The Maximum Weight Independent Set of Polygons problem is a fundamental problem in computational geometry. Given a set of weighted polygons in the 2-dimensional plane, the goal is to find a set of pairwise non-overlapping polygons with maximum total weight. Due to its wide range of applications, 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 well-understood. Currently the best known polynomial time algorithm achieves an approximation ratio of n^(epsilon) [Fox and Pach, SODA 2011], and it is not even clear whether the problem is APX-hard. We present a (1+epsilon)-approximation algorithm, assuming that each polygon in the input has at most a polylogarithmic number of vertices. Our algorithm has quasi-polynomial running time. 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 axis-parallel 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 more complex than the original one, and it can handle the arising difficulties due to the arbitrary angles of the 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.
For the general case, the best known result is a @math -approximation algorithm for collections of curves due to Fox and Pach @cite_16 , which assumes that any two curves intersect only @math times, i.e., that they are @math -intersecting. Prior to this, Agarwal and Mustafa presented an algorithm for the special case of straight line segments that computes an independent set of size @math @cite_22 , which yields a worst case approximation ratio of @math . Note that it is @math -hard to decide whether a given graph can be represented as an intersection graph of a collection of curves (i.e., is a string graph) @cite_2 .
@cite_16 @cite_22 @cite_2
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Computing the maximum number of disjoint elements in a collection C of geometric objects is a classical problem in computational geometry with applications ranging from frequency assignment in cellular networks to map labeling in computational cartography. The problem is equivalent to finding the independence number, α(GC), of the intersection graph GC of C, obtained by connecting two elements of C with an edge if and only if their intersection is nonempty. This is known to be an NP-hard task even for systems of segments in the plane with at most two different slopes. The best known polynomial time approximation algorithm for systems of arbitrary segments is due to Agarwal and Mustafa, and returns in the worst case an n1 2+o(1)-approximation for α. Using extensions of the Lipton-Tarjan separator theorem, we improve this result and present, for every e > 0, a polynomial time algorithm for computing α(GC) with approximation ratio at most ne. In contrast, for general graphs, for any e > 0 it is NP-hard to approximate the independence number within a factor of n1−e. We also give a subexponential time exact algorithm for computing the independence number of intersection graphs of arcwise connected sets in the plane. We prove the approximation ratio 8 5 for the metric s,t -path-TSP, and more generally for shortest connected T-joins. The algorithm that achieves this ratio is the simple "Best of Many" version of Christofides' algorithm (1976), suggested by An, Kleinberg and Shmoys (2012), which consists in determining the best Christofides s,t -tour out of those constructed from a family @math of trees having a convex combination dominated by an optimal solution x* of the Held-Karp relaxation. They give the approximation guarantee @math for such an s,t -tour, which is the first improvement after the 5 3 guarantee of Hoogeveen's Christofides type algorithm (1991). Cheriyan, Friggstad and Gao (2012) extended this result to a 13 8-approximation of shortest connected T-joins, for |T|≥4. The ratio 8 5 is proved by simplifying and improving the approach of An, Kleinberg and Shmoys that consists in completing x* 2 in order to dominate the cost of "parity correction" for spanning trees. We partition the edge-set of each spanning tree in @math into an s,t -path (or more generally, into a T-join) and its complement, which induces a decomposition of x*. This decomposition can be refined and then efficiently used to complete x* 2 without using linear programming or particular properties of T, but by adding to each cut deficient for x* 2 an individually tailored explicitly given vector, inherent in x*. A simple example shows that the Best of Many Christofides algorithm may not find a shorter s,t -tour than 3 2 times the incidentally common optima of the problem and of its fractional relaxation. The intersection graph of a set of geometric objects is defined as a graph G = (S, E) in which there is an edge between two nodes si, sj ∈ S if si ∩ sj ≠ 0. The problem of computing a maximum independent set in the intersection graph of a set of objects is known to be NP-complete for most cases in two and higher dimensions. We present approximation algorithms for computing a maximum independent set of intersection graphs of convex objects in R2. Specifically, given (i) a set of n line segments in the plane with maximum independent set of size α, we present algorithms that find an independent set of size at least (α (2 log(2n α)))1 2 in time O(n3) and (α (2log(2n α)))1 4 in time O(n4 3 logcn), (ii) a set of n convex objects with maximum independent set of size α, we present an algorithm that finds an independent set of size at least (α (2log(2n α)))1 3 in time O(n3 + τ (S)), assuming that S can be preprocessed in time τ(S) to answer certain primitive operations on these convex sets, and (iii) a set of n rectangles with maximum independent set of size βn, for β≤ 1, we present an algorithm that computes an independent set of size Ω (β2n). All our algorithms use the notion of partial orders that exploit the geometric structure of the convex objects. We present an O(nlogn)-approximation algorithm for the problem of finding the sparsest spanner of a given directed graph G on n vertices. A spanner of a graph is a sparse subgraph that approximately preserves distances in the original graph. More precisely, given a graph G=(V,E) with nonnegative edge lengths d:E->R^>=^0 and a stretchk>=1, a subgraph H=(V,E"H) is a k-spanner of G if for every edge (s,t)@?E, the graph H contains a path from s to t of length at most [email protected]?d(s,t). The previous best approximation ratio was [email protected]?(n^2^ ^3), due to Dinitz and Krauthgamer (STOC @?11). We also improve the approximation ratio for the important special case of directed 3-spanners with unit edge lengths from [email protected]?(n) to O(n^1^ ^3logn). The best previously known algorithms for this problem are due to Berman, Raskhodnikova and Ruan (FSTTCS @?10) and Dinitz and Krauthgamer. The approximation ratio of our algorithm almost matches Dinitz and [email protected]?s lower bound for the integrality gap of a natural linear programming relaxation. Our algorithm directly implies an O(n^1^ ^3logn)-approximation for the 3-spanner problem on undirected graphs with unit lengths. An easy O(n)-approximation algorithm for this problem has been the best known for decades. Finally, we consider the Directed Steiner Forest problem: given a directed graph with edge costs and a collection of ordered vertex pairs, find a minimum-cost subgraph that contains a path between every prescribed pair. We obtain an approximation ratio of O(n^2^ ^3^+^@e) for any constant @e>0, which improves the O(n^@[email protected]?min(n^4^ ^5,m^2^ ^3)) ratio due to Feldman, Kortsarz and Nutov ([email protected]?12). We study the Maximum Independent Set of Rectangles (MISR) problem: given a collection R of n axis-parallel rectangles, find a maximum-cardinality subset of disjoint rectangles. MISR is a special case of the classical Maximum Independent Set problem, where the input is restricted to intersection graphs of axis-parallel rectangles. Due to its many applications, ranging from map labeling to data mining, MISR has received a significant amount of attention from various research communities. Since the problem is NP-hard, the main focus has been on the design of approximation algorithms. Several groups of researches have independently suggested O(log n)-approximation algorithms for MISR, and this remained the best currently known approximation factor for the problem. The main result of our paper is an O(log log n)-approximation algorithm for MISR. Our algorithm combines existing approaches for solving special cases of the problem, in which the input set of rectangles is restricted to containing specific intersection types, with new insights into the combinatorial structure of sets of intersecting rectangles in the plane. We also consider a generalization of MISR to higher dimensions, where rectangles are replaced by d-dimensional hyper-rectangles. Our results for MISR imply an O((log n)d−2 log log n)-approximation algorithm for this problem, improving upon the best previously known O((log n)d−1)-approximation. We present two new approximation algorithms for the problem of finding a k-node connected spanning subgraph (directed or undirected) of minimum cost. The best known approximation guarantees for this problem were @math for both directed and undirected graphs, and @math for undirected graphs with @math , where @math is the number of nodes in the input graph. Our first algorithm has approximation ratio @math , which is @math except for very large values of @math , namely, @math . This algorithm is based on a new result on @math -connected @math -critical graphs, which is of independent interest in the context of graph theory. Our second algorithm uses the primal-dual method and has approximation ratio @math for all values of @math . Combining these two gives an algorithm with approximation ratio @math , which asymptotically improves the best known approximation guarantee for directed graphs for all values of @math , and for undirected graphs for @math . Moreover, this is the first algorithm that has an approximation guarantee better than @math for all values of @math . Our approximation ratio also provides an upper bound on the integrality gap of the standard LP-relaxation. par>We prove some non-approximability results for restrictions of basic combinatorial optimization problems to instances of bounded “degreeror bounded “width.” Specifically: We prove that the Max 3SAT problem on instances where each variable occurs in at most B clauses, is hard to approximate to within a factor @math , unless @math . H stad [18] proved that the problem is approximable to within a factor @math in polynomial time, and that is hard to approximate to within a factor @math . Our result uses a new randomized reduction from general instances of Max 3SAT to bounded-occurrences instances. The randomized reduction applies to other Max SNP problems as well. We observe that the Set Cover problem on instances where each set has size at most B is hard to approximate to within a factor @math unless @math . The result follows from an appropriate setting of parameters in Feige's reduction [11]. This is essentially tight in light of the existence of @math -approximate algorithms [20, 23, 9] We present a new PCP construction, based on applying parallel repetition to the inner verifier,'' and we provide a tight analysis for it. Using the new construction, and some modifications to known reductions from PCP to Hitting Set, we prove that Hitting Set with sets of size B is hard to approximate to within a factor @math . The problem can be approximated to within a factor B [19], and it is the Vertex Cover problem for B =2. The relationship between hardness of approximation and set size seems to have not been explored before. We observe that the Independent Set problem on graphs having degree at most B is hard to approximate to within a factor @math , unless P = NP . This follows from a comination of results by Clementi and Trevisan [28] and Reingold, Vadhan and Wigderson [27]. It had been observed that the problem is hard to approximate to within a factor @math unless P = NP [1]. An algorithm achieving factor @math is also known [21, 2, 30, 16 . We present the best known algorithms for approximating the minimum-size undirected @math -edge connected spanning subgraph. For simple graphs our approximation ratio is @math . The more precise version of this bound requires @math , and for all such @math it improves the long-standing performance ratio of Cheriyan and Thurimella [SIAM J. Comput., 30 (2000), pp. 528-560], @math . The improvement comes in two steps. First we show that for simple @math -edge connected graphs, any laminar family of degree @math sets is smaller than the general bound ( @math versus @math ). This immediately implies that iterated rounding improves the performance ratio of Cheriyan and Thurimella. The second step carefully chooses good edges for rounding. For multigraphs our approximation ratio is @math . This improves the previous ratio @math [H. N. Gabow, M. X. Goemans, E. Tardos, and D. P. Williamson, Networks, 53 (2009), pp. 345-357]. It is of interest since it is known that for some constant @math , an approximation ratio @math implies @math . Our approximation ratio extends to the minimum-size Steiner network problem, where @math denotes the average vertex demand. The algorithm exploits rounding properties of the first two linear programs in iterated rounding. We consider the problem of scheduling jobs that are given as groups of non-intersecting segments on the real line. Each job J j is associated with an interval, I j , which consists of up to t segments, for some t ≥ 1, a positive weight, w j , and two jobs are in conflict if any of their segments intersect. Such jobs show up in a wide range of applications, including the transmission of continuous-media data, allocation of linear resources (e.g. bandwidth in linear processor arrays), and in computational biology geometry. The objective is to schedule a subset of non-conflicting jobs of maximum total weight.In a single machine environment, our problem can be formulated as the problem of finding a maximum weight independent set in a t-interval graph (the special case of t = 1 is an ordinary interval graph). We show that, for t ≥ 2, this problem is APX-hard, even for highly restricted instances. Our main result is a 2t-approximation algorithm for general instances, based on a novel fractional version of the Local Ratio technique. Previously, the problem was considered only for proper union graphs, a restricted subclass of t-interval graphs, and the approximation factor achieved was (2t - 1 + 1 2t). A bi-criteria polynomial time approximation scheme (PTAS) is developed for the subclass of t-union graphs.In the online case, we consider uniform weight jobs that consist of at most two segments. We show that when the resulting 2-interval graph is proper, a simple greedy algorithm is 3-competitive, while any randomized algorithm has competitive ratio at least 2.5. For general instances, we give a randomized O(log2 R)-competitive (or O((log R)2+e)-competitive) algorithm, where R is the known (unknown) ratio between the longest and the shortest segment in the input sequence. Abstract In this paper we study the Steiner tree problem in graphs for the case when vertices as well as edges have weights associated with them. A greedy approximation algorithm based on “spider decompositions” was developed by Klein and Ravi for this problem. This algorithm provides a worst case approximation ratio of 2 ln k , where k is the number of terminals. However, the best known lower bound on the approximation ratio is (1− o (1)) ln k , assuming that NP ⊈ DTIME [ n O (log log n ) ], by a reduction from set cover. We show that for the unweighted case we can obtain an approximation factor of ln k . For the weighted case we develop a new decomposition theorem and generalize the notion of “spiders” to “branch-spiders” that are used to design a new algorithm with a worst case approximation factor of 1.5 ln k . We then generalize the method to yield an approximation factor of (1.35+ e ) ln k , for any constant e >0. These algorithms, although polynomial, are not very practical due to their high running time, since we need to repeatedly find many minimum weight matchings in each iteration. We also develop a simple greedy algorithm that is practical and has a worst case approximation factor of 1.6103 ln k . The techniques developed for this algorithm imply a method of approximating node weighted network design problems defined by 0–1 proper functions as well. These new ideas also lead to improved approximation guarantees for the problem of finding a minimum node weighted connected dominating set. The previous best approximation guarantee for this problem was 3 ln n by Guha and Khuller. By a direct application of the methods developed in this paper we are able to develop an algorithm with an approximation factor of (1.35+ e ) ln n for any fixed e >0. We present a new efficient algorithm for the search version of the approximate Closest Vector Problem with Preprocessing (CVPP). Our algorithm achieves an approximation factor of O(n sqrt log n ), improving on the previous best of O(n^ 1.5 ) due to Lag arias, Lenstra, and Schnorr hkzbabai . We also show, somewhat surprisingly, that only O(n) vectors of preprocessing advice are sufficient to solve the problem (with the slightly worse approximation factor of O(n)). We remark that this still leaves a large gap with respect to the decisional version of CVPP, where the best known approximation factor is O(sqrt n log n ) due to Aharonov and Regev AharonovR04 . To achieve these results, we show a reduction to the same problem restricted to target points that are close to the lattice and a more efficient reduction to a harder problem, Bounded Distance Decoding with preprocessing (BDDP). Combining either reduction with the previous best-known algorithm for BDDP by Liu, Lyubashevsky, and Micciancio LiuLM06 gives our main result. In the setting of CVP without preprocessing, we also give a reduction from (1+eps)gamma approximate CVP to gamma approximate CVP where the target is at distance at most 1+1 eps times the minimum distance (the length of the shortest non-zero vector) which relies on the lattice sparsification techniques of Dadush and Kun DadushK13 . As our final and most technical contribution, we present a substantially more efficient variant of the LLM algorithm (both in terms of run-time and amount of preprocessing advice), and via an improved analysis, show that it can decode up to a distance proportional to the reciprocal of the smoothing parameter of the dual lattice MR04 . We show that this is never smaller than the LLM decoding radius, and that it can be up to an wide tilde Omega (sqrt n ) factor larger. The notion of recoverable value was advocated in the work of Feige, Immorlica, Mirrokni and Nazerzadeh (APPROX 2009) as a measure of quality for approximation algorithms. There, this concept was applied to facility location problems. In the current work we apply a similar framework to the maximum independent set problem (MIS). We say that an approximation algorithm has recoverable factor ρ, if for every graph it recovers an independent set of size at least where d(v) is the degree of vertex v, and I ranges over all independent sets in G. Hence, in a sense, from every vertex v in the maximum independent set the algorithm recovers a value of at least toward the solution. This quality measure is most effective in graphs in which the maximum independent set is composed of low degree vertices. A simple greedy algorithm achieves . We design a new randomized algorithm for MIS that ensures an expected recoverable factor of at least . In passing, we prove that approximating MIS in graphs with a given k-coloring within a ratio larger than 2 k is unique-games hard. This rules out an alternative approach for obtaining . © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 142–159, 2015 Given an arc-weighted directed graph G = (V, A, spl lscr ) and a pair of nodes s, t, we seek to find an s-t walk of length at most B that maximizes some given function f of the set of nodes visited by the walk. The simplest case is when we seek to maximize the number of nodes visited: this is called the orienteering problem. Our main result is a quasi-polynomial time algorithm that yields an O(log OPT) approximation for this problem when f is a given submodular set function. We then extend it to the case when a node v is counted as visited only if the walk reaches v in its time window [R(v), D(v)]. We apply the algorithm to obtain several new results. First, we obtain an O(log OPT) approximation for a generalization of the orienteering problem in which the profit for visiting each node may vary arbitrarily with time. This captures the time window problem considered earlier for which, even in undirected graphs, the best approximation ratio known [Bansal, (2004)] is O(log sup 2 OPT). The second application is an O(log sup 2 k) approximation for the k-TSP problem in directed graphs (satisfying asymmetric triangle inequality). This is the first non-trivial approximation algorithm for this problem. The third application is an O(log sup 2 k) approximation (in quasi-poly time) for the group Steiner problem in undirected graphs where k is the number of groups. This improves earlier ratios (Garg, ) by a logarithmic factor and almost matches the inapproximability threshold on trees (Halperin and Krauthgamer, 2003). This connection to group Steiner trees also enables us to prove that the problem we consider is hard to approximate to a ratio better than spl Omega (log sup 1- spl epsi OPT), even in undirected graphs. Even though our algorithm runs in quasi-poly time, we believe that the implications for the approximability of several basic optimization problems are interesting. We give a ( 17 12 ) -approximation algorithm for the following NP- hard problem: Given a simple undirected graph, find a 2-edge connected span- ning subgraph that has the minimum number of edges. The best previous approximation guarantee was ( 3 2 ) . If the well known TSP ( 4 3 ) conjecture holds, then there is a ( 4 3 ) -approximation algorithm. Thus our main result gets half-way to this target. A 5 4-approximation algorithm is presented for the minimum cardinality 2-edge-connected spanning subgraph problem in undirected graphs. This improves the previous best approximation ratio of 4 3. It is shown that our ratio is tight with respect to current lower bounds, and any further improvement is possible only if new lower bounds are discovered. The MAXIMUM INDEPENDENT SET problem in d-box graphs, i.e., in the intersection graphs of axis-parallel rectangles in Rd, is a challenge open problem. For any fixed d ≥ 2 the problem is NP-hard and no approximation algorithm with ratio o(logd-1 n) is known. In some restricted cases, e.g., for d-boxes with bounded aspect ratio, a PTAS exists [17]. In this paper we prove APX-hardness (and hence non-existence of a PTAS, unless P = NP), of the MAXIMUM INDEPENDENT SET problem in d-box graphs for any fixed d ≥ 3. We state also first explicit lower bound 443 442 on efficient approximability in such case. Additionally, we provide a generic method how to prove APX-hardness for many NP-hard graph optimization problems in d-box graphs for any fixed d ≥ 3. In 2-dimensional case we give a generic approach to NP-hardness results for these problems in highly restricted intersection graphs of axis-parallel unit squares (alternatively, in unit disk graphs). In this paper we study a fundamental open problem in the area of probabilistic checkable proofs: What is the smallest s such that NP ⊆ naPCP1,s[O(log n),3]? In the language of hardness of approximation, this problem is equivalent to determining the smallest s such that getting an s-approximation for satisfiable 3-bit constraint satisfaction problems ("3-CSPs") is NP-hard. The previous best upper bound and lower bound for s are 20 27+µ by Khot and Saket [KS06], and 5 8 (assuming NP subseteq BPP) by Zwick [Zwi98]. In this paper we close the gap assuming Khot's d-to-1 Conjecture. Formally, we prove that if Khot's d-to-1 Conjecture holds for any finite constant integer d, then NP naPCP1,5 8+ µ[O(log n),3] for any constant µ > 0. Our conditional result also solves Hastad's open question [Has01] on determining the inapproximability of satisfiable Max-NTW ("Not Two") instances and confirms Zwick's conjecture [Zwi98] that the 5 8-approximation algorithm for satisfiable 3-CSPs is optimal. An important question in theoretical computer science is to determine the best possible running time for solving a problem at hand. For geometric optimization problems, we often understand their complexity on a rough scale, but not very well on a finer scale. One such example is the two-dimensional knapsack problem for squares. There is a polynomial time (1 + ϵ)-approximation algorithm for it (i.e., a PTAS) but the running time of this algorithm is triple exponential in 1 ϵ, i.e., Ω(n221 ϵ). A double or triple exponential dependence on 1 ϵ is inherent in how this and several other algorithms for other geometric problems work. In this paper, we present an EPTAS for knapsack for squares, i.e., a (1+ϵ)-approximation algorithm with a running time of Oϵ(1)·nO(1). In particular, the exponent of n in the running time does not depend on ϵ at all! Since there can be no FPTAS for the problem (unless P = NP) this is the best kind of approximation scheme we can hope for. To achieve this improvement, we introduce two new key ideas: We present a fast method to guess the Ω(221 ϵ) relatively large squares of a suitable near-optimal packing instead of using brute-force enumeration. Secondly, we introduce an indirect guessing framework to define sizes of cells for the remaining squares. In the previous PTAS each of these steps needs a running time of Ω(n221 ϵ) and we improve both to Oϵ(1) · nO(1). We complete our result by giving an algorithm for two-dimensional knapsack for rectangles under (1 + ϵ)-resource augmentation. In this setting, we also improve the best known running time of Ω(n1 ϵ1 ϵ) to Oϵ(1) · nO(1) and compute even a solution with optimal profit, in contrast to the best previously known polynomial time algorithm for this setting that computes only an approximation. We believe that our new techniques have the potential to be useful for other settings as well. We obtain improved algorithms for finding small vertex covers in bounded degree graphs and hypergraphs. We use semidefinite programming to relax the problems and introduce new rounding techniques for these relaxations. On graphs with maximum degree at most @math , the algorithm achieves a performance ratio of @math for large @math , which improves the previously known ratio of @math obtained by Halld orsson and Radhakrishnan. Using similar techniques, we also present improved approximations for the vertex cover problem in hypergraphs. For k-uniform hypergraphs with n vertices, we achieve a ratio of @math for large n, and for k-uniform hypergraphs with maximum degree at most @math the algorithm achieves a ratio of @math for large @math . These results considerably improve the previous best ratio of @math for bounded degree k-uniform hypergraphs, and @math for general k-uniform hypergraphs, both obtained by Krivelevich. Using similar techniques, we also obtain an approximation algorithm for the weighted independent set problem, matching a recent result of Halldorsson. Let @math be a @math -regular triangle-free graph with @math edges. We present an algorithm which finds a cut in @math with at least @math edges in expectation, improving upon Shearer's classic result. In particular, this implies that any @math -regular triangle-free graph has a cut of at least this size, and thus, we obtain a new lower bound for the maximum number of edges in a bipartite subgraph of @math . Our algorithm is simpler than Shearer's classic algorithm and it can be interpreted as a very efficient randomised distributed (local) algorithm : each node needs to produce only one random bit, and the algorithm runs in one round. The randomised algorithm itself was discovered using computational techniques . We show that for any fixed @math , there exists a weighted neighbourhood graph @math such that there is a one-to-one correspondence between heavy cuts of @math and randomised local algorithms that find large cuts in any @math -regular input graph. This turns out to be a useful tool for analysing the existence of cuts in @math -regular graphs: we can compute the optimal cut of @math to attain a lower bound on the maximum cut size of any @math -regular triangle-free graph.