77 research outputs found
Edge covering with budget constrains
We study two related problems: finding a set of k vertices and minimum number
of edges (kmin) and finding a graph with at least m' edges and minimum number
of vertices (mvms).
Goldschmidt and Hochbaum \cite{GH97} show that the mvms problem is NP-hard
and they give a 3-approximation algorithm for the problem. We improve
\cite{GH97} by giving a ratio of 2. A 2(1+\epsilon)-approximation for the
problem follows from the work of Carnes and Shmoys \cite{CS08}. We improve the
approximation ratio to 2. algorithm for the problem. We show that the natural
LP for \kmin has an integrality gap of 2-o(1). We improve the NP-completeness
of \cite{GH97} by proving the pronlem are APX-hard unless a well-known instance
of the dense k-subgraph admits a constant ratio. The best approximation
guarantee known for this instance of dense k-subgraph is O(n^{2/9})
\cite{BCCFV}. We show that for any constant \rho>1, an approximation guarantee
of \rho for the \kmin problem implies a \rho(1+o(1)) approximation for \mwms.
Finally, we define we give an exact algorithm for the density version of kmin.Comment: 17 page
On the Approximability and Hardness of the Minimum Connected Dominating Set with Routing Cost Constraint
In the problem of minimum connected dominating set with routing cost
constraint, we are given a graph , and the goal is to find the
smallest connected dominating set of such that, for any two
non-adjacent vertices and in , the number of internal nodes on the
shortest path between and in the subgraph of induced by is at most times that in . For general graphs, the only
known previous approximability result is an -approximation algorithm
() for by Ding et al. For any constant , we
give an -approximation
algorithm. When , we give an -approximation
algorithm. Finally, we prove that, when , unless , for any constant , the problem admits no
polynomial-time -approximation algorithm, improving
upon the bound by Du et al. (albeit under a stronger hardness
assumption)
{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
Distance and the pattern of intra-European trade
Given an undirected graph G = (V, E) and subset of terminals T â V, the element-connectivity Îș âČ G (u, v) of two terminals u, v â T is the maximum number of u-v paths that are pairwise disjoint in both edges and non-terminals V \ T (the paths need not be disjoint in terminals). Element-connectivity is more general than edge-connectivity and less general than vertex-connectivity. Hind and Oellermann [21] gave a graph reduction step that preserves the global element-connectivity of the graph. We show that this step also preserves local connectivity, that is, all the pairwise element-connectivities of the terminals. We give two applications of this reduction step to connectivity and network design problems. âą Given a graph G and disjoint terminal sets T1, T2,..., Tm, we seek a maximum number of elementdisjoint Steiner forests where each forest connects each Ti. We prove that if each Ti is k element k connected then there exist âŠ( log hlog m) element-disjoint Steiner forests, where h = | i Ti|. If G is planar (or more generally, has fixed genus), we show that there exist âŠ(k) Steiner forests. Our proofs are constructive, giving poly-time algorithms to find these forests; these are the first non-trivial algorithms for packing element-disjoint Steiner Forests. âą We give a very short and intuitive proof of a spider-decomposition theorem of Chuzhoy and Khanna [12] in the context of the single-sink k-vertex-connectivity problem; this yields a simple and alternative analysis of an O(k log n) approximation. Our results highlight the effectiveness of the element-connectivity reduction step; we believe it will find more applications in the future
The rectilinear Steiner tree problem with given topology and length restrictions
We consider the problem of embedding the Steiner points of a Steiner tree
with given topology into the rectilinear plane. Thereby, the length of the path
between a distinguished terminal and each other terminal must not exceed given
length restrictions. We want to minimize the total length of the tree.
The problem can be formulated as a linear program and therefore it is
solvable in polynomial time. In this paper we analyze the structure of feasible
embeddings and give a combinatorial polynomial time algorithm for the problem.
Our algorithm combines a dynamic programming approach and binary search and
relies on the total unimodularity of a matrix appearing in a sub-problem.Comment: 14 page
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 -edge-connected graph. A long line of
results on TAP culminated in the previously best known approximation guarantee
of 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
-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 can be achieved, by presenting a
-approximation that is based on several new techniques
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 âŠ(n) integrality gap. Furthermore, in directed graphs, we show that single pair Cap-SNDP is 2log1â3 n-hard to approximate for any fixed constant ÎŽ \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 âŠ(log log n)-hard to approximate even for the single-source version of the problem
On the Computational Complexity of Measuring Global Stability of Banking Networks
Threats on the stability of a financial system may severely affect the
functioning of the entire economy, and thus considerable emphasis is placed on
the analyzing the cause and effect of such threats. The financial crisis in the
current and past decade has shown that one important cause of instability in
global markets is the so-called financial contagion, namely the spreading of
instabilities or failures of individual components of the network to other,
perhaps healthier, components. This leads to a natural question of whether the
regulatory authorities could have predicted and perhaps mitigated the current
economic crisis by effective computations of some stability measure of the
banking networks. Motivated by such observations, we consider the problem of
defining and evaluating stabilities of both homogeneous and heterogeneous
banking networks against propagation of synchronous idiosyncratic shocks given
to a subset of banks. We formalize the homogeneous banking network model of
Nier et al. and its corresponding heterogeneous version, formalize the
synchronous shock propagation procedures, define two appropriate stability
measures and investigate the computational complexities of evaluating these
measures for various network topologies and parameters of interest. Our results
and proofs also shed some light on the properties of topologies and parameters
of the network that may lead to higher or lower stabilities.Comment: to appear in Algorithmic
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 â V such that the connectivity (or flow) Ï(S, v) from S to any node v is at least the demand dv of v. In many SL problems Ï(S, v) = dv if v â 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 âbonus â pv †dv if it is selected to S, namely, Ï(S, v) = pv + Îș(S \ {v}, v) if v â S and Ï(S, v) = Îș(S, v) otherwise, where Îș(S, v) is the maximum number of internally disjoint (S, v)-paths. While the approximability of many SL problems was seemingly settled to Î(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 = maxvâV dv is the maximum demand. We improve this by achieving ratio min{p â ln k, k} · O(ln(k/qâ)) for a more general version with node capacities, where p â = maxvâV pv is the maximum bonus and q â = minvâV qv is the minimum capacity. In particular, for the most natural case p â = 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 âŠ(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 â E} and flow-cost bounds {bv: v â 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) â b(V)).
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