Conference Paper

# Length-Bounded Cuts and Flows

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## Abstract

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. ...
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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. ...
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... 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. ...
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... 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. ...
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... 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. ...
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... . 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). ...
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... 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. ...
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... 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. ...
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