Covering problems in edge- and node-weighted graphs

This paper discusses the graph covering problem in which a set of edges in an edge- and node-weighted graph is chosen to satisfy some covering constraints while minimizing the sum of the weights. In this problem, because of the large integrality gap of a natural linear programming (LP) relaxation, LP rounding algorithms based on the relaxation yield poor performance. Here we propose a stronger LP relaxation for the graph covering problem. The proposed relaxation is applied to designing primal-dual algorithms for two fundamental graph covering problems: the prize-collecting edge dominating set problem and the multicut problem in trees. Our algorithms are an exact polynomial-time algorithm for the former problem, and a 2-approximation algorithm for the latter problem, respectively. These results match the currently known best results for purely edge-weighted graphs.Comment: To appear in SWAT 201

Improved Approximation for Tree Augmentation: Saving by Rewiring

The Tree Augmentation Problem (TAP) is a fundamental network design problem in which we are given a tree and a set of additional edges, also called \emph{links}. The task is to find a set of links, of minimum size, whose addition to the tree leads to a $2$-edge-connected graph. A long line of results on TAP culminated in the previously best known approximation guarantee of $1.5$ achieved by a combinatorial approach due to Kortsarz and Nutov [ACM Transactions on Algorithms 2016], and also by an SDP-based approach by Cheriyan and Gao [Algorithmica 2017]. Moreover, an elegant LP-based $(1.5+\epsilon)$-approximation has also been found very recently by Fiorini, Gro\ss, K\"onemann, and Sanit\'a [SODA 2018]. In this paper, we show that an approximation factor below $1.5$ can be achieved, by presenting a $1.458$-approximation that is based on several new techniques

On Generalizations of Network Design Problems with Degree Bounds

Iterative rounding and relaxation have arguably become the method of choice in dealing with unconstrained and constrained network design problems. In this paper we extend the scope of the iterative relaxation method in two directions: (1) by handling more complex degree constraints in the minimum spanning tree problem (namely, laminar crossing spanning tree), and (2) by incorporating `degree bounds' in other combinatorial optimization problems such as matroid intersection and lattice polyhedra. We give new or improved approximation algorithms, hardness results, and integrality gaps for these problems.Comment: v2, 24 pages, 4 figure

Approximability of Capacitated Network Design

In the capacitated survivable network design problem (Cap- SNDP), we are given an undirected multi-graph where each edge has a capacity and a cost. The goal is to find a minimum cost subset of edges that satisfies a given set of pairwise minimum-cut requirements. Unlike its classical special case of SNDP when all capacities are unit, the approximability of Cap-SNDP is not well understood; even in very restricted settings no known algorithm achieves a o(m) approximation, where m is the number of edges in the graph. In this paper, we obtain several new results and insights into the approximability of Cap-SNDP. We give an O(log n) approximation for a special case of Cap-SNDP where the global minimum cut is required to be at least R, by rounding the natural cut-based LP relaxation strengthened with valid knapsackcover inequalities. We then show that as we move away from global connectivity, the single pair case (that is, when only one pair (s, t) has positive connectivity requirement) captures much of the difficulty of Cap-SNDP: even strengthened with KC inequalities, the LP has an Ω(n) integrality gap. Furthermore, in directed graphs, we show that single pair Cap-SNDP is 2log1−3 n-hard to approximate for any fixed constant δ \u3e 0. We also consider a variant of the Cap-SNDP in which multiple copies of an edge can be bought: we give an O(log k) approximation for this case, where k is the number of vertex pairs with non-zero connectivity requirement. This improves upon the previously known O(min{k, log Rmax})-approximation for this problem when the largest minimumcut requirement, namely Rmax, is large. On the other hand, we observe that the multiple copy version of Cap-SNDP is Ω(log log n)-hard to approximate even for the single-source version of the problem

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)).

Approximation algorithm for k-node connected subgraphs via critical graphs

We present two new approximation algorithms for the problem of finding a k-node connected spanning subgraph (directed or undirected) of minimum cost. The best known ap-n proximation guarantees for this problem were O(min{k, for both directed and undirected graphs, and O(ln k) for undirected graphs with n ≥ 6k 2, where n is the number of nodes in the input graph. Our first algorithm has approximation ratio O ( k n−k ln2 k), which is O(ln 2 k) except for very large values of k, namely, k = n − o(n). This algorithm is based on a new result on ℓ-connected p-critical graphs, which is of independent interest in the context of graph theory. Our second algorithm uses the primal-dual method and has approximation ratio O ( √ n ln k) for all values of n, k. Combining these two gives an algorithm with approximation ratio O(ln k · min { √ k k, ln k}), which asymptotically im-n−k proves the best known approximation guarantee for directed graphs for all values of n, k, and for undirected graphs for k> n/6. Moreover, this is the first algorithm that has an n−k approximation guarantee better than Θ(k) for all values of n, k. Our approximation ratio also provides an upper bound on the integrality gap of the standard LP-relaxation to the problem. As a byproduct, we also get the following result which is of independent interest. To get a faster implementation of our algorithms, we consider the problem of adding a minimumcost edge set to increase the outconnectivity of a directed graph by ∆; a graph is said to be ℓ-outconnected from its node r if it contains ℓ internally disjoint paths from r to any other node. The best known time complexity for the later problem is O(m 3). For the particular case of ∆ = 1, we give a primal-dual algorithm with running time O(m 2). Categories and Subject Descriptor

Approximation algorithms for disjoint st-paths with minimum activation cost

In network activation problems we are given a directed or undirected graph G = (V,E) with a family {f (x, x) : (u,v) ∈ E} of monotone non-decreasing activation functions from D to {0,1}, where D is a constant-size domain. The goal is to find activation values x for all v ∈ V of minimum total cost Σ x such that the activated set of edges satisfies some connectivity requirements. Network activation problems generalize several problems studied in the network literature such as power optimization problems. We devise an approximation algorithm for the fundamental problem of finding the Minimum Activation Cost Pair of Node-Disjoint st-Paths (MA2NDP). The algorithm achieves approximation ratio 1.5 for both directed and undirected graphs. We show that a ρ-approximation algorithm for MA2NDP with fixed activation values for s and t yields a ρ-approximation algorithm for the Minimum Activation Cost Pair of Edge-Disjoint st-Paths (MA2EDP) problem. We also study the MA2NDP and MA2EDP problems for the special case |D| = 2