Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userNew Algorithms for Maximum Disjoint Paths Based on Tree-Likeness
155
0
0.0
(
0
)
Added by
Krzysztof Fleszar
Publication date
2016
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
We study the classical NP-hard problems of finding maximum-size subsets from given sets of $k$ terminal pairs that can be routed via edge-disjoint paths (MaxEDP) or node-disjoint paths (MaxNDP) in a given graph. The approximability of MaxEDP/NDP is currently not well understood; the best known lower bound is $Omega(log^{1/2-epsilon}{n})$, assuming NP$~ otsubseteq~$ZPTIME$(n^{mathrm{poly}log n})$. This constitutes a significant gap to the best known approximation upper bound of $O(sqrt{n})$ due to Chekuri et al. (2006) and closing this gap is currently one of the big open problems in approximation algorithms. In their seminal paper, Raghavan and Thompson (Combinatorica, 1987) introduce the technique of randomized rounding for LPs; their technique gives an $O(1)$-approximation when edges (or nodes) may be used by $O(frac{log n}{loglog n})$ paths. In this paper, we strengthen the above fundamental results. We provide new bounds formulated in terms of the feedback vertex set number $r$ of a graph, which measures its vertex deletion distance to a forest. In particular, we obtain the following. * For MaxEDP, we give an $O(sqrt{r}cdot log^{1.5}{kr})$-approximation algorithm. As $rleq n$, up to logarithmic factors, our result strengthens the best known ratio $O(sqrt{n})$ due to Chekuri et al. * Further, we show how to route $Omega(mathrm{OPT})$ pairs with congestion $O(frac{log{kr}}{loglog{kr}})$, strengthening the bound obtained by the classic approach of Raghavan and Thompson. * For MaxNDP, we give an algorithm that gives the optimal answer in time $(k+r)^{O(r)}cdot n$. If $r$ is at most triple-exponential in $k$, this improves the best known algorithm for MaxNDP with parameter $k$, by Kawarabayashi and Wollan (STOC 2010). We complement these positive results by proving that MaxEDP is NP-hard even for $r=1$, and MaxNDP is W$[1]$-hard for parameter $r$.
rate research
Read More
In the classical Node-Disjoint Paths (NDP) problem, the input consists of an undirected $n$-vertex graph $G$, and a collection $mathcal{M}={(s_1,t_1),ldots,(s_k,t_k)}$ of pairs of its vertices, called source-destination, or demand, pairs. The goal is to route the largest possible number of the demand pairs via node-disjoint paths. The best current approximation for the problem is achieved by a simple greedy algorithm, whose approximation factor is $O(sqrt n)$, while the best current negative result is an $Omega(log^{1/2-delta}n)$-hardness of approximation for any constant $delta$, under standard complexity assumptions. Even seemingly simple special cases of the problem are still poorly understood: when the input graph is a grid, the best current algorithm achieves an $tilde O(n^{1/4})$-approximation, and when it is a general planar graph, the best current approximation ratio of an efficient algorithm is $tilde O(n^{9/19})$. The best currently known lower bound on the approximability of both the
In this paper we revisit the classical Edge Disjoint Paths (EDP) problem, where one is given an undirected graph G and a set of terminal pairs P and asks whether G contains a set of pairwise edge-disjoint paths connecting every terminal pair in P. Our focus lies on structural parameterizations for the problem that allow for efficient (polynomial-time or fpt) algorithms. As our first result, we answer an open question stated in Fleszar, Mnich, and Spoerhase (2016), by showing that the problem can be solved in polynomial time if the input graph has a feedback vertex set of size one. We also show that EDP parameterized by the treewidth and the maximum degree of the input graph is fixed-parameter tractable. Having developed two novel algorithms for EDP using structural restrictions on the input graph, we then turn our attention towards the augmented graph, i.e., the graph obtained from the input graph after adding one edge between every terminal pair. In constrast to the input graph, where EDP is known to remain NP-hard even for treewidth two, a result by Zhou et al. (2000) shows that EDP can be solved in non-uniform polynomial time if the augmented graph has constant treewidth; we note that the possible improvement of this result to an fpt-algorithm has remained open since then. We show that this is highly unlikely by establishing the W[1]-hardness of the problem parameterized by the treewidth (and even feedback vertex set) of the augmented graph. Finally, we develop an fpt-algorithm for EDP by exploiting a novel structural parameter of the augmented graph.
We study the classical Node-Disjoint Paths (NDP) problem: given an $n$-vertex graph $G$ and a collection $M={(s_1,t_1),ldots,(s_k,t_k)}$ of pairs of vertices of $G$ called demand pairs, find a maximum-cardinality set of node-disjoint paths connecting the demand pairs. NDP is one of the most basic routing problems, that has been studied extensively. Despite this, there are still wide gaps in our understanding of its approximability: the best currently known upper bound of $O(sqrt n)$ on its approximation ratio is achieved via a simple greedy algorithm, while the best current negative result shows that the problem does not have a better than $Omega(log^{1/2-delta}n)$-approximation for any constant $delta$, under standard complexity assumptions. Even for planar graphs no better approximation algorithms are known, and to the best of our knowledge, the best negative bound is APX-hardness. Perhaps the biggest obstacle to obtaining better approximation algorithms for NDP is that most currently known approximation algorithms for this type of problems rely on the standard multicommodity flow relaxation, whose integrality gap is $Omega(sqrt n)$ for NDP, even in planar graphs. In this paper, we break the barrier of $O(sqrt n)$ on the approximability of the NDP problem in planar graphs and obtain an $tilde O(n^{9/19})$-approximation. We introduce a new linear programming relaxation of the problem, and a number of new techniques, that we hope will be helpful in designing more powerful algorithms for this and related problems.
In the Disjoint Paths problem, the input is an undirected graph $G$ on $n$ vertices and a set of $k$ vertex pairs, ${s_i,t_i}_{i=1}^k$, and the task is to find $k$ pairwise vertex-disjoint paths connecting $s_i$ to $t_i$. The problem was shown to have an $f(k)n^3$ algorithm by Robertson and Seymour. In modern terminology, this means that Disjoint Paths is fixed parameter tractable (FPT), parameterized by the number of vertex pairs. This algorithm is the cornerstone of the entire graph minor theory, and a vital ingredient in the $g(k)n^3$ algorithm for Minor Testing (given two undirected graphs, $G$ and $H$ on $n$ and $k$ vertices, respectively, the objective is to check whether $G$ contains $H$ as a minor). All we know about $f$ and $g$ is that these are computable functions. Thus, a challenging open problem in graph algorithms is to devise an algorithm for Disjoint Paths where $f$ is single exponential. That is, $f$ is of the form $2^{{sf poly}(k)}$. The algorithm of Robertson and Seymour relies on topology and essentially reduces the problem to surface-embedded graphs. Thus, the first major obstacle that has to be overcome in order to get an algorithm with a single exponential running time for Disjoint Paths and {sf Minor Testing} on general graphs is to solve Disjoint Paths in single exponential time on surface-embedded graphs and in particular on planar graphs. Even when the inputs to Disjoint Paths are restricted to planar graphs, a case called the Planar Disjoint Paths problem, the best known algorithm has running time $2^{2^{O(k)}}n^2$. In this paper, we make the first step towards our quest for designing a single exponential time algorithm for Disjoint Paths by giving a $2^{O(k^2)}n^{O(1)}$-time algorithm for Planar Disjoint Paths.
The problem of finding the maximum number of vertex-disjoint uni-color paths in an edge-colored graph (called MaxCDP) has been recently introduced in literature, motivated by applications in social network analysis. In this paper we investigate how the complexity of the problem depends on graph parameters (namely the number of vertices to remove to make the graph a collection of disjoint paths and the size of the vertex cover of the graph), which makes sense since graphs in social networks are not random and have structure. The problem was known to be hard to approximate in polynomial time and not fixed-parameter tractable (FPT) for the natural parameter. Here, we show that it is still hard to approximate, even in FPT-time. Finally, we introduce a new variant of the problem, called MaxCDDP, whose goal is to find the maximum number of vertex-disjoint and color-disjoint uni-color paths. We extend some of the results of MaxCDP to this new variant, and we prove that unlike MaxCDP, MaxCDDP is already hard on graphs at distance two from disjoint paths.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
