Approximating Source Location and Star Survivable Network Problems

In Source Location (SL) problems the goal is to select a mini-mum cost source set $S \subseteq V$ such that the connectivity (or flow) $\psi(S,v)$ from $S$ to any node $v$ is at least the demand $d_v$ of $v$. In many SL problems $\psi(S,v)=d_v$ if $v \in S$, namely, the demand of nodes selected to $S$ is completely satisfied. In a node-connectivity variant suggested recently by Fukunaga, every node $v$ gets a "bonus" $p_v \leq d_v$ if it is selected to $S$. Fukunaga showed that for undirected graphs one can achieve ratio $O(k \ln k)$ for his variant, where $k=\max_{v \in V}d_v$ is the maximum demand. We improve this by achieving ratio \min\{p^*\lnk,k\}\cdot O(\ln (k/q^*)) for a more general version with node capacities, where $p^*=\max_{v \in V} p_v$ is the maximum bonus and $q^*=\min_{v \in V} q_v$ is the minimum capacity. In particular, for the most natural case $p^*=1$ considered by Fukunaga, we improve the ratio from $O(k \ln k)$ to $O(\ln^2k)$. We also get ratio $O(k)$ for the edge-connectivity version, for which no ratio that depends on $k$ only was known before. To derive these results, we consider a particular case of the Survivable Network (SN) problem when all edges of positive cost form a star. We give ratio $O(\min\{\ln n,\ln^2 k\})$ for this variant, improving over the best ratio known for the general case $O(k^3 \ln n)$ of Chuzhoy and Khanna

Approximating Directed Steiner Problems via Tree Embedding

In the k-edge connected directed Steiner tree (k-DST) problem, we are given a directed graph G on n vertices with edge-costs, a root vertex r, a set of h terminals T and an integer k. The goal is to find a min-cost subgraph H of G that connects r to each terminal t by k edge-disjoint r,t-paths. This problem includes as special cases the well-known directed Steiner tree (DST) problem (the case k = 1) and the group Steiner tree (GST) problem. Despite having been studied and mentioned many times in literature, e.g., by Feldman et al. [SODA'09, JCSS'12], by Cheriyan et al. [SODA'12, TALG'14] and by Laekhanukit [SODA'14], there was no known non-trivial approximation algorithm for k-DST for k >= 2 even in the special case that an input graph is directed acyclic and has a constant number of layers. If an input graph is not acyclic, the complexity status of k-DST is not known even for a very strict special case that k= 2 and |T| = 2. In this paper, we make a progress toward developing a non-trivial approximation algorithm for k-DST. We present an O(D k^{D-1} log n)-approximation algorithm for k-DST on directed acyclic graphs (DAGs) with D layers, which can be extended to a special case of k-DST on "general graphs" when an instance has a D-shallow optimal solution, i.e., there exist k edge-disjoint r,t-paths, each of length at most D, for every terminal t. For the case k= 1 (DST), our algorithm yields an approximation ratio of O(D log h), thus implying an O(log^3 h)-approximation algorithm for DST that runs in quasi-polynomial-time (due to the height-reduction of Zelikovsky [Algorithmica'97]). Consequently, as our algorithm works for general graphs, we obtain an O(D k^{D-1} log n)-approximation algorithm for a D-shallow instance of the k-edge-connected directed Steiner subgraph problem, where we wish to connect every pair of terminals by k-edge-disjoint paths

On rooted kk-connectivity problems in quasi-bipartite digraphs

We consider the directed Rooted Subset $k$-Edge-Connectivity problem: given a set $T \subseteq V$ of terminals in a digraph $G=(V+r,E)$ with edge costs and an integer $k$, find a min-cost subgraph of $G$ that contains $k$ edge disjoint $rt$-paths for all $t \in T$. The case when every edge of positive cost has head in $T$ admits a polynomial time algorithm due to Frank, and the case when all positive cost edges are incident to $r$ is equivalent to the $k$-Multicover problem. Recently, [Chan et al. APPROX20] obtained ratio $O(\ln k \ln |T|)$ for quasi-bipartite instances, when every edge in $G$ has an end in $T+r$. We give a simple proof for the same ratio for a more general problem of covering an arbitrary $T$-intersecting supermodular set function by a minimum cost edge set, and for the case when only every positive cost edge has an end in $T+r$

Parameterized Inapproximability of Independent Set in HH-Free Graphs

We study the Independent Set (IS) problem in $H$-free graphs, i.e., graphs excluding some fixed graph $H$ as an induced subgraph. We prove several inapproximability results both for polynomial-time and parameterized algorithms. Halld\'orsson [SODA 1995] showed that for every $\delta>0$ IS has a polynomial-time $(\frac{d-1}{2}+\delta)$-approximation in $K_{1,d}$-free graphs. We extend this result by showing that $K_{a,b}$-free graphs admit a polynomial-time $O(\alpha(G)^{1-1/a})$-approximation, where $\alpha(G)$ is the size of a maximum independent set in $G$. Furthermore, we complement the result of Halld\'orsson by showing that for some $\gamma=\Theta(d/\log d),$ there is no polynomial-time $\gamma$-approximation for these graphs, unless NP = ZPP. Bonnet et al. [IPEC 2018] showed that IS parameterized by the size $k$ of the independent set is W[1]-hard on graphs which do not contain (1) a cycle of constant length at least $4$, (2) the star $K_{1,4}$, and (3) any tree with two vertices of degree at least $3$ at constant distance. We strengthen this result by proving three inapproximability results under different complexity assumptions for almost the same class of graphs (we weaken condition (2) that $G$ does not contain $K_{1,5}$). First, under the ETH, there is no $f(k)\cdot n^{o(k/\log k)}$ algorithm for any computable function $f$. Then, under the deterministic Gap-ETH, there is a constant $\delta>0$ such that no $\delta$-approximation can be computed in $f(k) \cdot n^{O(1)}$ time. Also, under the stronger randomized Gap-ETH there is no such approximation algorithm with runtime $f(k)\cdot n^{o(k)}$. Finally, we consider the parameterization by the excluded graph $H$, and show that under the ETH, IS has no $n^{o(\alpha(H))}$ algorithm in $H$-free graphs and under Gap-ETH there is no $d/k^{o(1)}$-approximation for $K_{1,d}$-free graphs with runtime $f(d,k) n^{O(1)}$.Comment: Preliminary version of the paper in WG 2020 proceeding

Approximating Source Location and Star Survivable Network Problems

Abstract. In Source Location (SL) problems the goal is to select a minimum cost source set S ⊆ V such that the connectivity (or flow) ψ(S, v) from S to any node v is at least the demand dv of v. In many SL problems ψ(S, v) = dv if v ∈ S, namely, the demand of nodes se-lected to S is completely satisfied. In a node-connectivity variant sug-gested recently by Fukunaga [6], every node v gets a “bonus ” pv ≤ dv if it is selected to S, namely, ψ(S, v) = pv + κ(S \ {v}, v) if v ∈ S and ψ(S, v) = κ(S, v) otherwise, where κ(S, v) is the maximum number of internally disjoint (S, v)-paths. While the approximability of many SL problems was seemingly settled to Θ(ln d(V)) in [18], Fukunaga [6] showed that for undirected graphs one can achieve ratio O(k ln k) for his variant, where k = maxv∈V dv is the maximum demand. We improve this by achieving ratio min{p ∗ ln k, k} · O(ln(k/q∗)) for a more general version with node capacities, where p ∗ = maxv∈V pv is the maximum bonus and q ∗ = minv∈V qv is the minimum capacity. In particular, for the most natural case p ∗ = 1 considered in [6] we improve the ratio from O(k ln k) to O(ln2 k). Our result also implies ratio k for the edge-connectivity version. To derive these results, we consider a particular case of the Survivable Network (SN) problem when all edges of positive cost form a star. We give ratio O(min{lnn, ln2 k}) for this variant, improving over the best ratio known for the general case O(k3 lnn) of Chuzhoy and Khanna [3]. In addition, we show that directed SL with unit costs is Ω(logn)-hard to approximate even for 0, 1 demands, while SL with uniform demands can be solved in polynomial time. Finally, we consider a generalization of SL where we also have edge-costs {ce: e ∈ E} and flow-cost bounds {bv: v ∈ V}, and require that for every node v, the minimum cost of a flow of value dv from S to v is at most bv. We show that this problem admits approximation ratio O(ln d(V) + ln(nc(E) − b(V)).

The Complexity of Network Design for s-t Eff ective Resistance

We consider a new problem of designing a network with small $s$-$t$ effective resistance. In this problem, we are given an undirected graph $G=(V,E)$ where each edge $e$ has a cost $c_e$ and a resistance $r_e$, two designated vertices $s,t \in V$, and a cost budget $k$. Our goal is to choose a subgraph to minimize the $s$-$t$ effective resistance, subject to the constraint that the total cost in the subgraph is at most $k$. This problem has applications in electrical network design and is an interpolation between the shortest path problem and the minimum cost flow problem. We present algorithmic and hardness results for this problem. On the hardness side, we show that the problem is NP-hard by reducing the 3-dimensional matching problem to our problem. On the algorithmic side, we use dynamic programming to obtain a fully polynomial time approximation scheme when the input graph is a series-parallel graph. Finally, we propose a greedy algorithm for general graphs in which we add a path at each iteration and we conjecture that the algorithm is a $3.95$-approximation algorithm for the problem

A Spectral Approach to Network Design and Experimental Design

Over the last decade, the spectral sparsification technique has become a powerful tool in designing fast graph algorithms for various problems with numerous applications. In this thesis, we extend this spectral approach, and show that it is also very powerful in designing approximation algorithms for classical network design and experimental design problems. The central piece in this thesis is a problem called spectral rounding, which is inspired by spectral sparsification and studied in an earlier work on experimental design. In this problem, we are given vectors \vv_1, \ldots, \vv_m each with a non-negative cost, and a fractional solution \vx \in [0,1]^m. The task is to find an integral solution \vz \in \{0,1\}^m such that the spectrum of the integral solution is similar to the one of the fractional solution, i.e.~\sum_i \vz(i) \cdot \vv_i \vv_i^\top \approx \sum_i \vx(i) \cdot \vv_i \vv_i^\top, and the integral cost is approximately equal to the fractional cost. We observe that the spectral rounding problem underlies a large family of network design and experimental design problems. With this perspective, we bring new insights into these well-studied problems. For network design, we show that the spectral rounding technique provides a novel and general approach to significantly extend the scope of problems that can be solved efficiently. For experimental design, we show that the spectral rounding technique provides a unified and elegant framework that matches and improves all known existing algorithmic results. There are two key techniques that we will use in this thesis. The first one is regret minimization, which is well-known to the online optimization community and has been used for spectral sparsification. We use it to control the spectrum of the integral solution in the spectral rounding problem. The second key technique is concentration inequalities for analyzing adaptive random sampling processes, which enable us to satisfy spectral and linear constraints simultaneously with high probability