$p$ES), have come to play a central role in approximation algorithms. This is due both to their practical importance, and their usefulness as a tool for solving and establishing approximation bounds for other problems. These two problems are not well understood, and it is widely believed that they do not an admit a subpolynomial approximation ratio (although the best known hardness results do not rule this out). In this paper we generalize both D$k$S and S$p$ES from graphs to hypergraphs. We consider the Densest $k$-Subhypergraph problem (given a hypergraph $(V, E)$, find a subset $W\subseteq V$ of $k$ vertices so as to maximize the number of hyperedges contained in $W$) and define the Minimum $p$-Union problem (given a hypergraph, choose $p$ of the hyperedges so as to minimize the number of vertices in their union). We focus in particular on the case where all hyperedges have size 3, as this is the simplest non-graph setting. For this case we provide an $O(n^{4(4-\sqrt{3})/13 + ε}) \leq O(n^{0.697831+ε})$-approximation (for arbitrary constant $ε> 0$) for Densest $k$-Subhypergraph and an $\tilde O(n^{2/5})$-approximation for Minimum $p$-Union. We also give an $O(\sqrt{m})$-approximation for Minimum $p$-Union in general hypergraphs. Finally, we examine the interesting special case of interval hypergraphs (instances where the vertices are a subset of the natural numbers and the hyperedges are intervals of the line) and prove that both problems admit an exact polynomial time solution on these instances.

cs.DS↗

arXiv2016

A Tight Algorithm for Strongly Connected Steiner Subgraph On Two Terminals With Demands

Given an edge-weighted directed graph $G=(V,E)$ on $n$ vertices and a set $T=\{t_1, t_2, \ldots, t_p\}$ of $p$ terminals, the objective of the \scss ($p$-SCSS) problem is to find an edge set $H\subseteq E$ of minimum weight such that $G[H]$ contains an $t_{i}\rightarrow t_j$ path for each $1\leq i\neq j\leq p$. In this paper, we investigate the computational complexity of a variant of $2$-SCSS where we have demands for the number of paths between each terminal pair. Formally, the \sharinggeneral problem is defined as follows: given an edge-weighted directed graph $G=(V,E)$ with weight function $ω: E\rightarrow \mathbb{R}^{\geq 0}$, two terminal vertices $s, t$, and integers $k_1, k_2$ ; the objective is to find a set of $k_1$ paths $F_1, F_2, \ldots, F_{k_1}$ from $s\leadsto t$ and $k_2$ paths $B_1, B_2, \ldots, B_{k_2}$ from $t\leadsto s$ such that $\sum_{e\in E} ω(e)\cdot ϕ(e)$ is minimized, where $ϕ(e)= \max \Big\{|\{i\in [k_1] : e\in F_i\}|\ ,\ |\{j\in [k_2] : e\in B_j\}|\Big\}$. For each $k\geq 1$, we show the following: The \sharing problem can be solved in $n^{O(k)}$ time. A matching lower bound for our algorithm: the \sharing problem does not have an $f(k)\cdot n^{o(k)}$ algorithm for any computable function $f$, unless the Exponential Time Hypothesis (ETH) fails. Our algorithm for \sharing relies on a structural result regarding an optimal solution followed by using the idea of a "token game" similar to that of Feldman and Ruhl. We show with an example that the structural result does not hold for the \sharinggeneral problem if $\min\{k_1, k_2\}\geq 2$. Therefore \sharing is the most general problem one can attempt to solve with our techniques.

cs.DS↗

arXiv2015

Low-Risk Mechanisms for the Kidney Exchange Game

In this paper we consider the pairwise kidney exchange game. This game naturally appears in situations that some service providers benefit from pairwise allocations on a network, such as the kidney exchanges between hospitals. Ashlagi et al. present a $2$-approximation randomized truthful mechanism for this problem. This is the best known result in this setting with multiple players. However, we note that the variance of the utility of an agent in this mechanism may be as large as $Ω(n^2)$, which is not desirable in a real application. In this paper we resolve this issue by providing a $2$-approximation randomized truthful mechanism in which the variance of the utility of each agent is at most $2+ε$. Interestingly, we could apply our technique to design a deterministic mechanism such that, if an agent deviates from the mechanism, she does not gain more than $2\lceil \log_2 m\rceil$. We call such a mechanism an almost truthful mechanism. Indeed, in a practical scenario, an almost truthful mechanism is likely to imply a truthful mechanism. We believe that our approach can be used to design low risk or almost truthful mechanisms for other problems.

cs.GT↗

arXiv2015

A simplified 1.5-approximation algorithm for augmenting edge-connectivity of a graph from 1 to 2

The Tree Augmentation Problem (TAP) is: given a connected graph $G=(V,{\cal E})$ and an edge set $E$ on $V$ find a minimum size subset of edges $F \subseteq E$ such that $(V,{\cal E} \cup F)$ is $2$-edge-connected. In the conference version \cite{EFKN-APPROX} was sketched a $1.5$-approximation algorithm for the problem. Since a full proof was very complex and long, the journal version was cut into two parts. In the first part \cite{EFKN-TALG} was only proved ratio $1.8$. An attempt to simplify the second part produced an error in \cite{EKN-IPL}. Here we give a correct, different, and self contained proof of the ratio $1.5$, that is also substantially simpler and shorter than the previous proofs.

cs.DS↗