Combinatorial Algorithms for the Unsplittable Flow Problem

In this paper, we present a constant-factor approximation algorithm for the unsplittable flow problem on a path. This improves on the previous best known approximation factor of O(log n). The approximation ratio of our algorithm is 7+e for any e>;0. In the unsplittable flow problem on a path, we are given a capacitated path P and n tasks, each task having a demand, a profit, and start and end vertices. The goal is to compute a maximum profit set of tasks, such that for each edge e of P, the total demand of selected tasks that use e does not exceed the capacity of e. This is a well-studied problem that occurs naturally in various settings, and therefore it has been studied under alternative names, such as resource allocation, bandwidth allocation, resource constrained scheduling, temporal knapsack and interval packing. Polynomial time constant factor approximation algorithms for the problem were previously known only under the no-bottleneck assumption (in which the maximum task demand must be no greater than the minimum edge capacity). We introduce several novel algorithmic techniques, which might be of independent interest: a framework which reduces the problem to instances with a bounded range of capacities, and a new geometrically inspired dynamic program which solves a special case of the maximum weight independent set of rectangles problem to optimality. In addition, we show that the problem is strongly NP-hard even if all edge capacities are equal and all demands are either 1, 2, or 3.

Let $G = (V, E)$ be an undirected graph and let $B \subseteq V \times V$ be a set of terminal pairs. A node/edge multicut is a subset of vertices/edges of $G$ whose removal destroys all the paths between every terminal pair in $B$. The problem of computing a {\em minimum} node/edge multicut is NP-hard and extensively studied from several viewpoints. In this paper, we study the problem of enumerating all {\em minimal} node multicuts. We give an incremental polynomial delay enumeration algorithm for minimal node multicuts, which extends an enumeration algorithm due to Khachiyan et al. (Algorithmica, 2008) for minimal edge multicuts. Important special cases of node/edge multicuts are node/edge {\em multiway cuts}, where the set of terminal pairs contains every pair of vertices in some subset $T \subseteq V$, that is, $B = T \times T$. We improve the running time bound for this special case: We devise a polynomial delay and exponential space enumeration algorithm for minimal node multiway cuts and a polynomial delay and space enumeration algorithm for minimal edge multiway cuts.

Given a graph with a source and a sink node, the NP-hard maximum k-splittable s,t-flow (MkSF) problem is to find a flow of maximum value from s to t with a flow decomposition using at most k paths. The multicommodity variant of this problem is a natural generalization of disjoint paths and unsplittable flow problems. Constructing a k-splittable flow requires two interdepending decisions. One has to decide on k paths (routing) and on the flow values for the paths (packing). We give efficient algorithms for computing exact and approximate solutions by decoupling the two decisions into a first packing step and a second routing step. Usually the routing is considered before the packing. Our main contributions are as follows: (i) We show that for constant k a polynomial number of packing alternatives containing at least one packing used by an optimal M kSF solution can be constructed in polynomial time. If k is part of the input, we obtain a slightly weaker result. In this case we can guarantee that, for any fixed e>0, the computed set of alternatives contains a packing used by a (1−e)-approximate solution. The latter result is based on the observation that (1−e)-approximate flows only require constantly many different flow values. We believe that this observation is of interest in its own right. (ii) Based on (i), we prove that, for constant k, the MkSF problem can be solved in polynomial time on graphs of bounded treewidth. If k is part of the input, this problem is still NP-hard and we present a polynomial time approximation scheme for it.

In this thesis, we study the Multidimensional Knapsack Problem (MKP) and two closely related special cases: the Unsplittable Flow Problem on Trees (UFPT), and the Unsplittable Flow Problem on Paths (UFPP). For these problems, when the natural integer programming formulation is relaxed to a linear program, the integrality gap is O(n). Previous work on UFPT and UFPP has established that the addition of rank constraints to the linear program can be effective in some cases at reducing the integrality gap. However, Friggstad and Gao proved that even with all rank constraints added, the integrality gap of UFPT is Ω(√(log n)). Faenza and Sanità showed that a formulation for these problems which approximates the integer hull arbitrarily well must either have exponentially many facets or be described as an extended formulation (i.e., described in a higher dimensional space). We are interested in polynomially sized formulations, so we focus our study on extended formulations. Our first contribution is a greedy algorithm which finds an optimal solution to the linear programming relaxation for the special case of UFPT where all requests share a common endpoint. We apply this to tighten the analysis of hard instances described in the literature. Following this, we describe two approximate extended formulations for MKP using disjunctive programming. The first is a formulation which improves upon the integrality gap of the linear relaxation using only a small number of extra variables and constraints. The second is a polyhedral (1 - ε)-approximate extended formulation for MKP for any 0 < ε ≤ 1, which was originally given by Pritchard. We then introduce a new hierarchy of strengthened formulations for MKP, which we study in the context of UFPT. The strengthened formulations are NP-Hard to separate over, but they can be (1 - ε)-approximated using the aforementioned result. We conclude by evaluating the strength of this hierarchy when applied to the known hard instances from the literature. Our results suggest that this hierarchy may be most useful in conjunction with the rank constraints, but open questions remain.

Minimal multicut and maximal integer multiflow: A survey

The Disjoint Connecting Paths problem and its capacitated generalization, called Unsplittable Flow problem, play an important role in practical applications such as communication network design and routing. These tasks are NP-hard in general, but various polynomial-time approximations are known. Nevertheless, the approximations tend to be either too loose (allowing large deviation from the optimum), or too complicated, often rendering them impractical in large, complex networks. Therefore, our goal is to present a solution that provides a relatively simple, efficient algorithm for the unsplittable flow problem in large directed graphs, where the task is NP-hard, and is known to remain NP-hard even to approximate up to a large factor. The efficiency of our algorithm is achieved by sacrificing a small part of the solution space. This also represents a novel paradigm for approximation. Rather than giving up the search for an exact solution, we restrict the solution space to a subset that is the most important for applications, and excludes only a small part that is marginal in some well-defined sense. Specifically, the sacrificed part only contains scenarios where some edges are very close to saturation. Since nearly saturated links are undesirable in practical applications, therefore, excluding near saturation is quite reasonable from the practical point of view. We refer the solutions that contain no nearly saturated edges as safe solutions, and call the approach safe approximation. We prove that this safe approximation can be carried out efficiently. That is, once we restrict ourselves to safe solutions, we can find the exact optimum by a randomized polynomial time algorithm.

The Multicut problem, given a graph G, a set of terminal pairs \(\ensuremath{\mathcal{T}}=\{(s_i,t_i)\ |\ 1\leq i\leq r\}\) and an integer p, asks whether one can find a cutset consisting of at most p non-terminal vertices that separates all the terminal pairs, i.e., after removing the cutset, t i is not reachable from s i for each 1 ≤ i ≤ r. The fixed-parameter tractability of Multicut in undirected graphs, parameterized by the size of the cutset only, has been recently proven by Marx and Razgon [2] and, independently, by Bousquet et al. [3], after resisting attacks as a long-standing open problem. In this paper we prove that Multicut is fixed-parameter tractable on directed acyclic graphs, when parameterized both by the size of the cutset and the number of terminal pairs. We complement this result by showing that this is implausible for parameterization by the size of the cutset only, as this version of the problem remains W[1]-hard.KeywordsDirected Acyclic GraphReduction RuleDegree ReductionShadow RemovalTerminal PairThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The minimum-cost multicommodity flow problem involves simultaneously shipping multiple commodities through a single network so that the total flow obeys arc capacity constraints and has minimum cost. Multicommodity flow problems can be expressed as linear programs, and most theoretical and practical algorithms use linear-programming algorithms specialized for the problems'' structures. Combinatorial approximation algorithms yield flows with costs slightly larger than the minimum cost and use capacities slightly larger than the given capacities. Theoretically, the running times of these algorithms are much less than that of linear-programming-based algorithms. We combine and modify the theoretical ideas in these approximation algorithms to yield a fast, practical implementation solving the minimum-cost multicommodity flow problem. Experimentally, the algorithm solved our problem instances (to 1% accuracy) two to three orders of magnitude faster than the linear-programming package CPLEX and the linear-programming based multicommodity flow program PPRN.

In this paper we consider the problem of (k, υ)-balanced graph partitioning - dividing the vertices of a graph into k almost equal size components (each of size less than υ • nk) so that the capacity of edges between different components is minimized. This problem is a natural generalization of several other problems such as minimum bisection, which is the (2,1)-balanced partitioning problem. We present a bicriteria polynomial time approximation algorithm with an O(log2n)-approximation for any constant υ > 1. For υ = 1 we show that no polytime approximation algorithm can guarantee a finite approximation ratio unless P=NP. Previous work has only considered the (k, υ)-balanced partitioning problem for υ ≥ 2.

We revisit a classical problem in transportation, known as the (bilevel) continuous network design problem, CNDP for short. Given a graph for which the latency of each edge depends on the ratio of the edge flow and the capacity installed, the goal is to find an optimal investment in edge capacities so as to minimize the sum of the routing costs of the induced Wardrop equilibrium and the investment costs for installing the edge's capacities. While this problem is considered to be challenging in the literature, its complexity status was still unknown. We close this gap, showing that CNDP is strongly $\mathsf{NP}$-hard and $\mathsf{APX}$-hard, both on directed and undirected networks and even for instances with affine latencies. As for the approximation of the problem, we first provide a detailed analysis for a heuristic studied by Marcotte for the special case of monomial latency functions [P. Marcotte, Math. Prog., 34 (1986), pp. 142--162]. We derive a closed form expression of its approximation guarantee for arbitrary sets of latency functions. We then propose a different approximation algorithm and show that it has the same approximation guarantee. Then, we prove that using the better of the two approximation algorithms results in a strictly improved approximation guarantee for which we derive a closed form expression. For affine latencies, for example, this best-of-two approach achieves an approximation factor of $49/41\approx 1.195$, which improves on the factor of $5/4$ that has been shown before by Marcotte.

Community search on attributed networks has recently attracted great deal of research interest. However, most of existing works require query users to specify some community structure parameters. This may not be always practical as sometimes a user does not have the knowledge and experience to decide the suitable parameters. In this paper, we propose a novel parameter-free contextual community model for attributed community search. The proposed model only requires a query context, i.e., a set of keywords describing the desired matching community context, while the community returned is both structure and attribute cohesive w.r.t. the provided query context. We theoretically show that both our exact and approximate contextual community search algorithms can be executed in worst case polynomial time. The exact algorithm is based on an elegant parametric maximum flow technique and the approximation algorithm that significantly improves the search efficiency is analyzed to have an approximation factor of 1/3. In the experiment, we use six real networks with ground-truth communities to evaluate the effectiveness of our contextual community model. Experimental results demonstrate that the proposed model can find near ground-truth communities. We also test both our exact and approximate algorithms using eight large real networks to demonstrate the high efficiency of the proposed algorithms.

Khuller and Raghavachari [J. Algorithms, 21 (1996), pp. 434--450] present an approximation algorithm (the KR algorithm) for finding the smallest k-edge connected spanning subgraph (k-ECSS) of an undirected multigraph. They prove the KR algorithm has an approximation ratio < 1.85. We improve this bound to $\le 1+\sqrt{1/e}<1.61$ (for odd k we modify the base case of the KR algorithm). This is the best-known performance bound for a combinatorial approximation algorithm for the smallest k-ECSS problem for arbitrary k. Our analysis also gives the best-known combinatorial performance bound for any fixed value of $k\ge 3$, e.g., for even k the approximation ratio is $\le 1+(1-{1\over k})^{k/2}$. Our analysis is based on a laminar family of sets (similar to families used in related contexts) which gives a better accounting of edges added in previous iterations of the algorithm. We also present a polynomial time implementation of the KR algorithm on multigraphs, running in the time for O(nm) maximum flow computations, where n (m) is the number of vertices (edges, not counting parallel copies), respectively. This complements the implementation of Khuller and Raghavachari [J. Algorithms, 21 (1996), pp. 434--450] which uses time O((kn)2) and is efficient for small k.

The Minimum Capacity s-t Cut Problem (MinCut) is an intensively studied problem in combinatorial optimization. A natural extension is the problem of choosing a minimum capacity s-t cut when arc capacities are unknown but confined to known intervals. This motivates the Robust Minimum Capacity s-t Cut Problem (RobuCut), which has applications such as open-pit mining and project scheduling. In this paper, we show how RobuCut can be reduced to solving a sequence of maximum flow problems and provide an efficient algorithm for rapidly solving this sequence of problems. We demonstrate that our algorithm solves instances of RobuCut in seconds that would require hours if a standard maximum flow solver is iteratively used as a black-box subroutine.

The maximum indicators of the functionality of a multiuser communication system are investigated under a change in the capacity of network edges. Based on the transmission routes of the maximum flows of the same type, the minimum cuts of the network are determined. The influence of changes in the capacity of the cut edges on the maximum allowable internodal flows under monopoly control modes is studied. The maximum values of the internodal flows correspond to the corners of the set of allowable flows satisfying the constraints of the model. A convex combination of vectors corresponding to the corners is used to estimate the relative changes in network characteristics during deformation of the set of feasible flows. Multi-criteria guaranteed estimates of the maximum feasible deviations from the reference indicators of the network functioning are formulated. Summary diagrams for networks with various structural features are provided.

We formulate and analyze a fundamental downlink transmission scheduling problem in a wireless communication system, composed of a base station and a set of users, each requesting a packet to be served within a time window. Some packets are requested by several users and can be served simultaneously due to the broadcast nature of the wireless medium. The base station can choose from a set of transmission strategies, e.g., in terms of combination of data rate and coding scheme, to serve the users. Each request can be served by a subset of transmission strategies. We seek a downlink transmission scheduling algorithm generating maximum aggregated system-wide utility. In this paper, we develop approximation algorithms for the formulated downlink transmission scheduling problem in both offline and online settings. We first establish the NP-hardness of the offline scheduling problem and the approximation hardness and bound of the online scheduling problem in non-preemptive and preemptive-restart models, respectively. We then design approximation algorithms and derive the approximation and competitive ratios of our online and offline scheduling algorithms. Numerical simulations are performed to further evaluate our algorithms in a wide range of network settings.