2.8 Combinatorial Optimization
2.8.1 Approximation Algorithms for Hard Optimization Problems
One of the most basic methods to deal with NP-hard optimization problems is to design polynomial-time algorithms that find a solution which is provably not far from the optimum.
As we will see, our work spans different areas, including graph problems, computational economics, computational geometry, metric embeddings, mathematical programming, and scheduling.
Small Hop-diameter Sparse Spanners for Doubling Metrics Investigator: Hubert Chan
The study of finite metrics and their properties has been a very fruitful area of research, with applications to many different problems. Many commonly arising problems (e.g., clustering, near-neighbor finding, network routing, just to name a few) deal with sets of points on which a distance function has been defined, and one wants to store and process this metric in different ways.
We focus on obtaining sparse representations of metrics: these are called spanners, and they have been studied extensively for both general and Euclidean metrics. Formally, a t-spanner for a metric M = (V, d) is a weighted undirected graph G = (V, E) such that the distances according to dG (the shortest-path metric of G) are close to the distances in d: specifically, d(u, v) ≤ dG(u, v) ≤ t d(u, v). Clearly, one can take a complete graph and obtain t = 1, and hence the quality of the spanner is typically measured by how few edges can G contain whilst maintaining a stretch of at most t. The notion of spanners has been widely studied for general metrics (see, e.g. [1, 6, 8]), and for geometric distances (see, e.g., [2, 3, 9, 10]).
Very recently, there have been good constructions of spanners for doubling metrics as well:
given a metric with doubling dimension7dim, the results of Chan et al. [4], and independently, those of Har-Peled and Mendel [7] show how to construct (1 + ε)-spanners with n(1 + 1/ε)O(dim) edges, where n = |V | is the number of points in the metric.
In the paper [5], we extend these results to find spanners that also have small hop diameter.
A t-spanner has hop diameter D if every pair u, v ∈ V are connected by some path in G
7that is, one in which every ball can be covered by a constant number of balls of half the radius
having length at most t d(u, v), and furthermore there are at most D edges on this path.
We show that given any metric with constant doubling dimension k, and any 0 < ε < 1, one can find a (1 + ε)-spanner for the metric with a nearly linear number of edges (i.e., only O(n log∗n + nε−O(k)) edges) and constant hop diameter; we can also obtain a (1 + ε)-spanner with a linear number of edges (i.e., only nε−O(k) edges) that achieves a hop diameter that grows like the functional inverse of Ackermann’s function. Moreover, we prove that such tradeoffs between the number of edges and the hop diameter are asymptotically optimal.
References
[1] I. Alth¨ofer, G. Das, D. Dobkin, D. Joseph, and J. Soares. On sparse spanners of weighted graphs.
Discrete Comput. Geom., 9(1):81–100, 1993.
[2] S. Arya, G. Das, D. M. Mount, J. S. Salowe, and M. H. M. Smid. Euclidean spanners: short, thin, and lanky. In Proceedings of the 27th annual ACM symposium on Theory of computing (STOC), New York, NY, USA, 1995, pp. 489–498. ACM.
[3] P. B. Callahan and S. R. Kosaraju. A decomposition of multidimensional point sets with applica-tions to k-nearest-neighbors and n-body potential fields. J. Assoc. Comput. Mach., 42(1):67–90, 1995.
[4] H. T.-H. Chan, A. Gupta, B. M. Maggs, and S. Zhou. On hierarchical routing in doubling met-rics. In Proceedings of the 16th annual ACM-SIAM symposium on Discrete algorithms (SODA), Philadelphia, PA, USA, 2005, pp. 762–771. Society for Industrial and Applied Mathematics.
• [5] T.-H. H. Chan. Small hop-diameter sparse spanners for doubling metrics. Discrete and Compu-tational Geometry, 41(1):28–44, 2009.
[6] B. Chandra, G. Das, G. Narasimhan, and J. Soares. New sparseness results on graph spanners.
Internat. J. Comput. Geom. Appl., 5(1-2):125–144, 1995. Eighth Annual ACM Symposium on Computational Geometry (Berlin, 1992).
[7] S. Har-Peled and M. Mendel. Fast construction of nets in low dimensional metrics, and their applications. Symposium on Computational Geometry, pp. 150–158, 2005.
[8] D. Peleg and A. A. Sch¨affer. Graph spanners. In J. Graph Theory 13, 1989, pp. 99–116.
[9] J. S. Salowe. Constructing multidimensional spanner graphs. In Internat. J. Comput. Geom.
Appl. 1, 1991, pp. 99–107.
[10] P. M. Vaidya. A sparse graph almost as good as the complete graph on points in k dimensions.
In Discrete Comput. Geom. 6, 1991, pp. 369–381.
A QPTAS for TSP with Fat Weakly Disjoint Neighborhoods in Doubling Metrics Investigators: Hubert Chan and Khaled Elbassioni
We consider the Traveling Salesman Problem with Neighborhoods (TSPN) in a metric space (V, d). An instance of the problem is given by a collection W of n subsets {P1, P2, . . . , Pn} in V . Each subset Pj ⊂ V is known as a neighborhood or region. The objective is to find a minimum length tour that visits at least one point from each region.
This problem generalizes the well-known Traveling Salesman Problem (TSP), for which there are PTAS’s for low-dimensional Euclidean metrics [8, 2, 10], and a QPTAS for doubling metrics [11]. The neighborhood version of the problem was first introduced by Arkin and
Hassin [1], who gave constant approximation for the case when the regions are in the plane and “well-behaved” (e.g., disks, parallel and similar length segments, bounded ratio between the largest and smallest diameters). The general version of the problem was shown to have an inapproximability threshold of Ω(log2−n) for any > 0 by Halperin and Krauthgamer [7].
There is an almost matching upper bound of O(log N log k log n)-approximation, using the results of Garg et al. [6] and Fakcharoenphol et al. [5], where N is the total number of points in V and k is the maximum number of points in each region.
The best previously known result for getting a (1 + ε)-approximation is by Mitchell [9], who obtained a PTAS for the Euclidean plane, where the regions are fat and almost disjoint.
This result is obtained by the “guillotine subdivision” technique, which unfortunately only works for 2 dimensions. On the other hand, the hierarchical decomposition technique by Arora [2] and Talwar [11] is applicable to more general metrics. However, as pointed out by Mitchell [9], previous attempts in applying this technique have led to only limited success.
In [3], we obtain some partial results. In particular, we give a (1 + ε)-approximation for instances on metrics with bounded doubling dimension. This includes low-dimensional Eu-clidean metrics, and hence is a generalization of Mitchell’s result [9] for 3 or more dimensions.
Moreover, since the doubling dimension is well defined for any metric, our framework covers metrics that do not have any geometric structure, and the regions need not be convex or even connected, where such notions might not even be applicable in the first place.
Nevertheless, we still need to place some restrictions on the regions, because the problem is APX-hard in general on the plane [4], which has bounded doubling dimension. We combine the notions of diameter variation, fatness and disjointness for geometric spaces, and define for regions in general metrics the notion of α-fat weak disjointness. We assume that the regions have a bounded number ∆ types of radii. For the regions within the same type, there is some ρ > 0 such that there is a ρ-packing8 consisting of one point from each region, and all the regions have diameters at least ρ and at most O(αρ).
Our definition allows very general regions. Intuitively, all we require is that regions of similar diameters should each designate a point within, such that these points are far away from one another; the regions can otherwise intersect arbitrarily. The assumption that there are only a bounded number ∆ of types of region diameters is also necessary, as otherwise the problem remains APX-hard.9 Of course, the catch with working on such weak assumptions is that the running time of our algorithm is only quasi-polynomial. This is not surprising, because there is only a QPTAS known even for TSP on doubling metrics by Talwar [11].
References
[1] E. M. Arkin and R. Hassin. Approximation algorithms for the geometric covering salesman problem. Discrete Applied Mathematics, 55(3):197–218, 1994.
[2] S. Arora. Approximation algorithms for geometric TSP. In The traveling salesman problem and its variations, Comb. Optim., vol. 12, pp. 207–221. Kluwer Acad. Publ., Dordrecht, 2002.
[3] H. Chan and K. Elbassioni. A QPTAS for TSP with fat weakly disjoint neighborhoods in doubling metrics. Technical Report 2009-01, DIMACS Technical Reports, 2009.
8A ρ-packing is a set of points with inter-point distance larger than ρ.
9However, this assumption is not necessary in the case of Euclidean metric.
[4] M. Dror and J. B. Orlin. Combinatorial optimization with explicit delineation of the ground set by a collection of subsets. In SIAM J. Discrete Math. 21(4), 2008, pp. 1019–1034.
[5] J. Fakcharoenphol, S. Rao, and K. Talwar. A tight bound on approximating arbitrary metrics by tree metrics. In Proceedings of the 35th ACM symposium on Theory of computing (STOC), 2003, pp. 448–455. ACM Press.
[6] N. Garg, G. Konjevod, and R. Ravi. A polylogarithmic approximation algorithm for the group Steiner tree problem. Journal of Algorithms, 37(1):66–84, 2000. (Preliminary version in 9th SODA, pages 253–259, 1998).
[7] E. Halperin and R. Krauthgamer. Polylogarithmic inapproximability. In Proceedings of the thirty-fifth ACM symposium on Theory of computing, 2003, pp. 585–594. ACM Press.
[8] J. S. B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: a simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems. SIAM J. Comput., 28(4):1298–1309, 1999.
[9] J. S. B. Mitchell. A PTAS for TSP with neighborhoods among fat regions in the plane. In N. Bansal, K. Pruhs, and C. Stein, eds., Proceedings of the 18th annual ACM-SIAM symposium on Discrete algorithms (SODA), 2007, pp. 11–18. SIAM.
[10] S. B. Rao and W. D. Smith. Approximating geometrical graphs via “spanners” and “banyans”.
In Proceedings on 30th Annual ACM Symposium on Theory of Computing (STOC), pp. 540–550.
ACM, New York, 1998.
[11] K. Talwar. Bypassing the embedding: algorithms for low dimensional metrics. In Proceedings of the 36th annual ACM symposium on Theory of computing (STOC), New York, NY, USA, 2004, pp. 281–290. ACM.
Metric Embeddings with Relaxed Guarantees Investigator: Hubert Chan
Over the past decade, the field of metric embeddings has gained much importance in algo-rithm design. The central genre of problem in this area is the mapping of a given metric space into a “simpler” one, in such a way that the distances between points do not change too much. More formally, an embedding of a finite metric space (V, d) into a target metric space (V0, d0) is a map ϕ : V → V0. Recent work on embeddings has used distortion as the fundamental measure of quality; the distortion of an embedding is the worst multiplicative factor by which distances are increased by the embedding10. The popularity of distortion has been driven by its applicability to approximation algorithms: if the embedding ϕ : V → V0 has a distortion of D, then the cost of solutions to some optimization problems on (V, d) and on (ϕ(V ), d0) can only differ by some function of D; this idea has led to numerous approximation algorithms [3].
In the context of some networking applications, however, distortion as defined above has turned out to be too demanding an objective function. Instead, the recent networking work
10Formally, for an embedding ϕ : V → V0, the distortion is the smallest D so that ∃ α, β ≥ 1 with α · β ≤ D such that α1d(x, y) ≤ d0(ϕ(x), ϕ(y)) ≤ β d(x, y) for all pairs x, y ∈ V × V . Note that this definition of distortion is slightly non-standard—since α, β ≥ 1, it is no longer invariant under arbitrary scaling;
however, this is merely for notational convenience, and all our results can be cast in the usual definitions of distortion.
has provided empirical guarantees of the following form: if we allow a small fraction of all distances to be arbitrarily distorted, we can embed the remainder with constant distortion in constant-dimensional Euclidean space. Such guarantees are natural for the underlying networking applications; essentially, a very small fraction of the location-based lookups may yield poor performance (due to the arbitrary distortion), but for the rest the quality of the embedding will be very good.
These types of results form a suggestive contrast with the theoretical work on embeddings.
In particular, are the strong empirical guarantees for Internet latencies the result of fortuitous artifacts of this particular set of distances, or is something more general going on? To address this, Kleinberg, Slivkins, and Wexler [5] defined the notion of embeddings with slack : in addition to the metrics (V, d) and (V0, d0) in the initial formulation above, we are also given a slack parameter ε, and we want to find a map ϕ whose distortion is bounded by some quantity D(ε) on all but an ε fraction of the pairs of points in V × V . (Note that we allow the distortion on the remaining εn2 pairs of points to be arbitrarily large.) Roughly, Kleinberg et. al. [5] showed that any metric of bounded doubling dimension can be embedded with constant distortion into constant-dimensional Euclidean space, allowing a constant slack ε.
Such metrics, which have been extensively studied in their own right, have also been proposed on several occasions as candidates for tractable abstractions of the set of Internet latencies (see e.g. [2, 4, 6, 7]).
In the paper [1], we answer some of the open questions posed in [5]. In particular, we show that provable guarantees of this type can in fact be achieved in general: any finite metric can be embedded, with constant slack and constant distortion, into constant-dimensional Euclidean space. We then show that there exist stronger embeddings into `1 which exhibit gracefully degrading distortion: there is a single embedding into `1 that achieves distortion at most O(log1ε) on all but at most an ε fraction of distances, simultaneously for all ε > 0.
We extend this with distortion O(log1ε)1/p to maps into general `p, p ≥ 1 for several classes of metrics, including those with bounded doubling dimension and those arising from the shortest-path metric of a graph with an excluded minor. Finally, we show that many of our constructions are tight, and give a general technique to obtain lower bounds for ε-slack embeddings from lower bounds for low-distortion embeddings.
References
• [1] T.-H. H. Chan. Metric embeddings with relaxed guarantees. SIAM Journal on Computing, 38:2303–2329, 2009.
[2] M. Fomenkov, k. claffy, B. Huffaker, and D. Moore. Macroscopic Internet topology and perfor-mance measurements from the DNS root name servers. In Usenix LISA, 2001.
[3] P. Indyk. Algorithmic aspects of geometric embeddings. In Proceedings of the 42nd Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2001.
[4] D. R. Karger and M. Ruhl. Finding nearest neighbors in growth-restricted metrics. In Proceedings of the 34th Annual ACM Symposium on the Theory of Computing, 2002, pp. 63–66.
[5] J. Kleinberg, A. Slivkins, and T. Wexler. Triangulation and embedding using small sets of bea-cons. In Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2004.
[6] T. Ng and H. Zhang. Predicting Internet network distance with coordinates-based approaches.
In Proceedings of the 21st Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM), 2002.
[7] C. G. Plaxton, R. Rajaraman, and A. W. Richa. Accessing nearby copies of replicated objects in a distributed environment. Theory Comput. Syst., 32(3), 1999.
1.5D Terrain Guarding
Investigators: Khaled Elbassioni and Juli´an Mestre in cooperation with Erik Krohn, Domagoj Matijevi´c, and Domagoj ˇSeverdija
In the 1.5D terrain guarding problem we are given a polygonal region in the plane determined by an x-monotone polygonal chain, and the objective is to find the minimum number of guards to place on the chain such that every point in the polygonal region is guarded.
This kind of guarding problems and its generalizations to 3-dimensions are motivated by optimal placement of antennas for communication networks; for more details see [2, 1] and the references therein.
In [4] we presented a 4-approximation algorithm for the problem of placing the fewest guards on a 1.5D terrain so that every point of the terrain is seen by at least one guard. This improves on the currently best approximation factor of 5 [3]. Unlike most of the previous techniques, our method is based on rounding the linear programming relaxation of the cor-responding covering problem. Besides the simplicity of the analysis, which mainly relies on decomposing the constraint matrix of the LP into totally balanced matrices, our algorithm, unlike previous work, generalizes to the weighted and partial versions of the basic problem.
References
[1] B. Ben-Moshe, M. J. Katz, and J. S. B. Mitchell. A constant-factor approximation algorithm for optimal 1.5D terrain guarding. SIAM Journal on Computing, 36(6):1631–1647, 2007.
[2] D. Z. Chen, V. Estivill-Castro, and J. Urrutia. Optimal guarding of polygons and monotone chains. In Proceedings of the 7th Canadian Conference on Computational Geometry, 1995, pp.
133–138.
[3] J. King. A 4-approximation algorithm for guarding 1.5-dimensional terrains. In Proceedings of the 13th Latin American Symposium on Theoretical Informatics, 2006, pp. 629–640.
• [4] E. Krohn, K. Elbassioni, D. Matijevic, J. Mestre, and D. Severdija. Improved approximation algorithms for 1.5D terrain guarding. In 26th International Symposium on Theoretical As-pects of Computer Science (STACS), Freiburg - Germany, 2009. Internationales Begegnungs und Forschungszentrum f”ur Informatik (IBFI).
Scheduling to Minimize Lateness
Investigator: Juli´an Mestre in cooperation with Samir Khuller
In [1] we re-examined the classical problem of minimizing maximum lateness which is defined as follows: given a collection of n jobs with processing times and due dates, in what order should they be processed on a single machine to minimize maximum lateness? The lateness
of a job is defined as its completion time minus its due date. This problem can be solved easily by ordering the jobs in non-decreasing due date order.
Consider the following question: which subset of k jobs should we reject to reduce the maximum lateness by the largest amount? While this problem can be solved optimally in polynomial time, we showed the following surprising result: there is a fixed ordering of the jobs, such that for all k, if we reject the first k jobs from this ordering, we derive an optimal solution for the problem in which we are allowed to reject k jobs. This allows for an incremental solution in which we can keep incrementally rejecting jobs if we need a solution with lower maximum lateness value. Moreover, we also developed an optimal O(n log n) time algorithm to find this ordering.
References
• [1] S. Khuller and J. Mestre. An optimal incremental algorithm for minimizing lateness with rejection.
In 16th Annual European Symposium on Algorithms (ESA), Karlsruhe, Germany, 2008, LNCS 5193, pp. 601–610. Springer.
Scheduling with Limited Machine Availability
Investigators: Nicole Megow in cooperation with Jose Verschae, Alberto Marchetti-Spaccamela, Martin Skutella, and Leen Stougie
In classical scheduling theory it is assumed that resources are available continuously through-out the entire planning period. This is hardly the case in practice since there are working shifts, planned maintenance periods, or even unexpected machine breakdowns. Therefore, research on scheduling with limitations in the resource availability is certainly of practical importance, which is also reflected by the large number of publications on this field.
We consider scheduling on a single machine with limited availability to minimize the sum of weighted completion times. In [2], we consider the special case in which there is only a single non-available time period. We provide a preemptive algorithm with an approximation ratio arbitrarily close to the Golden Ratio, (1 +√
5)/2 + , which improves on a previously best known 2-approximation. The non-preemptive version of the same algorithm yields a (2 +
)-approximation. In [1] we improve these results by presenting a fully polynomial-time approximation scheme for both problems.
In [1] we affirmatively answer the longstanding open question whether a constant ap-proximation algorithm exists for the problem with arbitrary unavailability periods. Here, we consider the much more general model of full uncertainty about the breakdowns. We design a polynomial time deterministic algorithm that finds a robust prefixed scheduling sequence with a solution value within 4 times the value an optimal clairvoyant algorithm can achieve, knowing the disruptions in advance and even being allowed to interrupt jobs at any moment.
A randomized version of this algorithm attains in expectation a ratio of e w.r.t. a clairvoyant optimum. We show that such a ratio can never be achieved by any deterministic algorithm by proving that the price of robustness of any such algorithm is at least 1 +√
3 ≈ 2.73205 > e.
References
[1] A. Marchetti-Spaccamela, N. Megow, M. Skutella, and L. Stougie. Robust sequencing on a single machine. Submitted, 2009.
[2] N. Megow and J. Verschae. A note on scheduling on a single machine with one non-availability period. Submitted, 2008.
On Eulerian Extension Problems and their Application to Sequencing Problems Investigators: Nicole Megow in cooperation with Wiebke H¨ohn and Tobias Jacobs
We present in [2] a new technique for analyzing sequencing problems such as the variant of the Traveling Salesman Problem (TSP) studied first by Gilmore and Gomory [1] and re-lated flowshop scheduling problems. We show that those sequencing problems have a natural interpretation as Eulerian Extension Problems which leads to new structural insights and solution methods. On a high level view, for an instance of a sequencing problem we find a particular Eulerian graph in which all existing Eulerian circuits represent sequencing
We present in [2] a new technique for analyzing sequencing problems such as the variant of the Traveling Salesman Problem (TSP) studied first by Gilmore and Gomory [1] and re-lated flowshop scheduling problems. We show that those sequencing problems have a natural interpretation as Eulerian Extension Problems which leads to new structural insights and solution methods. On a high level view, for an instance of a sequencing problem we find a particular Eulerian graph in which all existing Eulerian circuits represent sequencing