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**Published in:** LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

The Connected Max Cut (CMC) problem takes in an undirected graph G(V,E) and finds a subset S ⊆ V such that the induced subgraph G[S] is connected and the number of edges connecting vertices in S to vertices in V⧵S is maximized. This problem is closely related to the Max Leaf Degree (MLD) problem. The input to the MLD problem is an undirected graph G(V,E) and the goal is to find a subtree of G that maximizes the degree (in G) of its leaves. [Gandhi et al. 2018] observed that an α-approximation for the MLD problem induces an 𝒪(α)-approximation for the CMC problem.
We present an 𝒪(log log |V|)-approximation algorithm for the MLD problem via local search. This implies an 𝒪(log log |V|)-approximation algorithm for the CMC problem. Thus, improving (exponentially) the best known 𝒪(log |V|) approximation of the Connected Max Cut problem [Hajiaghayi et al. 2015].

Baruch Schieber and Soroush Vahidi. Approximating Connected Maximum Cuts via Local Search. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 93:1-93:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{schieber_et_al:LIPIcs.ESA.2023.93, author = {Schieber, Baruch and Vahidi, Soroush}, title = {{Approximating Connected Maximum Cuts via Local Search}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {93:1--93:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. 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.2023.93}, URN = {urn:nbn:de:0030-drops-187466}, doi = {10.4230/LIPIcs.ESA.2023.93}, annote = {Keywords: approximation algorithms, graph theory, max-cut, local search} }

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APPROX

**Published in:** LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

We consider scheduling real-time jobs in the classic flow shop model. The input is a set of n jobs, each consisting of m segments to be processed on m machines in the specified order, such that segment I_i of a job can start processing on machine M_i only after segment I_{i-1} of the same job completed processing on machine M_{i-1}, for 2 ≤ i ≤ m. Each job also has a release time, a due date, and a weight. The objective is to maximize the throughput (or, profit) of the n jobs, i.e., to find a subset of the jobs that have the maximum total weight and can complete processing on the m machines within their time windows. This problem has numerous real-life applications ranging from manufacturing to cloud and embedded computing platforms, already in the special case where m = 2. Previous work in the flow shop model has focused on makespan, flow time, or tardiness objectives. However, little is known for the flow shop model in the real-time setting. In this work, we give the first nontrivial results for this problem and present a pseudo-polynomial time (2m+1)-approximation algorithm for the problem on m ≥ 2 machines, where m is a constant. This ratio is essentially tight due to a hardness result of Ω(m/(log m)) for the approximation ratio. We further give a polynomial-time algorithm for the two-machine case, with an approximation ratio of (9+ε) where ε = O(1/n). We obtain better bounds for some restricted subclasses of inputs with two machines. To the best of our knowledge, this fundamental problem of throughput maximization in the flow shop scheduling model is studied here for the first time.

Lior Ben Yamin, Jing Li, Kanthi Sarpatwar, Baruch Schieber, and Hadas Shachnai. Maximizing Throughput in Flow Shop Real-Time Scheduling. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 48:1-48:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{benyamin_et_al:LIPIcs.APPROX/RANDOM.2020.48, author = {Ben Yamin, Lior and Li, Jing and Sarpatwar, Kanthi and Schieber, Baruch and Shachnai, Hadas}, title = {{Maximizing Throughput in Flow Shop Real-Time Scheduling}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)}, pages = {48:1--48:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-164-1}, ISSN = {1868-8969}, year = {2020}, volume = {176}, editor = {Byrka, Jaros{\l}aw and Meka, Raghu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.48}, URN = {urn:nbn:de:0030-drops-126510}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2020.48}, annote = {Keywords: Flow shop, real-time scheduling, throughput maximization, approximation algorithms} }

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**Published in:** LIPIcs, Volume 150, 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)

We revisit a classical scheduling model to incorporate modern trends in data center networks and cloud services. Addressing some key challenges in the allocation of shared resources to user requests (jobs) in such settings, we consider the following variants of the classic resource allocation problem (RAP). The input to our problems is a set J of jobs and a set M of homogeneous hosts, each has an available amount of some resource. A job is associated with a release time, a due date, a weight and a given length, as well as its resource requirement. A feasible schedule is an allocation of the resource to a subset of the jobs, satisfying the job release times/due dates as well as the resource constraints. A crucial distinction between classic RAP and our problems is that we allow preemption and migration of jobs, motivated by virtualization techniques.
We consider two natural objectives: throughput maximization (MaxT), which seeks a maximum weight subset of the jobs that can be feasibly scheduled on the hosts in M, and resource minimization (MinR), that is finding the minimum number of (homogeneous) hosts needed to feasibly schedule all jobs. Both problems are known to be NP-hard. We first present an Omega(1)-approximation algorithm for MaxT instances where time-windows form a laminar family of intervals. We then extend the algorithm to handle instances with arbitrary time-windows, assuming there is sufficient slack for each job to be completed. For MinR we study a more general setting with d resources and derive an O(log d)-approximation for any fixed d >= 1, under the assumption that time-windows are not too small. This assumption can be removed leading to a slightly worse ratio of O(log d log^* T), where T is the maximum due date of any job.

Kanthi Sarpatwar, Baruch Schieber, and Hadas Shachnai. The Preemptive Resource Allocation Problem. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 26:1-26:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{sarpatwar_et_al:LIPIcs.FSTTCS.2019.26, author = {Sarpatwar, Kanthi and Schieber, Baruch and Shachnai, Hadas}, title = {{The Preemptive Resource Allocation Problem}}, booktitle = {39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)}, pages = {26:1--26:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-131-3}, ISSN = {1868-8969}, year = {2019}, volume = {150}, editor = {Chattopadhyay, Arkadev and Gastin, Paul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2019.26}, URN = {urn:nbn:de:0030-drops-115886}, doi = {10.4230/LIPIcs.FSTTCS.2019.26}, annote = {Keywords: Machine Scheduling, Resource Allocation, Vector Packing, Approximation Algorithms} }

Document

**Published in:** LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

We study a variant of the generalized assignment problem (GAP) with group constraints. An instance of (Group GAP) is a set I of items, partitioned into L groups, and a set of m uniform (unit-sized) bins. Each item i in I has a size s_i >0, and a profit p_{i,j} >= 0 if packed in bin j. A group of items is satisfied if all of its items are packed. The goal is to find a feasible packing of a subset of the items in the bins such that the total profit from satisfied groups is maximized. We point to central applications of Group GAP in Video-on-Demand services, mobile Device-to-Device network caching and base station cooperation in 5G networks.
Our main result is a 1/6-approximation algorithm for Group GAP instances where the total size of each group is at most m/2. At the heart of our algorithm lies an interesting derivation of a submodular function from the classic LP formulation of GAP, which facilitates the construction of a high profit solution utilizing at most half the total bin capacity, while the other half is reserved for later use. In particular, we give an algorithm for submodular maximization subject to a knapsack constraint, which finds a solution of profit at least 1/3 of the optimum, using at most half the knapsack capacity, under mild restrictions on element sizes. Our novel approach of submodular optimization subject to a knapsack with reserved capacity constraint may find applications in solving other group assignment problems.

Ariel Kulik, Kanthi Sarpatwar, Baruch Schieber, and Hadas Shachnai. Generalized Assignment via Submodular Optimization with Reserved Capacity. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 69:1-69:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{kulik_et_al:LIPIcs.ESA.2019.69, author = {Kulik, Ariel and Sarpatwar, Kanthi and Schieber, Baruch and Shachnai, Hadas}, title = {{Generalized Assignment via Submodular Optimization with Reserved Capacity}}, booktitle = {27th Annual European Symposium on Algorithms (ESA 2019)}, pages = {69:1--69:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-124-5}, ISSN = {1868-8969}, year = {2019}, volume = {144}, editor = {Bender, Michael A. and Svensson, Ola 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.2019.69}, URN = {urn:nbn:de:0030-drops-111906}, doi = {10.4230/LIPIcs.ESA.2019.69}, annote = {Keywords: Group Generalized Assignment Problem, Submodular Maximization, Knapsack Constraints, Approximation Algorithms} }

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**Published in:** LIPIcs, Volume 116, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)

We study the generalized assignment problem with time-sensitive item groups (chi-AGAP). It has central applications in advertisement placement on the Internet, and in virtual network embedding in Cloud data centers. We are given a set of items, partitioned into n groups, and a set of T identical bins (or, time-slots). Each group 1 <= j <= n has a time-window chi_j = [r_j, d_j]subseteq [T] in which it can be packed. Each item i in group j has a size s_i>0 and a non-negative utility u_{it} when packed into bin t in chi_j. A bin can accommodate at most one item from each group and the total size of the items in a bin cannot exceed its capacity. The goal is to find a feasible packing of a subset of the items in the bins such that the total utility from groups that are completely packed is maximized. Our main result is an Omega(1)-approximation algorithm for chi-AGAP. Our approximation technique relies on a non-trivial rounding of a configuration LP, which can be adapted to other common scenarios of resource allocation in Cloud data centers.

Kanthi Sarpatwar, Baruch Schieber, and Hadas Shachnai. Generalized Assignment of Time-Sensitive Item Groups. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 24:1-24:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{sarpatwar_et_al:LIPIcs.APPROX-RANDOM.2018.24, author = {Sarpatwar, Kanthi and Schieber, Baruch and Shachnai, Hadas}, title = {{Generalized Assignment of Time-Sensitive Item Groups}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)}, pages = {24:1--24:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-085-9}, ISSN = {1868-8969}, year = {2018}, volume = {116}, editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.24}, URN = {urn:nbn:de:0030-drops-94287}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.24}, annote = {Keywords: Approximation Algorithms, Packing and Covering problems, Generalized Assignment problem} }

Document

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

We consider the problem of maintaining a maximal independent set (MIS) in a dynamic graph subject to edge insertions and deletions. Recently, Assadi, Onak, Schieber and Solomon (STOC 2018) showed that an MIS can be maintained in sublinear (in the dynamically changing number of edges) amortized update time. In this paper we significantly improve the update time for uniformly sparse graphs. Specifically, for graphs with arboricity alpha, the amortized update time of our algorithm is O(alpha^2 * log^2 n), where n is the number of vertices. For low arboricity graphs, which include, for example, minor-free graphs as well as some classes of "real world" graphs, our update time is polylogarithmic. Our update time improves the result of Assadi et al. for all graphs with arboricity bounded by m^{3/8 - epsilon}, for any constant epsilon > 0. This covers much of the range of possible values for arboricity, as the arboricity of a general graph cannot exceed m^{1/2}.

Krzysztof Onak, Baruch Schieber, Shay Solomon, and Nicole Wein. Fully Dynamic MIS in Uniformly Sparse Graphs. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 92:1-92:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{onak_et_al:LIPIcs.ICALP.2018.92, author = {Onak, Krzysztof and Schieber, Baruch and Solomon, Shay and Wein, Nicole}, title = {{Fully Dynamic MIS in Uniformly Sparse Graphs}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {92:1--92:14}, 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.2018.92}, URN = {urn:nbn:de:0030-drops-90968}, doi = {10.4230/LIPIcs.ICALP.2018.92}, annote = {Keywords: dynamic graph algorithms, independent set, sparse graphs, graph arboricity} }

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**Published in:** LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)

We introduce and study a network resource management problem that is a special case of non-metric k-median, naturally arising in cross platform scheduling and cloud computing. In the continuous d-dimensional container selection problem, we are given a set C of input points in d-dimensional Euclidean space, for some d >= 2, and a budget k. An input point p can be assigned to a "container point" c only if c dominates p in every dimension. The assignment cost is then equal to the L1-norm of the container point. The goal is to find k container points in the d-dimensional space, such that the total assignment cost for all input points is minimized. The discrete variant of the problem has one key distinction, namely, the container points must be chosen from a given set F of points.
For the continuous version, we obtain a polynomial time approximation scheme for any fixed dimension d>= 2. On the negative side, we show that the problem is NP-hard for any d>=3. We further show that the discrete version is significantly harder, as it is NP-hard to approximate without violating the budget k in any dimension d>=3. Thus, we focus on obtaining bi-approximation algorithms. For d=2, the bi-approximation guarantee is (1+epsilon,3), i.e., for any epsilon>0, our scheme outputs a solution of size 3k and cost at most (1+epsilon) times the optimum. For fixed d>2, we present a (1+epsilon,O((1/epsilon)log k)) bi-approximation algorithm.

Viswanath Nagarajan, Kanthi K. Sarpatwar, Baruch Schieber, Hadas Shachnai, and Joel L. Wolf. The Container Selection Problem. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 416-434, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{nagarajan_et_al:LIPIcs.APPROX-RANDOM.2015.416, author = {Nagarajan, Viswanath and Sarpatwar, Kanthi K. and Schieber, Baruch and Shachnai, Hadas and Wolf, Joel L.}, title = {{The Container Selection Problem}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)}, pages = {416--434}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-89-7}, ISSN = {1868-8969}, year = {2015}, volume = {40}, editor = {Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.416}, URN = {urn:nbn:de:0030-drops-53153}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2015.416}, annote = {Keywords: non-metric k-median, geometric hitting set, approximation algorithms, cloud computing, cross platform scheduling.} }

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**Published in:** LIPIcs, Volume 8, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)

We consider the following fundamental scheduling problem. The input consists of $n$ jobs to be scheduled on a set of machines of bounded capacities. Each job is associated with a release time, a due date, a processing time and demand for machine capacity. The goal is to schedule all of the jobs non-preemptively in their release-time-deadline windows, subject to machine capacity constraints, such that the total busy time of the machines is minimized. Our problem has important applications in power-aware scheduling, optical network design and unit commitment in power systems. Scheduling to minimize busy times is APX-hard already in the special case where all jobs have the same (unit) processing times and can be scheduled in a fixed time interval.
Our main result is a $5$-approximation algorithm for general instances. We extend this result to obtain an algorithm with the same approximation ratio for the problem of scheduling moldable jobs, that requires also to determine, for each job, one of several processing-time vs. demand configurations. Better bounds and exact algorithms are derived for several special cases, including proper interval graphs, intervals forming a clique and laminar families of intervals.

Rohit Khandekar, Baruch Schieber, Hadas Shachnai, and Tami Tamir. Minimizing Busy Time in Multiple Machine Real-time Scheduling. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010). Leibniz International Proceedings in Informatics (LIPIcs), Volume 8, pp. 169-180, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{khandekar_et_al:LIPIcs.FSTTCS.2010.169, author = {Khandekar, Rohit and Schieber, Baruch and Shachnai, Hadas and Tamir, Tami}, title = {{Minimizing Busy Time in Multiple Machine Real-time Scheduling}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)}, pages = {169--180}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-23-1}, ISSN = {1868-8969}, year = {2010}, volume = {8}, editor = {Lodaya, Kamal and Mahajan, Meena}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2010.169}, URN = {urn:nbn:de:0030-drops-28909}, doi = {10.4230/LIPIcs.FSTTCS.2010.169}, annote = {Keywords: real-time scheduling, busy time, preemption, approximation algorithm} }

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