arXiv subjects

Bundit Laekhanukit

Publications and source records attributed to Bundit Laekhanukit.

At least 19 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

On the Integrality Gap of Directed Steiner Tree LPs with Relatively Integral Solutions

The Directed Steiner Tree (DST) problem is defined on a directed graph $G=(V,E)$, where we are given a designated root vertex $r$ and a set of $k$ terminals $K \subseteq V \setminus {r}$. The goal is to find a minimum-cost subgraph that provides directed $r \rightarrow t$ paths for all terminals $t \in K$. The approximability of DST has long been a central open problem in network design. While there exist polylogarithmic-approximation algorithms with quasi-polynomial running times (Charikar et al. 1998; Grandoni, Laekhanukit, and Li 2019; Ghuge and Nagarajan 2020), the best known polynomial-time approximation until now has remained at $k^ε$, for any constant $ε> 0$. Whether a polynomial-time algorithm achieving a polylogarithmic approximation exists has remained unresolved. In this paper, we present a flow-based LP-relaxation for DST that admits a polylogarithmic integrality gap under the relative integral condition -- there exists a fractional solution in which each edge $e$ either carries a zero flow ($f^t_e=0$) or uses its full capacity ($f^t_e=x_e$), where $f^t_e$ denotes the flow variable and $x_e$ denotes the indicator variable treated as capacities. This stands in contrast to known lower bounds, as the standard flow-based relaxation is known to exhibit a polynomial integrality gap even under relatively integral solutions. In fact, this relatively integral property is shared by all the known integrality gap instances of DST [Halperin~et~al., SODA'07; Zosin-Khuller, SODA'02; Li-Laekhanukit, SODA'22]. We further provide a randomized polynomial-time algorithm that gives an $O(\log^3 k)$-approximation, assuming access to a relatively integral fractional solution.

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arXiv2024

Almost Tight Approximation Hardness for Single-Source Directed k-Edge-Connectivity

In the $k$-connected directed Steiner tree problem ($k$-DST), we are given an $n$-vertex directed graph $G=(V,E)$ with edge costs, a connectivity requirement $k$, a root $r\in V$ and a set of terminals $T\subseteq V$. The goal is to find a minimum-cost subgraph $H\subseteq G$ that has $k$ internally disjoint paths from the root vertex $r$ to every terminal $t\in T$. In this paper, we show the approximation hardness of $k$-DST for various parameters, which thus close some long-standing open problems. - $Ω\left(|T|/\log |T|\right)$-approximation hardness, which holds under the standard assumption $\mathrm{NP}\neq \mathrm{ZPP}$. The inapproximability ratio is tightened to $Ω\left(|T|\right)$ under the Strongish Planted Clique Hypothesis [Manurangsi, Rubinstein and Schramm, ITCS 2021]. The latter hardness result matches the approximation ratio of $|T|$ obtained by a trivial approximation algorithm, thus closing the long-standing open problem. - $Ω\left(\sqrt{2}^k / k\right)$-approximation hardness for the general case of $k$-DST under the assumption $\mathrm{NP}\neq\mathrm{ZPP}$. This is the first hardness result known for survivable network design problems with an inapproximability ratio exponential in $k$. - $Ω\left((k/L)^{L/4}\right)$-approximation hardness for $k$-DST on $L$-layered graphs for $L\le O\left(\log n\right)$. This almost matches the approximation ratio of $O(k^{L-1}\cdot L \cdot \log |T|)$ achieving in $O\left(n^L\right)$-time due to Laekhanukit [ICALP`16].

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arXiv2024

Simple Combinatorial Construction of the $k^{o(1)}$-Lower Bound for Approximating the Parameterized $k$-Clique

In the parameterized $k$-clique problem, or $k$-Clique for short, we are given a graph $G$ and a parameter $k\ge 1$. The goal is to decide whether there exist $k$ vertices in $G$ that induce a complete subgraph (i.e., a $k$-clique). This problem plays a central role in the theory of parameterized intractability as one of the first W[1]-complete problems. Existing research has shown that even an FPT-approximation algorithm for $k$-Clique with arbitrary ratio does not exist, assuming the Gap-Exponential-Time Hypothesis (Gap-ETH) [Chalermsook et al., FOCS'17 and SICOMP]. However, whether this inapproximability result can be based on the standard assumption of $\mathrm{W} 1\ne \mathrm{FPT}$ remains unclear. The recent breakthrough of Bingkai Lin [STOC'21] and subsequent works by Karthik C.S. and Khot [CCC'22], and by Lin, Ren, Sun Wang [ICALP'22] give a technique that bypasses Gap-ETH, thus leading to the inapproximability ratio of $O(1)$ and $k^{o(1)}$ under $\mathrm{W}[1]$-hardness (the first two) and ETH (for the latter one). All the work along this line follows the framework developed by Lin, which starts from the $k$-vector-sum problem and requires some involved algebraic techniques. This paper presents an alternative framework for proving the W[1]-hardness of the $k^{o(1)}$-FPT-inapproximability of $k$-Clique. Using this framework, we obtain a gap-producing self-reduction of $k$-Clique without any intermediate algebraic problem. More precisely, we reduce from $(k,k-1)$-Gap Clique to $(q^k, q^{k-1})$-Gap Clique, for any function $q$ depending only on the parameter $k$, thus implying the $k^{o(1)}$-inapproximability result when $q$ is sufficiently large. Our proof is relatively simple and mostly combinatorial. At the core of our construction is a novel encoding of $k$-element subset stemming from the theory of "network coding" and a "Sidon set" representation of a graph.

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arXiv2022

Polynomial Integrality Gap of Flow LP for Directed Steiner Tree

In the Directed Steiner Tree (DST) problem, we are given a directed graph $G=(V,E)$ on $n$ vertices with edge-costs $c \in \mathbb{R}_{\geq 0}^E$, a root vertex $r \in V$, and a set $K \subseteq V \setminus \{r\}$ of $k$ terminals. The goal is to find a minimum-cost subgraph of $G$ that contains a path from $r$ to every terminal $t \in K$. DST has been a notorious problem for decades as there is a large gap between the best-known polynomial-time approximation ratio of $O(k^ε)$ for any constant $ε> 0$, and the best quasi-polynomial-time approximation ratio of $O\left(\frac{\log^2 k}{\log \log k}\right)$. Towards understanding this gap, we study the integrality gap of the standard flow LP relaxation for the problem. We show that the LP has an integrality gap of $Ω(n^{0.0418})$. Previously, the integrality gap of the LP is only known to be $Ω\left(\frac{\log^2n}{\log\log n}\right)$ [Halperin~et~al., SODA'03 \& SIAM J.~Comput.] and $Ω(\sqrt{k})$ [Zosin-Khuller, SODA'02] in some instance with $\sqrt{k}=O\left(\frac{\log n}{\log \log n}\right)$. Our result gives the first known lower bound on the integrality gap of this standard LP that is polynomial in $n$, the number of vertices. Consequently, we rule out the possibility of developing a poly-logarithmic approximation algorithm for the problem based on the flow LP relaxation.

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arXiv2022

Survivable Network Design Revisited: Group-Connectivity

In the classical survivable network design problem (SNDP), we are given an undirected graph $G=(V,E)$ with costs on edges and a connectivity requirement $k(s,t)$ for each pair of vertices. The goal is to find a minimum-cost subgraph $H\subseteq V$ such that every pair $(s,t)$ are connected by $k(s,t)$ edge or (openly) vertex disjoint paths, abbreviated as EC-SNDP and VC-SNDP, respectively. The seminal result of Jain [FOCS'98, Combinatorica'01] gives a $2$-approximation algorithm for EC-SNDP, and a decade later, an $O(k^3\log n)$-approximation algorithm for VC-SNDP, where $k$ is the largest connectivity requirement, was discovered by Chuzhoy and Khanna [FOCS'09, Theory Comput.'12]. While there is rich literature on point-to-point settings of SNDP, the viable case of connectivity between subsets is still relatively poorly understood. This paper concerns the generalization of SNDP into the subset-to-subset setting, namely Group EC-SNDP. We develop the framework, which yields the first non-trivial (true) approximation algorithm for Group EC-SNDP. Previously, only a bicriteria approximation algorithm is known for Group EC-SNDP [Chalermsook, Grandoni, and Laekhanukit, SODA'15], and a true approximation algorithm is known only for the single-source variant with connectivity requirement $k(S,T)\in\{0,1,2\}$ [Gupta, Krishnaswamy, and Ravi, SODA'10; Khandekar, Kortsarz, and Nutov, FSTTCS'09 and Theor. Comput. Sci.'12].

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arXiv2021

On a Partition LP Relaxation for Min-Cost 2-Node Connected Spanning Subgraphs

Our motivation is to improve on the best approximation guarantee known for the problem of finding a minimum-cost 2-node connected spanning subgraph of a given undirected graph with nonnegative edge costs. We present an LP (Linear Programming) relaxation based on partition constraints. The special case where the input contains a spanning tree of zero cost is called 2NC-TAP. We present a greedy algorithm for 2NC-TAP, and we analyze it via dual-fitting for our partition LP relaxation. Keywords: 2-node connected graphs, approximation algorithms, connectivity augmentation, greedy algorithm, network design, partition relaxation

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arXiv2020

On the Approximability of the Traveling Salesman Problem with Line Neighborhoods

We study the variant of the Euclidean Traveling Salesman problem where instead of a set of points, we are given a set of lines as input, and the goal is to find the shortest tour that visits each line. The best known upper and lower bounds for the problem in $\mathbb{R}^d$, with $d\ge 3$, are $\mathrm{NP}$-hardness and an $O(\log^3 n)$-approximation algorithm which is based on a reduction to the group Steiner tree problem. We show that TSP with lines in $\mathbb{R}^d$ is APX-hard for any $d\ge 3$. More generally, this implies that TSP with $k$-dimensional flats does not admit a PTAS for any $1\le k \leq d-2$ unless $\mathrm{P}=\mathrm{NP}$, which gives a complete classification of the approximability of these problems, as there are known PTASes for $k=0$ (i.e., points) and $k=d-1$ (hyperplanes). We are able to give a stronger inapproximability factor for $d=O(\log n)$ by showing that TSP with lines does not admit a $(2-ε)$-approximation in $d$ dimensions under the unique games conjecture. On the positive side, we leverage recent results on restricted variants of the group Steiner tree problem in order to give an $O(\log^2 n)$-approximation algorithm for the problem, albeit with a running time of $n^{O(\log\log n)}$.

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arXiv2020

Vertex Sparsification for Edge Connectivity

Graph compression or sparsification is a basic information-theoretic and computational question. A major open problem in this research area is whether $(1+ε)$-approximate cut-preserving vertex sparsifiers with size close to the number of terminals exist. As a step towards this goal, we study a thresholded version of the problem: for a given parameter $c$, find a smaller graph, which we call connectivity-$c$ mimicking network, which preserves connectivity among $k$ terminals exactly up to the value of $c$. We show that connectivity-$c$ mimicking networks with $O(kc^4)$ edges exist and can be found in time $m(c\log n)^{O(c)}$. We also give a separate algorithm that constructs such graphs with $k \cdot O(c)^{2c}$ edges in time $mc^{O(c)}\log^{O(1)}n$. These results lead to the first data structures for answering fully dynamic offline $c$-edge-connectivity queries for $c \ge 4$ in polylogarithmic time per query, as well as more efficient algorithms for survivable network design on bounded treewidth graphs.

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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.

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arXiv2019

Polylogarithmic Approximation Algorithm for k-Connected Directed Steiner Tree on Quasi-Bipartite Graphs

In the k-Connected Directed Steiner Tree problem (k-DST), we are given a directed graph G=(V, E) with edge (or vertex) costs, a root vertex r, a set of q terminals T, and a connectivity requirement k>0; the goal is to find a minimum-cost subgraph H of G such that H has k internally disjoint paths from the root r to each terminal t . The k-DST problem is a natural generalization of the classical Directed Steiner Tree problem (DST) in the fault-tolerant setting in which the solution subgraph is required to have an r,t-path, for every terminal t, even after removing k-1 edges or vertices. Despite being a classical problem, there are not many positive results on the problem, especially for the case k >= 3. In this paper, we will present an O(log k log q)-approximation algorithm for k-DST when an input graph is quasi-bipartite, i.e., when there is no edge joining two non-terminal vertices. To the best of our knowledge, our algorithm is the only known non-trivial approximation algorithm for k-DST, for k >= 3, that runs in polynomial-time regardless of the structure of the optimal solution. In addition, our algorithm is tight for every constant k, due to the hardness result inherited from the Set Cover problem.

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arXiv2019

Mimicking Networks Parameterized by Connectivity

Given a graph $G=(V,E)$, capacities $w(e)$ on edges, and a subset of terminals $\mathcal{T} \subseteq V: |\mathcal{T}| = k$, a mimicking network for $(G,\mathcal{T})$ is a graph $(H,w')$ that contains copies of $\mathcal{T}$ and preserves the value of minimum cuts separating any subset $A, B \subseteq \mathcal{T}$ of terminals. Mimicking networks of size $2^{2^k}$ are known to exist and can be constructed algorithmically, while the best known lower bound is $2^{Ω(k)}$; therefore, an exponential size is required if one aims at preserving cuts exactly. In this paper, we study mimicking networks that preserve connectivity of the graph exactly up to the value of $c$, where $c$ is a parameter. This notion of mimicking network is sufficient for some applications, as we will elaborate. We first show that a mimicking of size $3^c \cdot k$ exists, that is, we can preserve cuts with small capacity using a network of size linear in $k$. Next, we show an algorithm that finds such a mimicking network in time $2^{O(c^2)} \operatorname{poly}(m)$.

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arXiv2018

On the Complexity of Closest Pair via Polar-Pair of Point-Sets

Every graph $G$ can be represented by a collection of equi-radii spheres in a $d$-dimensional metric $Δ$ such that there is an edge $uv$ in $G$ if and only if the spheres corresponding to $u$ and $v$ intersect. The smallest integer $d$ such that $G$ can be represented by a collection of spheres (all of the same radius) in $Δ$ is called the sphericity of $G$, and if the collection of spheres are non-overlapping, then the value $d$ is called the contact-dimension of $G$. In this paper, we study the sphericity and contact dimension of the complete bipartite graph $K_{n,n}$ in various $L^p$-metrics and consequently connect the complexity of the monochromatic closest pair and bichromatic closest pair problems.

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arXiv2018

$O(\log^2k/\log\log{k})$-Approximation Algorithm for Directed Steiner Tree: A Tight Quasi-Polynomial-Time Algorithm

In the Directed Steiner Tree (DST) problem we are given an $n$-vertex directed edge-weighted graph, a root $r$, and a collection of $k$ terminal nodes. Our goal is to find a minimum-cost arborescence that contains a directed path from $r$ to every terminal. We present an $O(\log^2 k/\log\log{k})$-approximation algorithm for DST that runs in quasi-polynomial-time. By adjusting the parameters in the hardness result of Halperin and Krauthgamer, we show the matching lower bound of $Ω(\log^2{k}/\log\log{k})$ for the class of quasi-polynomial-time algorithms. This is the first improvement on the DST problem since the classical quasi-polynomial-time $O(\log^3 k)$ approximation algorithm by Charikar et al. (The paper erroneously claims an $O(\log^2k)$ approximation due to a mistake in prior work.) Our approach is based on two main ingredients. First, we derive an approximation preserving reduction to the Label-Consistent Subtree (LCST) problem. The LCST instance has quasi-polynomial size and logarithmic height. We remark that, in contrast, Zelikovsky's heigh-reduction theorem used in all prior work on DST achieves a reduction to a tree instance of the related Group Steiner Tree (GST) problem of similar height, however losing a logarithmic factor in the approximation ratio. Our second ingredient is an LP-rounding algorithm to approximately solve LCST instances, which is inspired by the framework developed by Rothvoß. We consider a Sherali-Adams lifting of a proper LP relaxation of LCST. Our rounding algorithm proceeds level by level from the root to the leaves, rounding and conditioning each time on a proper subset of label variables. A small enough (namely, polylogarithmic) number of Sherali-Adams lifting levels is sufficient to condition up to the leaves.

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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.

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arXiv2018

Finding All Useless Arcs in Directed Planar Graphs

We present a linear-time algorithm for simplifying flow networks on directed planar graphs: Given a directed planar graph on $n$ vertices, a source vertex $s$ and a sink vertex $t$, our algorithm removes all the arcs that do not participate in any simple $s,t$-path in linear-time. The output graph produced by our algorithm satisfies the prerequisite needed by the $O(n\log n)$-time algorithm of Weihe [FOCS'94 \& JCSS'97] for computing maximum $s,t$-flow in directed planar graphs. Previously, Weihe's algorithm could not run in $O(n\log n)$-time due to the absence of the preprocessing step; all the preceding algorithms run in $\tildeΩ(n^2)$-time [Misiolek-Chen, COCOON'05 \& IPL'06; Biedl, Brejov{á} and Vinar, MFCS'00]. Consequently, this provides an alternative $O(n\log n)$-time algorithm for computing maximum $s,t$-flow in directed planar graphs in addition to the known $O(n\log n)$-time algorithms [Borradaile-Klein, SODA'06 \& J.ACM'09; Erickson, SODA'10]. Our algorithm can be seen as a (truly) linear-time $s,t$-flow sparsifier for directed planar graphs, which runs faster than any maximum $s,t$-flow algorithm (which can also be seen of as a sparsifier). The simplified structures of the resulting graph might be useful in future developments of maximum $s,t$-flow algorithms in both directed and undirected planar graphs.

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arXiv2018

On the Parameterized Complexity of Approximating Dominating Set

We study the parameterized complexity of approximating the $k$-Dominating Set (DomSet) problem where an integer $k$ and a graph $G$ on $n$ vertices are given as input, and the goal is to find a dominating set of size at most $F(k) \cdot k$ whenever the graph $G$ has a dominating set of size $k$. When such an algorithm runs in time $T(k) \cdot poly(n)$ (i.e., FPT-time) for some computable function $T$, it is said to be an $F(k)$-FPT-approximation algorithm for $k$-DomSet. We prove the following for every computable functions $T, F$ and every constant $\varepsilon > 0$: $\bullet$ Assuming $W[1]\neq FPT$, there is no $F(k)$-FPT-approximation algorithm for $k$-DomSet. $\bullet$ Assuming the Exponential Time Hypothesis (ETH), there is no $F(k)$-approximation algorithm for $k$-DomSet that runs in $T(k) \cdot n^{o(k)}$ time. $\bullet$ Assuming the Strong Exponential Time Hypothesis (SETH), for every integer $k \geq 2$, there is no $F(k)$-approximation algorithm for $k$-DomSet that runs in $T(k) \cdot n^{k - \varepsilon}$ time. $\bullet$ Assuming the $k$-Sum Hypothesis, for every integer $k \geq 3$, there is no $F(k)$-approximation algorithm for $k$-DomSet that runs in $T(k) \cdot n^{\lceil k/2 \rceil - \varepsilon}$ time. Our results are obtained by establishing a connection between communication complexity and hardness of approximation, generalizing the ideas from a recent breakthrough work of Abboud et al. [FOCS 2017]. Specifically, we show that to prove hardness of approximation of a certain parameterized variant of the label cover problem, it suffices to devise a specific protocol for a communication problem that depends on which hypothesis we rely on. Each of these communication problems turns out to be either a well studied problem or a variant of one; this allows us to easily apply known techniques to solve them.

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arXiv2018

Survivable Network Design for Group Connectivity in Low-Treewidth Graphs

In the Group Steiner Tree problem (GST), we are given a (vertex or edge)-weighted graph $G=(V,E)$ on $n$ vertices, a root vertex $r$ and a collection of groups $\{S_i\}_{i\in[h]}: S_i\subseteq V(G)$. The goal is to find a min-cost subgraph $H$ that connects the root to every group. We consider a fault-tolerant variant of GST, which we call Restricted (Rooted) Group SNDP. In this setting, each group $S_i$ has a demand $k_i\in[k],k\in\mathbb N$, and we wish to find a min-cost $H\subseteq G$ such that, for each group $S_i$, there is a vertex in $S_i$ connected to the root via $k_i$ (vertex or edge) disjoint paths. While GST admits $O(\log^2 n\log h)$ approximation, its high connectivity variants are Label-Cover hard, and for the vertex-weighted version, the hardness holds even when $k=2$. Previously, positive results were known only for the edge-weighted version when $k=2$ [Gupta et al., SODA 2010; Khandekar et al., Theor. Comput. Sci., 2012] and for a relaxed variant where the disjoint paths may end at different vertices in a group [Chalermsook et al., SODA 2015]. Our main result is an $O(\log n\log h)$ approximation for Restricted Group SNDP that runs in time $n^{f(k, w)}$, where $w$ is the treewidth of $G$. This nearly matches the lower bound when $k$ and $w$ are constant. The key to achieving this result is a non-trivial extension of the framework in [Chalermsook et al., SODA 2017], which embeds all feasible solutions to the problem into a dynamic program (DP) table. However, finding the optimal solution in the DP table remains intractable. We formulate a linear program relaxation for the DP and obtain an approximate solution via randomized rounding. This framework also allows us to systematically construct DP tables for high-connectivity problems. As a result, we present new exact algorithms for several variants of survivable network design problems in low-treewidth graphs.

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