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Register a new userFinding Colorful Paths in Temporal Graphs
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Added by
Riccardo Dondi
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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The problem of finding paths in temporal graphs has been recently considered due to its many applications. In this paper we consider a variant of the problem that, given a vertex-colored temporal graph, asks for a path whose vertices have distinct colors and include the maximum number of colors. We study the approximation complexity of the problem and we provide an inapproximability lower bound. Then we present a heuristic for the problem and an experimental evaluation of our heuristic, both on synthetic and real-world graphs.
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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.
Paths $P_1,ldots,P_k$ in a graph $G=(V,E)$ are mutually induced if any two distinct $P_i$ and $P_j$ have neither common vertices nor adjacent vertices (except perhaps their end-vertices). The Induced Disjoint Paths problem is to decide if a graph $G$ with $k$ pairs of specified vertices $(s_i,t_i)$ contains $k$ mutually induced paths $P_i$ such that each $P_i$ connects $s_i$ and $t_i$. This is a classical graph problem that is NP-complete even for $k=2$. We study it for AT-free graphs. Unlike its subclasses of permutation graphs and cocomparability graphs, the class of AT-free graphs has no geometric intersection model. However, by a new, structural analysis of the behaviour of Induced Disjoint Paths for AT-free graphs, we prove that it can be solved in polynomial time for AT-free graphs even when $k$ is part of the input. This is in contrast to the situation for other well-known graph classes, such as planar graphs, claw-free graphs, or more recently, (theta,wheel)-free graphs, for which such a result only holds if $k$ is fixed. As a consequence of our main result, the problem of deciding if a given AT-free graph contains a fixed graph $H$ as an induced topological minor admits a polynomial-time algorithm. In addition, we show that such an algorithm is essentially optimal by proving that the problem is W[1]-hard with parameter $|V_H|$, even on a subclass of AT-free graph, namely cobipartite graphs. We also show that the problems $k$-in-a-Path and $k$-in-a-Tree are polynomial-time solvable on AT-free graphs even if $k$ is part of the input. These problems are to test if a graph has an induced path or induced tree, respectively, spanning $k$ given vertices.
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 $O(nlog n)$-time algorithm of Weihe [FOCS94 & JCSS97] for computing maximum $s,t$-flow in directed planar graphs. Previously, Weihes algorithm could not run in $O(nlog n)$-time due to the absence of the preprocessing step; all the preceding algorithms run in $tilde{Omega}(n^2)$-time [Misiolek-Chen, COCOON05 & IPL06; Biedl, Brejov{{a}} and Vinar, MFCS00]. Consequently, this provides an alternative $O(nlog n)$-time algorithm for computing maximum $s,t$-flow in directed planar graphs in addition to the known $O(nlog n)$-time algorithms [Borradaile-Klein, SODA06 & J.ACM09; Erickson, SODA10]. 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.
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.
For two positive integers $k$ and $ell$, a $(k times ell)$-spindle is the union of $k$ pairwise internally vertex-disjoint directed paths with $ell$ arcs between two vertices $u$ and $v$. We are interested in the (parameterized) complexity of several problems consisting in deciding whether a given digraph contains a subdivision of a spindle, which generalize both the Maximum Flow and Longest Path problems. We obtain the following complexity dichotomy: for a fixed $ell geq 1$, finding the largest $k$ such that an input digraph $G$ contains a subdivision of a $(k times ell)$-spindle is polynomial-time solvable if $ell leq 3$, and NP-hard otherwise. We place special emphasis on finding spindles with exactly two paths and present FPT algorithms that are asymptotically optimal under the ETH. These algorithms are based on the technique of representative families in matroids, and use also color-coding as a subroutine. Finally, we study the case where the input graph is acyclic, and present several algorithmic and hardness results.
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