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Pricing on Paths: A PTAS for the Highway Problem
"... In the highway problem, we are given an nedge line graph (the highway), and a set of paths (the drivers), each one with its own budget. For a given assignment of edge weights (the tolls), the highway owner collects from each driver the weight of the associated path, when it does not exceed the budg ..."
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In the highway problem, we are given an nedge line graph (the highway), and a set of paths (the drivers), each one with its own budget. For a given assignment of edge weights (the tolls), the highway owner collects from each driver the weight of the associated path, when it does not exceed the budget of the driver, and zero otherwise. The goal is choosing weights so as to maximize the profit. A lot of research has been devoted to this apparently simple problem. The highway problem was shown to be strongly NPhard only recently [Elbassioni,Raman,Ray,Sitters’09]. The bestknown approximation is O(log n / log log n) [Gamzu,Segev’10], which improves on the previousbest O(log n) approximation [Balcan,Blum’06]. Better approximations are known for a number of special cases. Finding a constant (or better!) approximation algorithm for the general case is a challenging open problem. In this paper we present a PTAS for the highway problem, hence closing the complexity status of the problem. Our result is based on a novel randomized dissection approach, which has some points in common with Arora’s quadtree dissection for Euclidean network design [Arora’98]. The basic idea is enclosing the highway in a bounding path, such that both the size of the bounding path and the position of the highway in it are random variables. Then we consider a recursive O(1)ary dissection of the bounding path, in subpaths of uniform optimal weight. Since the optimal weights are unknown, we construct the dissection in a bottomup fashion via dynamic programming, while computing the approximate solution at the same time. Our algorithm can be easily derandomized. The same basic approach provides PTASs also for two generalizations of the problem: the tollbooth problem with a constant number of leaves and the maximumfeasibility subsystem problem on interval matrices. In both cases the previous best approximation factors are polylogarithmic [Gamzu,Segev’10,Elbassioni,Raman,Ray,Sitters’09].
Unsplittable flow in paths and trees and columnrestricted packing integer programs
 IN PROCEEDINGS, INTERNATIONAL WORKSHOP ON APPROXIMATION ALGORITHMS FOR COMBINATORIAL OPTIMIZATION PROBLEMS
, 2009
"... We consider the unsplittable flow problem (UFP) and the closely related columnrestricted packing integer programs (CPIPs). In UFP we are given an edgecapacitated graph G = (V, E) and k request pairs R1,..., Rk, where each Ri consists of a sourcedestination pair (si, ti), a demand di and a weigh ..."
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Cited by 11 (0 self)
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We consider the unsplittable flow problem (UFP) and the closely related columnrestricted packing integer programs (CPIPs). In UFP we are given an edgecapacitated graph G = (V, E) and k request pairs R1,..., Rk, where each Ri consists of a sourcedestination pair (si, ti), a demand di and a weight wi. The goal is to find a maximum weight subset of requests that can be routed unsplittably in G. Most previous work on UFP has focused on the nobottleneck case in which the maximum demand of the requests is at most the smallest edge capacity. Inspired by the recent work of Bansal et al. [3] on UFP on a path without the above assumption, we consider UFP on paths as well as trees. We give a simple O(log n) approximation for UFP on trees when all weights are identical; this yields an O(log 2 n) approximation for the weighted case. These are the first nontrivial approximations for UFP on trees. We develop an LP relaxation for UFP on paths that has an integrality gap of O(log 2 n); previously there was no relaxation with o(n) gap. We also consider UFP in general graphs and CPIPs without the nobottleneck assumption and obtain new and useful results.
Partial convexification of general MIPs by DantzigWolfe reformulation
 INTEGER PROGRAMMING AND COMBINATORIAL OPTIMIZATION, VOLUME 6655 OF LECT. NOTES COMPUT. SCI
, 2011
"... DantzigWolfe decomposition is wellknown to provide strong dual bounds for specially structured mixed integer programs (MIPs) in practice. However, the method is not implemented in any stateoftheart MIP solver: it needs tailoring to the particular problem; the decomposition must be determined f ..."
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Cited by 9 (6 self)
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DantzigWolfe decomposition is wellknown to provide strong dual bounds for specially structured mixed integer programs (MIPs) in practice. However, the method is not implemented in any stateoftheart MIP solver: it needs tailoring to the particular problem; the decomposition must be determined from the typical bordered blockdiagonal matrix structure; the resulting column generation subproblems must be solved efficiently; etc. We provide a computational proofofconcept that the process can be automated in principle, and that strong dual bounds can be obtained on general MIPs for which a solution by a decomposition has not been the first choice. We perform an extensive computational study on the 01 dynamic knapsack problem (without blockdiagonal structure) and on general MIPLIB2003 instances. Our results support that DantzigWolfe reformulation may hold more promise as a generalpurpose tool than previously acknowledged by the research community.
A QPTAS for maximum weight independent set of polygons with polylogarithmically many vertices
 In Proceedings of the TwentyFifth Annual ACMSIAM Symposium on Discrete Algorithms, SODA 2014
"... The Maximum Weight Independent Set of Polygons (MWISP) problem is a fundamental problem in computational geometry. Given a set of weighted polygons in the twodimensional plane, the goal is to find a set of pairwise nonoverlapping polygons with maximum total weight. Due to its wide range of applica ..."
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The Maximum Weight Independent Set of Polygons (MWISP) problem is a fundamental problem in computational geometry. Given a set of weighted polygons in the twodimensional plane, the goal is to find a set of pairwise nonoverlapping polygons with maximum total weight. Due to its wide range of applications and connections to other problems, the MWISP problem and its special cases have been extensively studied both in the approximation algorithms and the computational geometry community. Despite a lot of research, its general case is not wellunderstood yet. Currently the best known polynomial time algorithm achieves an approximation ratio of n [Fox and Pach, SODA 2011], and it is not even clear whether the problem is APXhard. We present a (1+)approximation algorithm, assuming that each polygon in the input has at most a polylogarithmic number of vertices. Our algorithm has quasipolynomial running time, i.e., it runs in time 2poly(logn,1/). In particular, our result implies that for this setting the problem is not APXhard, unless NP ⊆ DTIME(2poly(logn)). We use a recently introduced framework for approximating maximum weight independent set in geometric intersection graphs. The framework has been used to construct a QPTAS in the much simpler case of axisparallel rectangles. We extend it in two ways, to adapt it to our much more general setting. First, we show that its technical core can be reduced to the case when all input polygons are triangles. Secondly, we replace its key technical ingredient which is a method to partition the plane using only few edges such that the objects stemming from the optimal solution are evenly distributed among the resulting faces and each object is intersected only a few times. Our new procedure for this task is not even more complex than the original one and, importantly, it can handle the arising difficulties due to the arbitrary angles of the input polygons. Note that already this obstacle makes the known analysis for the above framework fail. Also, in general it is not well understood how to handle this difficulty by efficient approximation algorithms.
Uncommon DantzigWolfe reformulation for the temporal knapsack problem
 INFORMS Journal on Computing
"... We study a natural generalization of the knapsack problem, in which each item exists only for a given time interval. One has to select a subset of the items (as in the classical case), guaranteeing that for each time instant the set of existing selected items has total weight not larger than the kn ..."
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Cited by 5 (2 self)
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We study a natural generalization of the knapsack problem, in which each item exists only for a given time interval. One has to select a subset of the items (as in the classical case), guaranteeing that for each time instant the set of existing selected items has total weight not larger than the knapsack capacity. We focus on the exact solution of the problem, noting that prior to our work the best method was the straightforward application of a generalpurpose solver to the natural ILP formulation. Our results indicate that much better results can be obtained by using the same generalpurpose solver to tackle a nonstandard DantzigWolfe reformulation in which subproblems are associated with groups of constraints. This is also interesting since the more natural DantzigWolfe reformulation of single constraints performs extremely poorly in practice. 1
Resource Allocation for Covering Time Varying Demands
"... Abstract. We consider the problem of allocating resources to satisfy demand requirements varying over time. The input specifies a demand for each timeslot. Each resource is specified by a starttime, endtime, an associated cost and a capacity. A feasible solution is a multiset of resources such tha ..."
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Cited by 4 (2 self)
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Abstract. We consider the problem of allocating resources to satisfy demand requirements varying over time. The input specifies a demand for each timeslot. Each resource is specified by a starttime, endtime, an associated cost and a capacity. A feasible solution is a multiset of resources such that at any point of time, the sum of the capacities offered by the resources is at least the demand requirement at that point of time. The goal is to minimize the total cost of the resources included in the solution. This problem arises naturally in many scenarios such as workforce management, sensor networks, cloud computing, energy management and distributed computing. We study this problem under the partial cover setting and the zeroone setting. In the former scenario, the input also includes a number k and the goal is to choose a minimum cost solution that satisfies the demand requirements of at least k timeslots. For this problem, we present a 16approximation algorithm; we show that there exist “wellstructured ” nearoptimal solutions and that such a solution can be found in polynomial time via dynamic programming. In the zeroone setting, a feasible solution is allowed to pick at most one copy of any resource. For this case, we present a 4approximation algorithm; our algorithm uses a novel LP relaxation involving flowcover inequalities. 1
Constant Integrality Gap LP Formulations of Unsplittable Flow on a Path
, 2013
"... The Unsplittable Flow Problem on a Path (UFPP) isacore problem in many important settings such as network flows, bandwidth allocation, resource constraint scheduling, and interval packing. We are given a path with capacities on the edges and a set of tasks, each task having a demand, a profit, a sou ..."
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Cited by 4 (4 self)
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The Unsplittable Flow Problem on a Path (UFPP) isacore problem in many important settings such as network flows, bandwidth allocation, resource constraint scheduling, and interval packing. We are given a path with capacities on the edges and a set of tasks, each task having a demand, a profit, a source and a destination vertex on the path. The goal is to compute a subset of tasks of maximum profit that does not violate the edge capacities. In practical applications generic approaches such as integer programming (IP) methods are desirable. Unfortunately, no IPformulation is known for the problem whose LPrelaxation has an integrality gap that is provably constant. For the unweighted case, we show that adding a few constraints to the standard LP of the problem is sufficient to make the integrality gap drop from Ω(n) to O(1). This positively answers an open question in [Chekuri et al., APPROX 2009]. For the general (weighted) case, we present an extended formulation with integrality gap bounded by 7+ε. This matches the best known approximation factor for the problem [Bonsma et al., FOCS 2011]. This result exploits crucially a technique for embedding dynamic programs into linear programs. We believe that this method could be useful to strengthen LPformulations for other problems as well and might eventually speed up computations due to stronger problem formulations.
A mazing 2+ε approximation for unsplittable flow on a path
 IN: PROCEEDINGS OF SODA 2014
, 2014
"... We study the unsplittable flow on a path problem (UFP), which arises naturally in many applications such as bandwidth allocation, job scheduling, and caching. Here we are given a path with nonnegative edge capacities and a set of tasks, which are characterized by a subpath, a demand, and a profit. ..."
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Cited by 3 (2 self)
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We study the unsplittable flow on a path problem (UFP), which arises naturally in many applications such as bandwidth allocation, job scheduling, and caching. Here we are given a path with nonnegative edge capacities and a set of tasks, which are characterized by a subpath, a demand, and a profit. The goal is to find the most profitable subset of tasks whose total demand does not violate the edge capacities. Not surprisingly, this problem has received a lot of attention in the research community. If the demand of each task is at most a small enough fraction δ of the capacity along its subpath (δsmall tasks), then it has been known for a long time [Chekuri et al., ICALP 2003] how to compute a solution of value arbitrarily close to the optimum via LP rounding. However, much remains unknown for the complementary case, that is, when the demand of each task is at least some fraction δ> 0 of the smallest capacity of its subpath (δlarge tasks). For this setting a constant factor approximation, improving on an earlier logarithmic approximation, was found only recently [Bonsma et al., FOCS 2011]. In this paper we present a PTAS for δlarge tasks, for any constant δ> 0. Key to this result is a complex geometrically inspired dynamic program. Each task is represented as a segment underneath the capacity curve, and we identify a proper mazelike structure so that each corridor of the maze is crossed by only O(1) tasks in the optimal solution. The maze has a tree topology, which guides our dynamic program. Our result implies a 2 + ε approximation for UFP, for any constant ε> 0, improving on the previously best 7 + ε approximation by Bonsma et al. We remark that our improved approximation algorithm matches the best
How UnsplittableFlowCovering Helps Scheduling with JobDependent Cost Functions,
"... Abstract. Generalizing many wellknown and natural scheduling problems, scheduling with jobspecific cost functions has gained a lot of attention recently. In this setting, each job incurs a cost depending on its completion time, given by a private cost function, and one seeks to schedule the jobs ..."
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Abstract. Generalizing many wellknown and natural scheduling problems, scheduling with jobspecific cost functions has gained a lot of attention recently. In this setting, each job incurs a cost depending on its completion time, given by a private cost function, and one seeks to schedule the jobs to minimize the total sum of these costs. The framework captures many important scheduling objectives such as weighted flow time or weighted tardiness. Still, the general case as well as the mentioned special cases are far from being very well understood yet, even for only one machine. Aiming for better general understanding of this problem, in this paper we focus on the case of uniform job release dates on one machine for which the state of the art is a 4approximation algorithm. This is true even for a special case that is equivalent to the covering version of the wellstudied and prominent unsplittable flow on a path problem, which is interesting in its own right. For that covering problem, we present a quasipolynomial time (1 + ε)approximation algorithm that yields an (e + ε)approximation for the above scheduling problem. Moreover, for the latter we devise the best possible resource augmentation result regarding speed: a polynomial time algorithm which computes a solution with optimal cost at 1 + ε speedup. Finally, we present an elegant QPTAS for the special case where the cost functions of the jobs fall into at most log n many classes. This algorithm allows the jobs even to have up to log n many distinct release dates. All proposed quasipolynomial time algorithms require the input data to be quasipolynomially bounded. 1