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**Published in:** LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

The sum of square roots is as follows: Given x_1,… ,x_n ∈ ℤ and a₁,… ,a_n ∈ ℕ decide whether E = ∑_{i=1}^n x_i √{a_i} ≥ 0. It is a prominent open problem (Problem 33 of the Open Problems Project), whether this can be decided in polynomial time. The state-of-the-art methods rely on separation bounds, which are lower bounds on the minimum nonzero absolute value of E. The current best bound shows that |E| ≥ (n ⋅ max_i (|x_i| ⋅√{a_i})) ^{-2ⁿ}, which is doubly exponentially small.
We provide a new bound of the form |E| ≥ γ ⋅ (n ⋅ max_i |x_i|)^{-2n} where γ is a constant depending on a₁,… ,a_n. This is singly exponential in n for fixed a_1,… ,a_n. The constant γ is not explicit and stems from the subspace theorem, a deep result in the geometry of numbers.

Friedrich Eisenbrand, Matthieu Haeberle, and Neta Singer. An Improved Bound on Sums of Square Roots via the Subspace Theorem. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 54:1-54:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{eisenbrand_et_al:LIPIcs.SoCG.2024.54, author = {Eisenbrand, Friedrich and Haeberle, Matthieu and Singer, Neta}, title = {{An Improved Bound on Sums of Square Roots via the Subspace Theorem}}, booktitle = {40th International Symposium on Computational Geometry (SoCG 2024)}, pages = {54:1--54:8}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-316-4}, ISSN = {1868-8969}, year = {2024}, volume = {293}, editor = {Mulzer, Wolfgang and Phillips, Jeff M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.54}, URN = {urn:nbn:de:0030-drops-199993}, doi = {10.4230/LIPIcs.SoCG.2024.54}, annote = {Keywords: Exact computing, Separation Bounds, Computational Geometry, Geometry of Numbers} }

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**Published in:** LIPIcs, Volume 204, 29th Annual European Symposium on Algorithms (ESA 2021)

We consider the problem of solving integer programs of the form min {c^⊺ x : Ax = b, x ∈ ℤ_{⩾ 0}}, where A is a multistage stochastic matrix in the following sense: the primal treedepth of A is bounded by a parameter d, which means that the columns of A can be organized into a rooted forest of depth at most d so that columns not bound by the ancestor/descendant relation do not have non-zero entries in the same row. We give an algorithm that solves this problem in fixed-parameter time f(d,‖A‖_{∞})⋅ nlog^{𝒪(2^d)} n, where f is a computable function and n is the number of rows of A. The algorithm works in the strong model, where the running time only measures unit arithmetic operations on the input numbers and does not depend on their bitlength. This is the first fpt algorithm for multistage stochastic integer programming to achieve almost linear running time in the strong sense. For two-stage stochastic integer programs, our algorithm works in time 2^{((r+s)‖A‖_∞)^{𝒪(r(r+s))}}⋅ nlog^{𝒪(rs)} n, which improves over previous methods both in terms of the polynomial factor and in terms of the dependence on r and s. In fact, for r = 1 the dependence on s is asymptotically almost tight assuming the Exponential Time Hypothesis. Our algorithm can be also parallelized: we give an implementation in the PRAM model that achieves running time f(d,‖A‖_{∞})⋅ log^{𝒪(2^d)} n using n processors.
The main conceptual ingredient in our algorithms is a new proximity result for multistage stochastic integer programs. We prove that if we consider an integer program P, say with a constraint matrix A, then for every optimum solution to the linear relaxation of P there exists an optimum (integral) solution to P that lies, in the 𝓁_{∞}-norm, within distance bounded by a function of ‖A‖_{∞} and the primal treedepth of A. On the way to achieve this result, we prove a generalization and considerable improvement of a structural result of Klein for multistage stochastic integer programs. Once the proximity results are established, this allows us to apply a treedepth-based branching strategy guided by an optimum solution to the linear relaxation.

Jana Cslovjecsek, Friedrich Eisenbrand, Michał Pilipczuk, Moritz Venzin, and Robert Weismantel. Efficient Sequential and Parallel Algorithms for Multistage Stochastic Integer Programming Using Proximity. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 33:1-33:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{cslovjecsek_et_al:LIPIcs.ESA.2021.33, author = {Cslovjecsek, Jana and Eisenbrand, Friedrich and Pilipczuk, Micha{\l} and Venzin, Moritz and Weismantel, Robert}, title = {{Efficient Sequential and Parallel Algorithms for Multistage Stochastic Integer Programming Using Proximity}}, booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)}, pages = {33:1--33:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-204-4}, ISSN = {1868-8969}, year = {2021}, volume = {204}, editor = {Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2021.33}, URN = {urn:nbn:de:0030-drops-146146}, doi = {10.4230/LIPIcs.ESA.2021.33}, annote = {Keywords: parameterized algorithm, multistage stochastic programming, proximity} }

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**Published in:** LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

We show that a constant factor approximation of the shortest and closest lattice vector problem w.r.t. any 𝓁_p-norm can be computed in time 2^{(0.802 +ε) n}. This matches the currently fastest constant factor approximation algorithm for the shortest vector problem w.r.t. 𝓁₂. To obtain our result, we combine the latter algorithm w.r.t. 𝓁₂ with geometric insights related to coverings.

Friedrich Eisenbrand and Moritz Venzin. Approximate CVP_p in Time 2^{0.802 n}. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 43:1-43:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{eisenbrand_et_al:LIPIcs.ESA.2020.43, author = {Eisenbrand, Friedrich and Venzin, Moritz}, title = {{Approximate CVP\underlinep in Time 2^\{0.802 n\}}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {43:1--43:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.43}, URN = {urn:nbn:de:0030-drops-129097}, doi = {10.4230/LIPIcs.ESA.2020.43}, annote = {Keywords: Shortest and closest vector problem, approximation algorithm, sieving, covering convex bodies} }

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**Published in:** LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

Diversity maximization is an important geometric optimization problem with many applications in recommender systems, machine learning or search engines among others. A typical diversification problem is as follows: Given a finite metric space (X,d) and a parameter k in N, find a subset of k elements of X that has maximum diversity. There are many functions that measure diversity. One of the most popular measures, called remote-clique, is the sum of the pairwise distances of the chosen elements. In this paper, we present novel results on three widely used diversity measures: Remote-clique, remote-star and remote-bipartition.
Our main result are polynomial time approximation schemes for these three diversification problems under the assumption that the metric space is doubling. This setting has been discussed in the recent literature. The existence of such a PTAS however was left open.
Our results also hold in the setting where the distances are raised to a fixed power q >= 1, giving rise to more variants of diversity functions, similar in spirit to the variations of clustering problems depending on the power applied to the pairwise distances. Finally, we provide a proof of NP-hardness for remote-clique with squared distances in doubling metric spaces.

Alfonso Cevallos, Friedrich Eisenbrand, and Sarah Morell. Diversity Maximization in Doubling Metrics. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 33:1-33:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{cevallos_et_al:LIPIcs.ISAAC.2018.33, author = {Cevallos, Alfonso and Eisenbrand, Friedrich and Morell, Sarah}, title = {{Diversity Maximization in Doubling Metrics}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {33:1--33:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-094-1}, ISSN = {1868-8969}, year = {2018}, volume = {123}, editor = {Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.33}, URN = {urn:nbn:de:0030-drops-99818}, doi = {10.4230/LIPIcs.ISAAC.2018.33}, annote = {Keywords: Remote-clique, remote-star, remote-bipartition, doubling dimension, grid rounding, epsilon-nets, polynomial time approximation scheme, facility location, information retrieval} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

We consider integer programming problems max {c^Tx : A x = b, l <= x <= u, x in Z^{nt}} where A has a (recursive) block-structure generalizing n-fold integer programs which recently received considerable attention in the literature. An n-fold IP is an integer program where A consists of n repetitions of submatrices A in Z^{r × t} on the top horizontal part and n repetitions of a matrix B in Z^{s × t} on the diagonal below the top part. Instead of allowing only two types of block matrices, one for the horizontal line and one for the diagonal, we generalize the n-fold setting to allow for arbitrary matrices in every block. We show that such an integer program can be solved in time n^2t^2 phi x (r s delta)^{O(rs^2+ sr^2)} (ignoring logarithmic factors). Here delta is an upper bound on the largest absolute value of an entry of A and phi is the largest binary encoding length of a coefficient of c. This improves upon the previously best algorithm of Hemmecke, Onn and Romanchuk that runs in time n^3t^3 phi x delta^{O(st(r+t))}. In particular, our algorithm is not exponential in the number t of columns of A and B.
Our algorithm is based on a new upper bound on the l_1-norm of an element of the Graver basis of an integer matrix and on a proximity bound between the LP and IP optimal solutions tailored for IPs with block structure. These new bounds rely on the Steinitz Lemma.
Furthermore, we extend our techniques to the recently introduced tree-fold IPs, where we again present a more efficient algorithm in a generalized setting.

Friedrich Eisenbrand, Christoph Hunkenschröder, and Kim-Manuel Klein. Faster Algorithms for Integer Programs with Block Structure. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 49:1-49:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{eisenbrand_et_al:LIPIcs.ICALP.2018.49, author = {Eisenbrand, Friedrich and Hunkenschr\"{o}der, Christoph and Klein, Kim-Manuel}, title = {{Faster Algorithms for Integer Programs with Block Structure}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {49:1--49:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.49}, URN = {urn:nbn:de:0030-drops-90537}, doi = {10.4230/LIPIcs.ICALP.2018.49}, annote = {Keywords: n-fold, Tree-fold, Integer Programming} }

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**Published in:** LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

Diversity maximization is an important concept in information retrieval, computational geometry and operations research. Usually, it is a variant of the following problem: Given a ground set, constraints, and a function f that measures diversity of a subset, the task is to select a feasible subset S such that f(S) is maximized. The sum-dispersion function f(S) which is the sum of the pairwise distances in S, is in this context a prominent diversification measure. The corresponding diversity maximization is the "max-sum" or "sum-sum" diversification. Many recent results deal with the design of constant-factor approximation algorithms of diversification problems involving sum-dispersion function under a matroid constraint.
In this paper, we present a PTAS for the max-sum diversity problem under a matroid constraint for distances d(.,.) of negative type. Distances of negative type are, for example, metric distances stemming from the l_2 and l_1 norms, as well as the cosine or spherical, or Jaccard distance which are popular similarity metrics in web and image search.
Our algorithm is based on techniques developed in geometric algorithms like metric embeddings and convex optimization. We show that one can compute a fractional solution of the usually non-convex relaxation of the problem which yields an upper bound on the optimum integer solution. Starting from this fractional solution, we employ a deterministic rounding approach which only incurs a small loss in terms of objective, thus leading to a PTAS. This technique can be applied to other previously studied variants of the max-sum dispersion function, including combinations of diversity with linear-score maximization, improving over the previous constant-factor approximation algorithms.

Alfonso Cevallos, Friedrich Eisenbrand, and Rico Zenklusen. Max-Sum Diversity Via Convex Programming. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 26:1-26:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{cevallos_et_al:LIPIcs.SoCG.2016.26, author = {Cevallos, Alfonso and Eisenbrand, Friedrich and Zenklusen, Rico}, title = {{Max-Sum Diversity Via Convex Programming}}, booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)}, pages = {26:1--26:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-009-5}, ISSN = {1868-8969}, year = {2016}, volume = {51}, editor = {Fekete, S\'{a}ndor and Lubiw, Anna}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.26}, URN = {urn:nbn:de:0030-drops-59186}, doi = {10.4230/LIPIcs.SoCG.2016.26}, annote = {Keywords: Geometric Dispersion, Embeddings, Approximation Algorithms, Convex Programming, Matroids} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 10211, Flexible Network Design (2010)

We investigate the diameter of a natural abstraction of the
$1$-skeleton of polyhedra. Even if this abstraction is more general than
other abstractions previously studied in the literature,
known upper bounds on the diameter of polyhedra continue to hold
here. On the other hand, we show that this abstraction has its
limits by providing an almost quadratic lower bound.

Friedrich Eisenbrand, Nicolai Hähnle, Alexander Razborov, and Thomas Rothvoß. Diameter of Polyhedra: Limits of Abstraction. In Flexible Network Design. Dagstuhl Seminar Proceedings, Volume 10211, pp. 1-5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{eisenbrand_et_al:DagSemProc.10211.2, author = {Eisenbrand, Friedrich and H\"{a}hnle, Nicolai and Razborov, Alexander and Rothvo{\ss}, Thomas}, title = {{Diameter of Polyhedra: Limits of Abstraction}}, booktitle = {Flexible Network Design}, pages = {1--5}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2010}, volume = {10211}, editor = {Anupam Gupta and Stefano Leonardi and Berthold V\"{o}cking and Roger Wattenhofer}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.10211.2}, URN = {urn:nbn:de:0030-drops-27247}, doi = {10.4230/DagSemProc.10211.2}, annote = {Keywords: Polyhedra, Graphs} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 10071, Scheduling (2010)

Consider a set of $n$ periodic tasks $ au_1,ldots, au_n$ where $ au_i$ is described
by an execution time $c_i$, a (relative) deadline $d_i$ and a period $p_i$.
We assume that jobs are released synchronously (i.e. at each multiple of $p_i$) and consider pre-emptive, uni-processor schedules.
We show that computing the response time of a task $ au_n$ in a Rate-monotonic schedule
i.e. computing
[
minleft{ r geq mid c_n + sum_{i=1}^{n-1} leftlceil frac{r}{p_i}
ight
ceil c_i leq r
ight}
]
is (weakly) $mathbf{NP}$-hard (where $ au_n$ has the lowest priority and the deadlines
are implicit, i.e. $d_i = p_i$).
Furthermore we obtain that verifying EDF-schedulability, i.e.
[
forall Q geq 0: sum_{i=1}^n left( leftlfloor frac{Q-d_i}{p_i}
ight
floor +1
ight)cdot c_i leq Q
]
for constrained-deadline tasks ($d_i leq p_i$) is weakly $mathbf{coNP}$-hard.

Friedrich Eisenbrand and Thomas Rothvoss. Recent Hardness Results for Periodic Uni-processor Scheduling. In Scheduling. Dagstuhl Seminar Proceedings, Volume 10071, pp. 1-7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{eisenbrand_et_al:DagSemProc.10071.10, author = {Eisenbrand, Friedrich and Rothvoss, Thomas}, title = {{Recent Hardness Results for Periodic Uni-processor Scheduling}}, booktitle = {Scheduling}, pages = {1--7}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2010}, volume = {10071}, editor = {Susanne Albers and Sanjoy K. Baruah and Rolf H. M\"{o}hring and Kirk Pruhs}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.10071.10}, URN = {urn:nbn:de:0030-drops-25458}, doi = {10.4230/DagSemProc.10071.10}, annote = {Keywords: Hardness, periodic scheduling, uni-processor scheduling} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 10071, Scheduling (2010)

We consider a real-time scheduling problem that occurs in the design
of software-based aircraft control. The goal is to distribute tasks
$ au_i=(c_i,p_i)$ on a minimum number of identical machines and to
compute offsets $a_i$ for the tasks such that no collision occurs. A
task $ au_i$ releases a job of running time $c_i$ at each time $a_i +
kcdot p_i, , k in mathbb{N}_0$ and a collision occurs if two jobs are
simultaneously active on the same machine.
We shed some light on the complexity and approximability landscape of this problem.
Although the problem cannot be approximated
within a factor of $n^{1-varepsilon}$ for any $varepsilon>0$, an interesting restriction
is much more tractable: If the periods are dividing (for each $i,j$ one has $p_i |
p_j$ or $p_j | p_i$), the problem allows for a better structured representation of solutions, which leads
to a 2-approximation. This result is tight, even asymptotically.

Friedrich Eisenbrand, Nicolai Hähnle, Martin Niemeier, Martin Skutella, Jose Verschae, and Andreas Wiese. Scheduling periodic tasks in a hard real-time environment. In Scheduling. Dagstuhl Seminar Proceedings, Volume 10071, pp. 1-3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{eisenbrand_et_al:DagSemProc.10071.13, author = {Eisenbrand, Friedrich and H\"{a}hnle, Nicolai and Niemeier, Martin and Skutella, Martin and Verschae, Jose and Wiese, Andreas}, title = {{Scheduling periodic tasks in a hard real-time environment}}, booktitle = {Scheduling}, pages = {1--3}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2010}, volume = {10071}, editor = {Susanne Albers and Sanjoy K. Baruah and Rolf H. M\"{o}hring and Kirk Pruhs}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.10071.13}, URN = {urn:nbn:de:0030-drops-25348}, doi = {10.4230/DagSemProc.10071.13}, annote = {Keywords: Real-Time Scheduling, Periodic scheduling problem, Periodic maintenance problem, Approximation hardness, Approximation algorithm} }