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An L-length-bounded cut in a graph G with source s, and sink t is a cut that destroys all s-t-paths of length at most L. An L-length-bounded flow is a flow in which only flow paths of length at most L are used. We show that the minimum length-bounded cut problem in graphs with unit edge lengths is NP-hard to approximate within a factor of at least 1.1377 for L ≧ 5 in the case of node-cuts and for L ≧ 4 in the case of edge-cuts. We also give approximation algorithms of ratio min{L, n/L} in the node case and min{L, n2/L2,√m} in the edge case, where n denotes the number of nodes and m denotes the number of edges. We discuss the integrality gaps of the LP relaxations of length-bounded flow and cut problems, analyze the structure of optimal solutions, and present further complexity results for special cases.

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... In each iteration of this algorithm, the algorithm needs to decide the following question: given an edge (u, v), is there a fault set F ⊆ E where |F | ≤ f and d H\F (u, v) ≤ (2k − 1) · d G\F (u, v)? Doing this in the obvious way (checking all sets of f or fewer edges) takes time exponential in f . And, unfortunately, it turns out that this question is equivalent to the Length Bounded Cut problem (LBC), which is known to be NP-hard [4]. ...

... A quick counting argument, standard in prior work (for example, [5]), shows that one introduces only O(n) new nodes in this way, and hence this again changes our claimed upper bounds on spanner size only by a constant factor. 4. We assume that all edges in H have distinct weights, so that we may unambiguously refer to the heaviest or lightest edge among a set of edges. ...

... Doing this in the obvious way (checking all sets of f or fewer edges) takes time exponential in f . And, unfortunately, it turns out that this question is equivalent to the Length Bounded Cut problem (LBC), which is known to be NP-hard [4]. ...

... A quick counting argument, standard in prior work (for example, [5]), shows that one introduces only O(n) new nodes in this way, and hence this again changes our claimed upper bounds on spanner size only by a constant factor. 4. We assume that all edges in H have distinct weights, so that we may unambiguously refer to the heaviest or lightest edge among a set of edges. ...

Recent work has established that, for every positive integer $k$, every $n$-node graph has a $(2k-1)$-spanner on $O(f^{1-1/k} n^{1+1/k})$ edges that is resilient to $f$ edge or vertex faults. For vertex faults, this bound is tight. However, the case of edge faults is not as well understood: the best known lower bound for general $k$ is $\Omega(f^{\frac12 - \frac{1}{2k}} n^{1+1/k} +fn)$. Our main result is to nearly close this gap with an improved upper bound, thus separating the cases of edge and vertex faults. For odd $k$, our new upper bound is $O_k(f^{\frac12 - \frac{1}{2k}} n^{1+1/k} + fn)$, which is tight up to hidden $poly(k)$ factors. For even $k$, our new upper bound is $O_k(f^{1/2} n^{1+1/k} +fn)$, which leaves a gap of $poly(k) f^{1/(2k)}$. Our proof is an analysis of the fault-tolerant greedy algorithm, which requires exponential time, but we also show that there is a polynomial-time algorithm which creates edge fault tolerant spanners that are larger only by factors of $k$.

... These edge-disjoint paths can be interpreted as a 0-1-valued flow in a network with unit capacity and unit edge length. Recently, Baier (2003) and Baier et al. (2006) studied a more general approach with fractional flow values. Here, the so-called length-bounded flow is feasible if and only if there exists a path decomposition such that each flow carrying path fulfills the resource constraint. ...

... Here, the so-called length-bounded flow is feasible if and only if there exists a path decomposition such that each flow carrying path fulfills the resource constraint. The authors of Baier (2003) and Baier et al. (2006) showed that it is already N P -complete to decide whether a given edge-flow can be decomposed into such paths. Still, the authors succeeded using an approximation algorithm for CSP to obtain an FPTAS for their flow problem. ...

Electric and hybrid vehicles are a big step towards a greener mobility, but they also open up completely new questions regarding the shortest path problem and the planning of trips. Since recharging an electric car will take much longer than refilling conventional fossil fuels, we have to balance between speed and range and we have to choose stops for charging wisely. For hybrid vehicles, a symbiosis between navigation system and power train control to choose a path with optimal phases for depleting and recharging the battery may yield much more energy-efficient paths. In this paper, we develop an appropriate model for finding shortest routes for these kinds of vehicles, which is mainly a constrained shortest path problem with convertible resources and charging stations. We study properties of solutions by classifying several types of cycles that may occur in the optimal route. We state sufficient conditions to exclude some of these cycle classes and we derive appropriate approximation schemes with provable quality and strict feasibility. We also study the related network flow problem for operating fleets of electric vehicles, e.g., shared vehicles or buses in urban areas.

... Several authors contributed to this topic, which has important applications to robust telecommunication networks, see e.g. [1, 12]. There is also a vast literature on bilevel network pricing problems, see e.g. ...

... We hence study a slightly more general model. For each commodity k i ∈ K we assume that a monotone detour willingness function ω i : [0, ∞) → [0, 1], satisfying ω i (0) = 0 and lim ∆→∞ ω i (∆) = 1, gives the fraction ω i (L − i ) of drivers from commodity k i who have a detour threshold ≤ L − i . Hence, the objective of the MAXCOV problem can be generalized to ...

We study an extension of the shortest path network interdiction problem and present a novel real-world application in this area. We consider the problem of determining optimal locations for toll control stations on the arcs of a transportation network. We handle the fact that drivers can avoid control stations on parallel secondary roads. The problem is formulated as a mixed integer program and solved using Benders decomposition. We present experimental results for the application of our models to German motorways.

... For finite values of s, the result follows from [15] and [20]. ▯ In fact, for any finite s ≥ 4, it is NP-hard even to approximate sep s G ðx; XÞ within a constant factor of 1.1377 [3]. Let G be a graph, k ∈ N, and s ∈ N þ . ...

... However, our parameters are polynomial-time computable for fugitives with unbounded speed or speed at most three, and NP-complete for all other finite speeds. As we mentioned in section 4, the results in [15], [20], and [3] imply that checking whether sep s G ðx; XÞ ≤ k is an NP-complete problem for every fixed s ≥ 4. Recent results in [14] imply that, for every fixed s, this problem can be solved by an s k · n Oð1Þ -step algorithm (i.e., an FPT-algorithm). Apparently, using the algorithm of section 6, the same type of algorithm can be derived for checking whether vlns s ðGÞ ≤ k or vlms s ðGÞ ≤ k. ...

Graph searching problems are described as games played on graphs, between a set of cops and a fugitive. Variants of the game
restrict the abilities of the cops and the fugitive and the corresponding search numbers (the least number of cops that have
a winning strategy) are related to several well-known parameters in graph theory. We study the case where the fugitive is
visible (the cops’ strategy can take into account his current position) and lazy (he moves only when the cops move to his
position). Our results are stated and proven in a general setting where the fugitive’s speed (i.e., the lengths of paths he
can move along) can be unbounded or bounded by some constant. We give a min-max characterization of the corresponding parameters,
which we show to be computable in polynomial time for fugitivess with unbounded speed and speed at most3 and to be NP-complete for all other finite speeds. This is in contrast to the other standard versions of the game, where the parameters
corresponding to fugitives with unbounded speed are NP-complete. Several consequences of our results are also discussed.

... For finite values of s, the result follows from [15] and [20]. ▯ In fact, for any finite s ≥ 4, it is NP-hard even to approximate sep s G ðx; XÞ within a constant factor of 1.1377 [3]. Let G be a graph, k ∈ N, and s ∈ N þ . ...

... However, our parameters are polynomial-time computable for fugitives with unbounded speed or speed at most three, and NP-complete for all other finite speeds. As we mentioned in section 4, the results in [15], [20], and [3] imply that checking whether sep s G ðx; XÞ ≤ k is an NP-complete problem for every fixed s ≥ 4. Recent results in [14] imply that, for every fixed s, this problem can be solved by an s k · n Oð1Þ -step algorithm (i.e., an FPT-algorithm). Apparently, using the algorithm of section 6, the same type of algorithm can be derived for checking whether vlns s ðGÞ ≤ k or vlms s ðGÞ ≤ k. ...

... Corollary 2. 10. The flows and weights returned by DECOMP(D,f init ,∅,1) yield a convex combination of f . ...

... Some heuristics for different problems imposing length bounds can be found in [85,17]. The first results for length-bounded flows were obtained by Mahjoub and McCormick [74] and Baier et al. [10,9]. Baier et al. present various complexity and algorithmic results, including that the integrality gap of the problem is in Ω( √ m) and that there is no polynomial algorithm which computes a length-bounded path-flow from a given edge-flow that is known to correspond to a length-bounded path-flow. ...

This thesis focuses on approximation algorithms and complexity assessments concerning network flows. It deals with various network flow problems with path restrictions. These restrictions cover the number of paths that are used to route commodities as well as the amount of flow that is routed along a single path or the path's length.
Concerning the first restriction we study the unsplittable flow problem-a generalization of the NP-hard edge-disjoint paths problem. Given a network with commodities that must be routed from their sources to their sinks, the unsplittable flow problem forbids each commodity to use more than one path. For this problem we prove a new lower bound on the performance guarantee of randomized rounding which so far belongs to the best approximation algorithms known for this problem. Further, we present an interesting relation between unsplittable flows and classical (splittable) multicommodity flows in the case that all commodities share a common source: Each single source multicommodity flow can be represented as a convex combination of unsplittable flows of congestion at most 2.
Further, we combine different path restrictions from the ones mentioned above. In the k-splittable flow problem with path capacities, we study the NP-hard problem that each commodity may be sent along a limited number of paths while the flow value of each path is bounded. This yields a generalization of the unsplittable flow problem, but we show how one can obtain the same asymptotic approximation ratios. For the length-bounded k-splittable flow problem, we consider the single commodity case and develop a constant factor approximation algorithm.
A crucial characteristic of network flows occurring in real-world applications is flow variation over time and the fact that flow does not travel instantaneously through a network but requires a certain amount of time to travel through each arc. Both characteristics are captured by "flows over time" which specify a flow rate for each arc and each point in time. We consider the quickest single commodity k-splittable flow problem and give a constant factor approximation algorithm for it. So far only results for k-splittable flows as well as for length-bounded flows and flows over time have been known, but nothing was known for combinations of them.
Bounding the flow value of each path is also interesting in the classical maximum s-t-flow problem. We study the case that each path may carry at most one unit of flow and prove that this restriction makes the maximum s-t-flow problem strongly NP-hard. In contrast to the classical maximum s-t-flow problem, the fractional and the integral problem diverge strongly with the new restriction. For the integral problem, we even prove APX-hardness. We develop an FPTAS for the fractional problem and an O(log m)-approximation algorithm for the integral one. (Here, m is the number of arcs in the network under consideration.) Similar results emerge for the multicommodity case. For the objective to find a maximum integral multicommodity flow our asymptotic approximation ratio of O(m^{0.5}) is proven to be best possible, unless P = NP.

... . The obvious way of doing this takes Ω(n f ) time in order to check all possible fault sets, which is not polynomial if f is superconstant (which is the interesting case, as we are studying f -dependence). One might hope to design a polynomial-time algorithm to perform this check, but even special cases of this problem are NP-hard: if the graph is unweighted then this is equivalent to the Length-Bounded Cut problem, which is known to be NP-hard [7]. The same problem arises in the context of VFT spanners, where the best-known algorithm is the VFT-greedy algorithm (exactly Algorithm 2 but without adding emulator edges). ...

A $k$-spanner of a graph $G$ is a sparse subgraph that preserves its shortest path distances up to a multiplicative stretch factor of $k$, and a $k$-emulator is similar but not required to be a subgraph of $G$. A classic theorem by Thorup and Zwick [JACM '05] shows that, despite the extra flexibility available to emulators, the size/stretch tradeoffs for spanners and emulators are equivalent. Our main result is that this equivalence in tradeoffs no longer holds in the commonly-studied setting of graphs with vertex failures. That is: we introduce a natural definition of vertex fault-tolerant emulators, and then we show a three-way tradeoff between size, stretch, and fault-tolerance for these emulators that polynomially surpasses the tradeoff known to be optimal for spanners. We complement our emulator upper bound with a lower bound construction that is essentially tight (within $\log n$ factors of the upper bound) when the stretch is $2k-1$ and $k$ is either a fixed odd integer or $2$. We also show constructions of fault-tolerant emulators with additive error, demonstrating that these also enjoy significantly improved tradeoffs over those available for fault-tolerant additive spanners.

... From both sets one can construct identically-sized BDD (which differ only by the arc labels), since the sets define a pair of dual monotone Boolean functions. We can apply Lemma 3 to this problem as follows: For fixed s and t and each α the set T M ≤ α contains the minimal sets of edges that need to be removed to destroy all shortest st-paths of length at most α (sometimes called α-lengthbounded cuts [3]). Clearly, each such set is a subset of some minimal st-cut. ...

Modeling decision-dependent scenario probabilities in stochastic programs is difficult and typically leads to large and highly non-linear MINLPs that are very difficult to solve. In this paper, we develop a new approach to obtain a compact representation of the recourse function using a set of binary decision diagrams (BDDs) that encode a nested cover of the scenario set. The resulting BDDs can then be used to efficiently characterize the decision-dependent scenario probabilities by a set of linear inequalities, which essentially factorizes the probability distribution and thus allows to reformulate the entire problem as a small mixed-integer linear program. The approach is applicable to a large class of stochastic programs with multivariate binary scenario sets, such as stochastic network design, network reliability, or stochastic network interdiction problems. Computational results show that the BDD-based scenario representation reduces the problem size, and hence the computation time, significant compared to previous approaches.

... The Firefighter problem is NP-complete even when the underlying graph is a tree [17], although [24] and [29] give 1/2 and (1 − 1/e) approximation algorithms for this case respectively, and [31] shows how to solve the problem in polynomial time for special cases of trees. The Firefighter problem with the MinBudget objective, has some structural similarity with the problem of length-bounded cuts [3], where the goal is to form a minimum s-t cut that destroys all paths of length at most L. If d is the length of the shortest path from the source of infection to node u, then all (s, u) paths of length at most d need to be 'cut' before or at time step d in order to save the node. However, as illustrated by the example above, the main difference here is that our paper deals with dynamic cuts i.e., cuts which arise over a period of time, while graph cuts are static in nature. ...

We provide approximation algorithms for several variants of the Firefighter problem on general graphs. The Fireghter problem models the case where a diusive process such as an infection (or an idea, a computer virus, a re) is spreading through a network, and our goal is to contain this infection by using targeted vaccinations. Specically, we are allowed to vaccinate at most a xed number (called the budget) of nodes per time-step, with the goal of minimizing the eect of the infection. The diculty of this problem comes from its temporal component, since we must choose nodes to vaccinate at every time-step while the infection is spreading through the network, leading to notions of \cuts over time". We consider two versions of the Fireghter problem: a \non-spreading" model, where vac- cinating a node means only that this node cannot be infected; and a \spreading" model where the vaccination itself is an infectious process, such as in the case where the infection is a harmful idea, and the vaccine to it is another infectious benecial idea. We look at two measures: the MaxSave measure in which we want to maximize the number of nodes which are not infected given a xed budget, and the MinBudget measure, in which we are given a set of nodes which we have to save and the goal is to minimize the budget. We study the approximability of these problems in both models.

... For each link from super source to source, we manipulate a length weight that is the source's length constraint minus the minimum length constraint (e.g., − ). Thus, the original problem becomes a NP maximum lengthbounded flow problem, which is to find a maximum flow between one source and one target where the length of all flows are bounded by a length constraint [26]. ...

3D-stacked ICs that employ through-silicon vias (TSVs) to connect multiple dies vertically have gained wide-spread interest in the semiconductor industry. In order to be commercially viable, the assembly yield for 3D-stacked ICs must be as high as possible, requiring TSVs to be reparable. Existing techniques typically assume TSV faults to be uniformly distributed and use neighboring TSVs to repair faulty ones, if any. In practice, however, clustered TSV faults are quite common due to the fact that the TSV bonding quality depends on surface roughness and cleaness of silicon dies, rendering prior TSV redundancy solutions less effective. To resolve this problem, we present a novel TSV repair framework, including a hardware architecture that enables faulty TSVs to be repaired by redundant TSVs that are farther apart, and the corresponding repair algorithm. By doing so, the manufacturing yield for 3D-stacked ICs can be dramatically improved, as demonstrated in our experimental results.

... , P k ) and corresponding flow values f1, . . . , f k , in which no path Pi ∈ P is longer than L. Baier [3] gives an extensive survey of what is known for length-bounded flows, and more recent results have been obtained for instance in [4] . Unfortunately, (multicommodity) network flows do not adequately model multicast problems, because the flow conservation condition does not apply to multicast problems: In multicast applications, intermediate relay nodes may send received data to several receivers, thus reducing the required bandwidth at the original sender. ...

Motivated by an application in distributed gaming, we define and study the latency-constrained total upload maximization problem . In this problem, a peer-to-peer overlay network is modeled as a complete graph and each node vi has an upload bandwidth capac- ity ci and a set of receivers R(i). Each sender-receiver pair (vi, vj), where vj ∈ R(i), is a request that should be satisfied, i.e., vi should send a data packet to each vj ∈ R(i). The goal is to find a set of at most n multicast-trees Ti of depth at most 2, such that each node can be part of multiple trees, all capacity constraints are m et, and the number of satisfied requests is maximized. In this paper, we prove that the problem is NP-complete, and we present an algo- rithm with approximation ratio 1 − 2/ √ cmin, where cmin is the minimum upload capacity. Finally, we also study the impact of network codingon the quality and approximability of the solution.

... Our approach employs an approximation algorithm, which selectively drops some cost-unfavorable sessions and maximizes the rates of the rest sessions. Path constrained flows have been studied in a wired setting, e.g., [5], [18], but the interference constraints, and the fact that we need both short paths and large flow values make our problem different. Our algorithm involves the following steps. ...

We study the problem of throughput maximization in multi-hop wireless networks with end-to-end delay constraints for each session. This problem has received much attention starting with the work of Grossglauser and Tse (2002), and it has been shown that there is a significant tradeoff between the end-to-end delays and the total achievable rate. We develop algorithms to compute such tradeoffs with provable performance guarantees for arbitrary instances, with general interference models. Given a target delay-bound Δ(c) for each session c, our algorithm gives a stable flow vector with a total throughput within a factor of O (logΔ<sub>m</sub>/log log Δ<sub>m</sub>) of the maximum, so that the per-session (end-to-end) delay is O ((logΔ<sub>m</sub>/log log Δ<sub>m</sub> Δ(c))<sup>2</sup>), where Δ<sub>m</sub> = max<sub>c</sub>{Δ(c)}; note that these bounds depend only on the delays, and not on the network size, and this is the first such result, to our knowledge.

... Figure 2 shows an instance of the max-1FP in which the maximum flow yields an integral flow value on all arcs, although it is fractional. A similar network is used in Baier et al. (2006) to prove the discrepancy of arc-wise and path-wise integrality for the length-bounded flow problem. ...

Since the seminal work of Ford and Fulkerson in the 1950s, network flow theory is one of the most important and most active
areas of research in combinatorial optimization. Coming from the classical maximum flow problem, we introduce and study an
apparently basic but new flow problem that features a couple of interesting peculiarities. We derive several results on the
complexity and approximability of the new problem. On the way we also discover two closely related basic covering and packing
problems that are of independent interest.
Starting from an LP formulation of the maximum s-t-flow problem in path variables, we introduce unit upper bounds on the amount of flow being sent along each path. The resulting
(fractional) flow problem is NP-hard; its integral version is strongly NP-hard already on very simple classes of graphs. For
the fractional problem we present an FPTAS that is based on solving the k shortest paths problem iteratively. We show that the integral problem is hard to approximate and give an interesting O(log m)-approximation algorithm, where m is the number of arcs in the considered graph. For the multicommodity version of the problem there is an
-approximation algorithm. We argue that this performance guarantee is best possible, unless P=NP.

We introduce and study Weighted Min (s, t)-Cut Prevention, where we are given a graph \(G=(V,E)\) with vertices s and t and an edge cost function and the aim is to choose an edge set D of total cost at most d such that G has no (s, t)-edge cut of capacity at most a that is disjoint from D. We show that Weighted Min (s, t)-Cut Prevention is NP-hard even on subcubcic graphs when all edges have capacity and cost one and provide a comprehensive study of the parameterized complexity of the problem. We show, for example W[1]-hardness with respect to d and an FPT algorithm for a.

We consider a cops and robber game where the cops are blocking edges of a graph, while the robber occupies its vertices. At each round of the game, the cops choose some set of edges to block and right after the robber is obliged to move to another vertex traversing at most s unblocked edges (s can be seen as the speed of the robber). Both parts have complete knowledge of the opponent's moves and the cops win when they occupy all edges incident to the robbers position. We introduce the capture cost on G against a robber of speed s. This defines a hierarchy of invariants, namely δe1,δe2,…,δe∞, where δe∞ is an edge-analogue of the admissibility graph invariant, namely the edge-admissibility of a graph. We prove that the problem asking wether δes(G)≤k, is polynomially solvable when s∈{1,2,3,∞} while, otherwise, it is NP-complete. Our main result is a structural theorem for graphs of bounded edge-admissibility. We prove that every graph of edge-admissibility at most k can be constructed using (≤k)-edge-sums, starting from graphs whose all vertices, except possibly from one, have degree at most k. Our structural result is approximately tight in the sense that graphs generated by this construction always have edge-admissibility at most 2k−1. Our proofs are based on a precise structural characterization of the graphs that do not contain θr as an immersion, where θr is the graph on two vertices and r parallel edges.

Emergency evacuation is a critical response to deadly disasters such as hurricanes, floods, and earthquakes, etc. However, mass emergency evacuation itself is a complex process that sometimes could lead to chaotic situations and unintended consequences. In many emergency scenarios, mass evacuation is necessary to cope with severe public threats within tight spatiotemporal ranges. To better understand complex phenomena like mass evacuation, and study possible consequences, agent-based models (ABMs) have been widely developed in previous work. Existing models simulate individual behaviors, posing computational challenges when applied to large geographic areas and sophisticated behaviors. A key strategy for resolving such computational challenges is to partition transportation networks into smaller regions and resolve corresponding computational costs by taking advantage of advanced cyberinfrastructure and cyberGIS. In this study, a novel network partition algorithm is developed to improve the scalability of agent-based modeling of mass evacuation based on a cutting-edge cyberGIS-enabled computational framework that exploits the spatial movement patterns of emergency evacuation. Specifically, the algorithm is termed as Voronoi Clustering based on Target-Shift, or ViCTS. It is enlightened by network Voronoi diagrams and designed to resolve computational scalability challenges caused by the unique characteristics of evacuation traffic. We conducted a set of computational experiments with real street network data in various evacuation scenarios to test the effectiveness and efficiency of the algorithm. Computational experiments show that ViCTS outperforms a widely used network partition algorithm for microscopic traffic simulation in terms of achieving optimal computational performance by balancing computational loads and reducing communications across high-performance parallel computing resources.

It was recently shown that a version of the greedy algorithm gives a construction of fault-tolerant spanners that is size-optimal, at least for vertex faults. However, the algorithm to construct this spanner is not polynomial-time, and the best-known polynomial time algorithm is significantly suboptimal. Designing a polynomial-time algorithm to construct (near-)optimal fault-tolerant spanners was given as an explicit open problem in the two most recent papers on fault-tolerant spanners ([Bodwin, Dinitz, Parter, Vassilevka Williams SODA '18] and [Bodwin, Patel PODC '19]). We give a surprisingly simple algorithm which runs in polynomial time and constructs fault-tolerant spanners that are extremely close to optimal (off by only a linear factor in the stretch) by modifying the greedy algorithm to run in polynomial time. To complement this result, we also give simple distributed constructions in both the LOCAL and CONGEST models.

Given a graph, a set of origin-destination (OD) pairs with communication requirements, and an integer k 2, the network design problem with vulnerability constraints (NDPVC) is to identify a subgraph with the minimum total edge costs such that, between each OD pair, there exist a hop-constrained primary path and a hop-constrained backup path after any k â' 1 edges of the graph fail. Formulations exist for single-edge failures (i.e., k = 2). To solve the NDPVC for an arbitrary number of edge failures, we develop two natural formulations based on the notion of length-bounded cuts. We compare their strengths and flexibilities in solving the problem for k 3. We study different methods to separate infeasible solutions by computing length-bounded cuts of a given size. Experimental results show that, for single-edge failures, our formulation increases the number of solved benchmark instances from 61% (obtained within a two-hour limit by the best published algorithm) to more than 95%, thus increasing the number of solved instances by 1,065. Our formulation also accelerates the solution process for larger hop limits and efficiently solves the NDPVC for general k. We test our best algorithm for two to five simultaneous edge failures and investigate the impact of multiple failures on the network design. Â

Because of the limited range of alternative fuel vehicles (AFVs) and the sparsity of the available alternative refueling stations (AFSs), AFV drivers cooperatively deviate from their paths to refuel. This deviation is bounded by the drivers' tolerance. Taking this behavior into account, the refueling station location problem with routing (RSLP-R) is defined as maximizing the AFV flow that can be accommodated in a road network by locating a given number of AFSs while respecting the range limitation of the vehicles and the deviation tolerance of the drivers. In this study, we develop a natural model for the RSLP-R based on the notion of length-bounded cuts, analyze the polyhedral properties of this model, and develop a branch-and-cut algorithm as an exact solution approach. Extensive computational experiments show that the algorithm significantly improves the solution times with respect to previously developed exact solution methods and extends the size of the instances solved to optimality. Using our methodology, we investigate the tradeoffs between covered vehicle flow and deviation tolerance of the drivers and present insights on deviation characteristics of drivers in a case study in California.

Let be a directed acyclic graph with arcs, a source and a sink . We introduce the cone of flow matrices, which is a polyhedral cone generated by the matrices , where is the incidence vector of the path . We show that several hard flow (or path) optimization problems, that cannot be solved by using the standard arc‐representation of a flow, reduce to a linear optimization problem over . This cone is intractable: we prove that the membership problem associated to is NP‐complete. However, the affine hull of this cone admits a nice description, and we give an algorithm which computes in polynomial‐time the decomposition of a matrix as a linear combination of some 's. Then, we provide two convergent approximation hierarchies, one of them based on a completely positive representation of . We illustrate this approach by computing bounds for the quadratic shortest path problem, as well as a maximum flow problem with pairwise arc‐capacities.

We consider a constrained shortest path problem with the possibility to refill the resource at certain nodes. This problem is motivated by routing electric vehicles with a comparatively short cruising range due to the limited battery capacity. Thus, for longer distances the battery has to be recharged on the way. Furthermore, electric vehicles can recuperate energy during downhill drive. We extend the common constrained shortest path problem to arbitrary costs on edges and we allow regaining resources at the cost of higher travel time. We show that this yields not shortest paths but shortest walks that may contain an arbitrary number of cycles. We study the structure of optimal solutions and develop approximation algorithms for finding short walks under mild assumptions on charging functions. We also address a corresponding network flow problem that generalizes these walks.

For a given number L, an L-length-bounded edge-cut (node-cut, respectively) in a graph G with source s and sink t is a set C of edges (nodes, respectively) such that no s-t-path of length at most L remains in the graph after removing the edges (nodes, respectively) in C. An L-length-bounded flow is a flow that can be decomposed into flow paths of length at most L. In contrast to classical flow theory, we describe instances for which the minimum L-length-bounded edge-cut (node-cut, respectively) is Θ(n2/3)-times (Θ(&sqrt;n)-times, respectively) larger than the maximum L-length-bounded flow, where n denotes the number of nodes; this is the worst case. We show that the minimum length-bounded cut problem is NP-hard to approximate within a factor of 1.1377 for L≥ 5 in the case of node-cuts and for L≥ 4 in the case of edge-cuts. We also describe algorithms with approximation ratio O(min{L,n/L}) ⊆ O&sqrt;n in the node case and O(min {L,n2/L2,&sqrt;m} ⊆ O2/3 in the edge case, where m denotes the number of edges. Concerning L-length-bounded flows, we show that in graphs with unit-capacities and general edge lengths it is NP-complete to decide whether there is a fractional length-bounded flow of a given value. We analyze the structure of optimal solutions and present further complexity results.

A path is said to be ℓ-bounded if it contains at most ℓ edges. We consider two types of ℓ-bounded disjoint paths problems. In the maximum edge- or node-disjoint path problems MEDP(ℓ) and MNDP(ℓ), the task is to find the maximum number of edge- or node-disjoint ℓ-bounded (s,t)-paths in a given graph G with source s and sink t, respectively. In the weighted edge- or node-disjoint path problems WEDP(ℓ) and WNDP(ℓ), we are also given an integer k ∈ ℕ and non-negative edge weights c
e
∈ ℕ, e ∈ E, and seek for a minimum weight subgraph of G that contains k edge- or node-disjoint ℓ-bounded (s,t)-paths. Both problems are of great practical relevance in the planning of fault-tolerant communication networks, for example.
Even though length-bounded cut and flow problems have been studied intensively in the last decades, the \(\mathcal{NP}\)-hardness of some 3- and 4-bounded disjoint paths problems was still open. In this paper, we settle the complexity status of all open cases showing that WNDP(3) can be solved in polynomial time, that MEDP(4) is \(\mathcal{AP\kern-1.5ptX}\)-complete and approximable within a factor of 2, and that WNDP(4) and WEDP(4) are \(\mathcal{AP\kern-1.5ptX}\)-hard and \(\mathcal{NPO}\)-complete, respectively.

For a given number L, an L-length-bounded edge-cut (node-cut, respectively) in a graph G with source s and sink t is a set C of edges (nodes, respectively) such that no s-t-path of length at most L remains in the graph after removing the edges (nodes, respectively) in C. An L-length-bounded flow is a flow that can be decomposed into flow paths of length at most L. In contrast to classical flow theory, we describe instances for which the minimum L-length-bounded edge-cut (node-cut, respectively) is Θ(n2/3)-times (Θ(&sqrt;n)-times, respectively) larger than the maximum L-length-bounded flow, where n denotes the number of nodes; this is the worst case. We show that the minimum length-bounded cut problem is NP-hard to approximate within a factor of 1.1377 for L≥ 5 in the case of node-cuts and for L≥ 4 in the case of edge-cuts. We also describe algorithms with approximation ratio O(min{L,n/L}) ⊆ O&sqrt;n in the node case and O(min {L,n2/L2,&sqrt;m} ⊆ O2/3 in the edge case, where m denotes the number of edges. Concerning L-length-bounded flows, we show that in graphs with unit-capacities and general edge lengths it is NP-complete to decide whether there is a fractional length-bounded flow of a given value. We analyze the structure of optimal solutions and present further complexity results.

We study the parameterized complexity of two families of problems: the bounded length disjoint paths problem and the bounded length cut problem. From Menger's theorem both problems are equiva- lent (and computationally easy) in the unbounded case for single source, single target paths. However, in the bounded case, they are combinato- rially distinct and are both NP-hard, even to approximate. Our results indicate that a more refined landscape appears when we study these problems with respect to their parameterized complexity. For this, we consider several parameterizations (with respect to the maximum length l of paths, the number k of paths or the size of a cut, and the treewidth of the input graph) of all variants of both problems (edge/vertex-disjoint paths or cuts, directed/undirected). We provide FPT-algorithms (for all variants) when parameterized by both k and l and hardness results when the parameter is only one of k and l. Our results indicate that the bounded length disjoint-path variants are structurally harder than their bounded length cut counterparts. Also, it appears that the edge variants are harder than their vertex-disjoint counterparts when parameterized by the treewidth of the input graph.

We consider the “flow on paths” versions of Max Flow and Min Cut when we restrict to paths having at most B arcs, and for versions where we allow fractional solutions or require integral solutions. We show that the continuous versions are polynomial even if B is part of the input, but that the integral versions are polynomial only when B ≤ 3. However, when B ≤ 3 we show how to solve the problems using ordinary Max Flow/Min Cut. We also give tight bounds on the integrality gaps between the integral and continuous objective values for both problems, and between the continuous objective values for the bounded-length paths version and the version allowing all paths. We give a primal–dual approximation algorithm for both problems whose approximation ratio attains the integrality gap, thereby showing that it is the best possible primal–dual approximation algorithm. oui

Let u and v be non-a~ljacent points in a connected graph G. A classical result known to all graph theorists is that called M~.so~.B's theorem. The point version of this result says that the maximum number of point-disjoint paths joining u and v is equal to the minimum number of points whose deletion destroys all paths joining u and v. The theorem may be proved purely in the language of graphs (probably the best known proof is indirect, and is due to DmAc [3 ] while a more neglected, but direct, proof may be found in ORE [ 7 ]). One may also prove the theorem by appeahng to flow theory (e.g. BERGs [1], p. 167). In many real-world situations which can be modeled by graphs certain paths joining two non-adjacent points may well exist, but may prove essentially useless because they are too long. Such considerations led the authors to study the following two parameters. Let u be any positive integer and let u and v be any two non-adjacent points in a graph G. Denote by An(u, v) the maximum number of point,disjoint paths joining u and v whose length (i.e., number of lines) does not exceed n. Analogously, let V,(u, v) be the minimum number of points in G the deletion of which destroys all paths joining u and v which do not exceed u in length. A special ease would obtain when ~ = p --IV(G)I, and we have by Menger's theorem, the equality An(u, v)= V,(u, v).

We study the problem of minimizing the maximum latency of flows in networks with congestion. We show that this problem is NP-hard, even when all arc latency functions are linear and there is a single source and sink. Still, an optimal flow and an equilibrium flow share a desirable property in this situation: all flow-carrying paths have the same length; i.e., these solutions are fair, which is in general not true for optimal flows in networks with nonlinear latency functions. In addition, the maximum latency of the Nash equilibrium, which can be computed efficiently, is within a constant factor of that of an optimal solution. That is, the so-called price of anarchy is bounded. In contrast, we present a family of instances with multiple sources and a single sink for which the price of anarchy is unbounded, even in networks with linear latencies. Furthermore, we show that an s-t-flow that is optimal with respect to the average latency objective is near optimal for the maximum latency objective, and it is close to being fair. Conversely, the average latency of a flow minimizing the maximum latency is also within a constant factor of that of a flow minimizing the average latency.

The problem of finding edge-disjoint paths in a planar graph such that each path connects two specified vertices on the outer face of the graph is well studied. The "classical" Eulerian case introduced by Okamura and Seymour [7] is solvable in linear time [10]. So far, the length of the paths were not considered. In this paper now, we prove that the problem of finding edge-disjoint paths of minimum total length in a planar graph is NP-hard, even if the graph fullfills the Eulerian condition and the maximum degree is four. Minimizing the length of the longest path is NP-hard as well. Efficient heuristics based on the algorithm from [10] are presented that determine edge-disjoint paths of small total length. We have implemented these heuristics and have studied their behaviour. It turns out that some of the heuristics are empirically very successful.

) SUMMARY OF RESULTS Consider a simple n-vertex undirected graph and assume there are edge-disjoint paths between two vertices u and v. We prove the following two results: ffl There are edge-disjoint paths between u and v, the average length of which is O(n= p ) ffl If all vertices have degree at least , there are edge-disjoint paths between u and v, each of which has length O(n=). These bounds are best possible. For directed graphs, the first result still holds but not the second. Some of the paths can be at leastOmegaGamma n) long. We also describe how to use a minimum cost flow algorithm to find the paths implied by the above results in time O(m). In a edge-connected graph, we define the concept of bistance (or bulk distance). The bistance between u and v is the minimum over all edge-disjoint paths between u and v of the maximum path length. We prove that bistance forms a metric. We give NP-hardness results on computing bistances in two cases. The third rem...

Graphs parameters such as connectivity and diameter have been studied extensively due to their intrinsic importance in graph theory, combinatorics and their relations to (and applications) fault tolerance and transmission delay in communications networks. The advent of VLSI technology and fiber optics material science has enabled us to design massively parallel processing computer systems and fast and complicated communications networks. All these systems increase their reliability by studying (among other) the existence of two (or more) disjoint paths connecting any two nodes. This paper addresses these issues by studying the width and length of containers in graphs and networks. In particular, the notions of w-distance and w-diameter on a graph are defined and studied which generalize both concepts of connectivity and diameter. These notions are also considered in finite groups. Other closely related parameters will be explored in the contexts of fault tolerance and routing. Known results are surveyed and open problems are offered for further investigation.

Introduction. The problem discussed in this paper was formulated by T. Harris as follows:
“Consider a rail network connecting two cities by way of a number of intermediate cities, where each link of the network has a number assigned to it representing its capacity. Assuming a steady state condition, find a maximal flow from one given city to the other.”

The following problem is considered: Given an integer K, a graph G with two distinct vertices s and t, find the maximum number of disjoint paths of length K from s to t. The problem has several variants: the paths may be vertex-disjoint or edge-disjoint, the lengths of the paths may be equal to K or bounded by K, the graph may be undirected or directed. It is shown that except for small values of K all the problems are NP-complete. Assuming P ≠ NP, for each problem, the largest value of K for which the problem is not NP-complete is found. Whenever a polynomial algorithm exists, an efficient algorithm is described.

In a recent paper Lovász, Neumann-Lara, and Plummer studied Mengerian theorems for paths of bounded length. Their study led to a conjecture concerning the extent to which Menger's theorem can fail when restricted to paths of bounded length. In this paper we offer counterexamples to this conjecture.

We consider the following problem: Given an integer k and a network G with two distinct vertices s and t, find a maximum number of vertex disjoint paths from s to t of length bounded by k. In a recent work [9] it was shown that for length greater than four this problem is NP-hard. In this paper we present a polynomial heuristic algorithm for the problem for general length. The algorithm is proved to give optimal solution for length less than five. Experiments show very good results for the algorithm.

We study the relation between a class of 0-1 integer linear programs and their rational relaxations. We give a randomized algorithm for transforming an optimal solution of a relaxed problem into a provably good solution for the 0-1 problem. Our technique can be a of extended to provide bounds on the disparity between the rational and 0-1 optima for a given problem instance.

In this paper we discuss the problem of finding edge-disjoint paths in a planar, undirected graph s.t. each path connects two specified vertices on the outer face boundary. We will focus on the
O( n5 \mathord/
\vphantom 5 3 3 ( loglogn )1 \mathord/
\vphantom 1 3 3 )\mathcal{O}\left( {n^{{5 \mathord{\left/{\vphantom {5 3}} \right.\kern-\nulldelimiterspace} 3}} \left( {\log \log n} \right)^{{1 \mathord{\left/{\vphantom {1 3}} \right.\kern-\nulldelimiterspace} 3}} } \right)
time, where n denotes the number of vertices. In this paper now, we introduce a new approach to this problem, which yields an
O( n )\mathcal{O}\left( n \right)
algorithm.

In this paper we consider the unsplittable flow problem (UFP): given a directed or undirected network G=(V,E) with edge capacities and a set of terminal pairs (or requests) with associated demands, find a subset of the pairs of maximum total demand for which a single flow path can be chosen for each pair so that for every edge, the sum of the demands of the paths crossing the edge does not exceed its capacity. We present a collection of new results for the UFP both in the offline (all requests are given from the beginning) and the online (requests arrive at the system one after the other) setting. A fundamental ingredient of our analysis is the introduction of a new graph parameter, the flow number, that aims to capture global communication properties of the network. With the help of the flow number we develop a general method for transforming arbitrary multicommodity flow solutions into solutions that use short paths only. This generalizes a well-known theorem of Leighton and Rao [J. ACM 46 (6) (1999) 787–832] that applies to uniform flows only. Both the parameter and the method may therefore be of independent interest.

We study the approximability of edge-disjoint paths and related problems. In the edge-disjoint paths (EDP) problem, we are given a network G with source–sink pairs , and the goal is to find a largest subset of source–sink pairs that can be simultaneously connected in an edge-disjoint manner. We show that in directed networks, for any ε>0, EDP is NP-hard to approximate within m1/2−ε. We also design simple approximation algorithms that achieve essentially matching approximation guarantees for some generalizations of EDP. Another related class of routing problems that we study concerns EDP with the additional constraint that the routing paths be of bounded length. We show that, for any ε>0, bounded length EDP is hard to approximate within m1/2−ε even in undirected networks, and give an -approximation algorithm for it. For directed networks, we show that even the single source–sink pair case (i.e. find the maximum number of paths of bounded length between a given source–sink pair) is hard to approximate within m1/2−ε, for any ε>0.

In this paper we discuss the problem of finding edge-disjoint paths in a planar, undirected graph such that each path connects
two specified vertices on the boundary of the graph. We will focus on the “classical” case where an instance additionally
fulfills the so-calledevenness-condition. The fastest algorithm for this problem known from the literature requiresO (n
5/3(loglogn)1/3) time, wheren denotes the number of vertices. In this paper now, we introduce a new approach to this problem, which results in anO(n) algorithm. The proof of correctness immediately yields an alternative proof of the Theorem of Okamura and Seymour, which
states a necessary and sufficient condition for solvability.

In a recent paper Lovász, Neumann-Lara and Plummer proved some Mengerian theorems for paths of bounded length. In this note the line connectivity analogue of their problem is considered.

We consider the “flow on paths” versions of Max Flow and Min Cut when we restrict to paths having at most B arcs, and for versions where we allow fractional solutions or require integral solutions. We show that the continuous versions are polynomial even if B is part of the input, but that the integral versions are polynomial only when B ≤ 3. However, when B ≤ 3 we show how to solve the problems using ordinary Max Flow/Min Cut. We also give tight bounds on the integrality gaps between the integral and continuous objective values for both problems, and between the continuous objective values for the bounded-length paths version and the version allowing all paths. We give a primal–dual approximation algorithm for both problems whose approximation ratio attains the integrality gap, thereby showing that it is the best possible primal–dual approximation algorithm. oui

This note discusses the problem of maximizing the rate of flow from one terminal to another, through a network which consists of a number of branches, each of which has a limited capacity. The main result is a theorem: The maximum possible flow from left to right through a network is equal to the minimum value among all simple cut-sets. This theorem is applied to solve a more general problem, in which a number of input nodes and a number of output nodes are used.

We prove the Minimum Vertex Cover problem to be NP-hard to approximate to within a factor of 1.3606, extending on previous PCP and hardness of approximation technique. To that end, one needs to develop a new proof framework, and borrow and extend ideas from several fields. 1

On the max flow min cut theorem of networks Linear Inequalities and Related Systems

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On line disjoint paths of bounded length Discrete Mathematics <b>44</b>

- G Exoo

On a generalization of Menger's Theorem. Acta Mathematica Universitatis Comenianae <b>42<

- L Niepel
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