Efficient approximation algorithms for computing k disjoint constrained shortest paths

Let G=(V, E) be a digraph with nonnegative integral cost and delay on each edge, s and t be two vertices, and D ∈ Z+/o be a delay bound, the k disjoint Restricted Shortest Path (k RSP) problem is to compute k disjoint paths between s and t with the total cost minimized and the total delay bounded by D. In this paper, we first present a pseudo-polynomial-time algorithm with a bifactor approximation ratio of (1,2), then improve the algorithm to polynomial time with a bifactor ratio of (1+e,2+e) for any fixed e>0, which is better than the current best approximation ratio (O(1+λ), O(1 + ln 1/λ)) for any fixed λʌ0. To the best of our knowledge, this is the first constant-factor algorithm that almost strictly obeys kRSP constraint.

In the maximum edge-disjoint paths problem, we are given a graph and a collection of pairs of vertices, and the objective is to find the maximum number of pairs that can be routed by edge-disjoint paths. An r-approximation algorithm for this problem is a polynomial time algorithm that finds at least OPT / r edge-disjoint paths, where OPT is the maximum possible. Currently, an O(n1/2)-approximation algorithm is best known for this problem even if a congestion of two is allowed, i.e., each edge is allowed to be used in at most two of the paths.In this paper, we give a randomized O(n3/7 • poly (log n))-approximation algorithm with congestion two. This is the first result that breaks the O(n1/2)-approximation algorithm. In particular, we prove the following. 1. If we have a (randomized) polynomial time algorithm for finding Ω(OPT1/p) edge-disjoint paths for some p>1, then, for some α >0, we can give a randomized O(n1/2-α)-approximation algorithm for the maximum edge-disjoint paths problem by using Rao-Zhou's algorithm. 2. Based on the well-linked set of Chekuri, Khanna, and Shepherd, we show that there is a randomized algorithm for finding Ω(OPT1/4) edge-disjoint paths connecting given terminal pairs with congestion two.Our framework for this algorithm is more general. Indeed, the above two ingredients also work for the maximum edge-disjoint paths problem (with congestion one) if the following conjecture is true. Conjecture: There is a (randomized) polynomial time algorithm for finding Ω(OPT1/p/β(n)) edge-disjoint paths connecting given terminal pairs, where β is a poly-logarithmic function.

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.

In this paper we consider the decremental single-source shortest paths (SSSP) problem, where given a graph G and a source node s the goal is to maintain shortest paths between s and all other nodes in G under a sequence of online adversarial edge deletions. In their seminal work, Even and Shiloach [JACM 1981] presented an exact solution to the problem with only O(mn) total update time over all edge deletions. Their classic algorithm was the best known result for the decremental SSSP problem for three decades, even when approximate shortest paths are allowed. The first improvement over the Even-Shiloach algorithm was given by Bernstein and Roditty [SODA 2011], who for the case of an unweighted and undirected graph presented an approximate (1+) algorithm with constant query time and a total update time of O(n2+O(1/√logn)). This work triggered a series of new results, culminating in a recent breakthrough of Henzinger, Krinninger and Nanongkai [FOCS 14], who presented a -approximate algorithm whose total update time is near linear O(m1+ O(1/√logn)). In this paper they posed as a major open problem the question of derandomizing their result. In fact, all known improvements over the Even-Shiloach algorithm are randomized. All these algorithms maintain some truncated shortest path trees from a small subset of nodes. While in the randomized setting it is possible to “hide” these nodes from the adversary, in the deterministic setting this is impossible: the adversary can delete all edges touching these nodes, thus forcing the algorithm to choose a new set of nodes and incur a new computation of shortest paths. In this paper we present the first deterministic decremental SSSP algorithm that breaks the Even-Shiloach bound of O(mn) total update time, for unweighted and undirected graphs. Our algorithm is (1 + є) approximate and achieves a total update time of O(n2). Our algorithm can also achieve the same bounds in the incremental setting. It is worth mentioning that for dense instances where m = Ω(n2 − 1/√log(n)), our algorithm is also faster than all existing randomized algorithms.

Approximating minimum size {1,2}-connected networks

We consider routing in survivable networks that provide protection against node or link failures. In these networks resilience against failures is provided by routing connections on pairs of disjoint paths called primary and backup paths. The primary path of a connection carries its traffic under normal circumstances and in the eventuality of a network failure effecting the primary path the connection traffic (all or some portion of it) is rerouted over its backup path. In an online setting as connection requests arrive a pair of disjoint primary and backup paths of least total cost (under some link cost metric) are selected to route the connections. In many situations the cost metric used for the primary path differs from the cost metric used for the backup path. Also in many realistic settings these two cost metrics are related to each other. In this paper we study the problem of finding a pair of edge or node disjoint paths of least total cost where the cost of the primary path is the total cost of its links while the cost for the backup path is α times the sum of the cost of its links, for some given α < 1. We show that the problem is hard to approximate to within a factor $$\Omega((\frac{1}{\alpha})^{1-\epsilon})$$ for any positive $$\epsilon \le 1$$ . In addition we show that the problem is complete for a set of hard to approximate problems. On the positive side we show that a simple algorithm achieves an approximation ratio of $$O(\frac{1}{\alpha})$$ for the problem.

Since the emergence of Artificial Intelligence as a field, planning under uncertainty has been viewed as one of its crucial subareas. Accordingly, the current lack of scalability of probabilistic planning techniques is a major reason why the grand vision of AI has not been fulfilled yet. Besides being an obstacle to advances in AI, scalability issues also hamper the applicability of probabilistic planning to real-world problems. A powerful framework for describing probabilistic planning problems is Markov Decision Processes (MDPs). Informally, an MDP specifies the objectives the agent is trying to achieve, the actions the agent can perform, and the states in which it can end up while working towards the objective. Solving an MDP means finding a policy, i.e. an assignment of actions to states, that allows the agent to achieve the objective. Optimal solution methods, those that look for the “best” policy according to some criterion, typically try to analyze all possible states or a large fraction of them. Since state space sizes of realistic scenarios can be astronomical, these algorithms quickly run out of memory. Fortunately, the mathematical structure of some classes of MDPs has allowed for inventing more efficient algorithms. This is the case for Stochastic Shortest Path (SSP) problems, whose mathematical properties gave rise to a family of algorithms called Find-and-Revise. When used in combination with a heuristic, the members of this family can find a near-optimal, or even optimal, policy while avoiding the analysis of many states. Nonetheless, the sheer number of states in real-world problems forces algorithms based on state-level analysis to exhaust their capabilities much too early, calling for a fundamentally different approach. Moreover, several expressive and potentially useful MDP classes are currently not known to have a mathematical structure for creating efficient approximation techniques. This dissertation advances the state of the art in probabilistic planning in three complementary ways. For SSP MDPs, it derives a novel class of approximate algorithms based on generalizing state analysis information across many states. This information is stored in the form of basis functions that are generated automatically and the number of which is much smaller than the number of states. As a result, the proposed algorithms have a very compact memory footprint and arrive at high-quality solutions faster than their state-based counterparts. In a parallel effort, the dissertation also builds up mathematical apparatus for classes of MDPs that previously had no applicable approximation algorithms. The developed theory enables the extension of the powerful Find-and-Revise paradigm to these MDP classes as well, providing the first memory-efficient algorithms for solving them. Last but not least, the dissertation will apply the proposed theoretical techniques to a large-scale real-world problem, urban traffic routing being one of the candidates.

Given a directed graph G with non-negative cost on the arcs, a directed tour cover T of G is a cycle (not necessary simple) in G such that either head or tail (or both of them) of every arc in G is touched by T . The minimum directed tour cover problem (DToCP) which is to find a directed tour cover of minimum cost, is NP -hard. It is thus interesting to design approximation algorithms with performance guarantee to solve this problem. Although its undirected counterpart (ToCP) has been studied in recent years [1,6], in our knowledge, the DTCP remains widely open. In this paper, we give a 2log2 (n )-approximation algorithm for the DTCP.

Disjoint paths and connected subgraphs for H-free graphs

Let G = (V, E, w) be a directed graph, where w : V ← R is an arbitrary weight function defined on its vertices. The bottleneck weight, or the capacity, of a path is the smallest weight of a vertex on the path. For two vertices u, v the bottleneck weight, or the capacity, from u to v, denoted c(u, v), is the maximum bottleneck weight of a path from u to v. In the All-Pairs Bottleneck Paths (APBP) problem we have to find the bottleneck weights for all ordered pairs of vertices. Our main result is an O(n2.575) time algorithm for the APBP problem. The exponent is derived from the exponent of fast matrix multiplication. Our algorithm is the first sub-cubic algorithm for this problem. Unlike the sub-cubic algorithm for the all-pairs shortest paths (APSP) problem, that only applies to bounded (or relatively small) integer edge or vertex weights, the algorithm presented for APBP problem works for arbitrary large vertex weights.The APBP problem has numerous applications, and several interesting problems that have recently attracted attention can be reduced to it, with no asymptotic loss in the running times of the known algorithms for these problems. Some examples are a result of Vassilevska and Williams [STOC 2006] on finding a triangle of maximum weight, a result of Bender et al. [SODA 2001] on computing least common ancestors in DAGs and a result of Kowaluk and Lingas [ICALP 2005] on finding maximum witnesses for boolean matrix multiplication. Thus, the APBP problem provides a uniform framework for these applications. For some of these problems, we can in fact show that their complexity is equivalent to that of the APBP problem.A slight modification of our algorithm enables us to compute shortest paths of maximum bottleneck weight. Let d(u, v) denote the (unweighted) distance from u to v, and let sc(u, v) denote the maximum bottleneck weight of a path from u to v having length d(u, v). The All-Pairs Bottleneck Shortest Paths (APBSP) problem is to compute sc(u, v) for all ordered pairs of vertices. We present an algorithm for the APBSP problem whose running time is O(n2.86).

Labeling methods for partially ordered paths

The Random Shortest Path Problem with the second moment criterion is discussed in this paper. After the formulation of the problem, exact algorithms, based on general concepts for solving the Multi-objective Shortest Path Problem, are described. Next, several approximate algorithms are proposed. It is shown that the complexity of the exact algorithms is exponential, while the complexity of the approximate algorithms is only polynomial. Computational results for the exact and approximate algorithms, which were performed on large graphs, are shown.

Finding a delay constrained least-cost path in a directed graph is a fundamental issue for quality of service provisioning in communication networks. Recently Feng and Turkay proposed a hybrid algorithm which uses a heuristic approach to boost the average performance of a well known approximation algorithm. In this paper we provide extensive simulation results to show that the hybrid algorithm runs significantly faster than the approximation algorithm, often by one or two orders of magnitude.

The Maximum Carpool Matching problem is a star packing problem in directed graphs. Formally, given a directed graph G = (V, A), a capacity function c: V -> N, and a weight function w: A -> R^+, a carpool matching is a subset of arcs, M subseteq A, such that every v in V satisfies: (i) d^{in}_M(v) cdot d^{out}_M(v) = 0, (ii) d^{in}_M(v) <= c(v), and (iii) d^{out}_M(v) <= 1. A vertex v for which d^{out}_M(v) = 1 is a passenger, and a vertex for which d^{out}_M(v) = 0 is a driver who has d^{in}_M(v) passengers. In the Maximum Carpool Matching problem the goal is to find a carpool matching M of maximum total weight. The problem arises when designing an online carpool service, such as Zimride, which tries to connect between users based on a similarity function. The problem is known to be NP-hard, even in the unweighted and uncapacitated case. The Maximum Group Carpool Matching problem, is an extension of Maximum Carpool Matching where each vertex represents an unsplittable group of passengers. Formally, each vertex u in V has a size s(u) in N, and the constraint d^{in}_M(v) <= c(v) is replaced with sum_{u:(u,v) in M} s(u) <= c(v). We show that Maximum Carpool Matching can be formulated as an unconstrained submodular maximization problem, thus it admits a 1/2-approximation algorithm. We show that the same formulation does not work for Maximum Group Carpool Matching, nevertheless, we present a local search (1/2 - epsilon)-approximation algorithm for Maximum Group Carpool Matching. For the unweighted variant of both problems when the maximum possible capacity, c_{max}, is bounded by a constant, we provide a local search (1/2 + 1/{2c_{max}} - epsilon)-approximation algorithm. We also show that the problem is APX-hard, even if the maximum degree and c_{max} are at most 3.

The shortest path problem (SPP) constitutes one of the most frequently encountered classes of problems in graph theory. It is certainly the most fundamental of components in the fields of transportation and communication networks. Shortest path problems may be encountered directly, possibly as a result of a clever formulation of a problem not at first sight involving shortest paths, or indirectly as a subproblem in the solution of a more complicated optimization problem. This use of shortest path problems as subroutines motivates the search for algorithms with good theoretical bounds on running time. We also seek computer implementations whose empirical performances are rapid in spite of perhaps weak theoretical bounds for the algorithms they implement. As testimony to the importance of shortest path and related problems, a large number of surveys, annotated bibliographies, and reviews have appeared over the past thirty years. Among them are those by Dreyfus (1969), Pierce (1975), Golden and Magnanti (1977), and Gallo and Pallottino (1988).