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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

We study the following basic problem called Bi-Covering. Given a graph G(V, E), find two (not necessarily disjoint) sets A subseteq V and B subseteq V such that A union B = V and that every edge e belongs to either the graph induced by A or to the graph induced by B. The goal is to minimize max{|A|, |B|}. This is the most simple case of the Channel Allocation problem [Gandhi et al., Networks, 2006]. A solution that outputs V,emptyset gives ratio at most 2. We show that under the similar Strong Unique Game Conjecture by [Bansal-Khot, FOCS, 2009] there is no 2 - epsilon ratio algorithm for the problem, for any constant epsilon > 0.
Given a bipartite graph, Max-bi-clique is a problem of finding largest k*k complete bipartite sub graph. For Max-bi-clique problem, a constant factor hardness was known under random 3-SAT hypothesis of Feige [Feige, STOC, 2002] and also under the assumption that NP !subseteq intersection_{epsilon > 0} BPTIME(2^{n^{epsilon}}) [Khot, SIAM J. on Comp., 2011]. It was an open problem in [Ambühl et. al., SIAM J. on Comp., 2011] to prove inapproximability of Max-bi-clique assuming weaker conjecture. Our result implies similar hardness result assuming the Strong Unique Games Conjecture.
On the algorithmic side, we also give better than 2 approximation for Bi-Covering on numerous special graph classes. In particular, we get 1.876 approximation for Chordal graphs, exact algorithm for Interval Graphs, 1 + o(1) for Minor Free Graph, 2 - 4*delta/3 for graphs with minimum degree delta*n, 2/(1+delta^2/8) for delta-vertex expander, 8/5 for Split Graphs, 2 - (6/5)*1/d for graphs with minimum constant degree d etc. Our algorithmic results are quite non-trivial. In achieving these results, we use various known structural results about the graphs, combined with the techniques that we develop tailored to getting better than 2 approximation.

Amey Bhangale, Rajiv Gandhi, Mohammad Taghi Hajiaghayi, Rohit Khandekar, and Guy Kortsarz. Bicovering: Covering Edges With Two Small Subsets of Vertices. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 6:1-6:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bhangale_et_al:LIPIcs.ICALP.2016.6, author = {Bhangale, Amey and Gandhi, Rajiv and Hajiaghayi, Mohammad Taghi and Khandekar, Rohit and Kortsarz, Guy}, title = {{Bicovering: Covering Edges With Two Small Subsets of Vertices}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {6:1--6:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.6}, URN = {urn:nbn:de:0030-drops-62728}, doi = {10.4230/LIPIcs.ICALP.2016.6}, annote = {Keywords: Bi-covering, Unique Games, Max Bi-clique} }

Document

**Published in:** LIPIcs, Volume 18, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)

We consider single processor preemptive scheduling with job-dependent setup times. In this model, a job-dependent setup time is incurred when a job is started for the first time, and each time it is restarted after preemption. This model is a common generalization of preemptive scheduling, and actually of non-preemptive scheduling as well. The objective is to minimize the sum of any general non-negative, non-decreasing cost functions of the completion times of the jobs -- this generalizes objectives of minimizing weighted flow time, flow-time squared, tardiness or the number of tardy jobs among many others. Our main result is a randomized polynomial time O(1)-speed O(1)-approximation algorithm for this problem. Without speedup, no polynomial time finite multiplicative approximation is possible unless P=NP.
We extend the approach of Bansal et al. (FOCS 2007) of rounding a linear programming relaxation which accounts for costs incurred due to the non-preemptive nature of the schedule. A key new idea used in the rounding is that a point in the intersection polytope of two matroids can be decomposed as a convex combination of incidence vectors of sets that are independent in both matroids. In fact, we use this for the intersection of a partition matroid and a laminar matroid, in which case the decomposition can be found efficiently using network flows.
Our approach gives a randomized polynomial time offline O(1)-speed O(1)-approximation algorithm for the broadcast scheduling problem with general cost functions as well.

Rohit Khandekar, Kirsten Hildrum, Deepak Rajan, and Joel Wolf. Scheduling with Setup Costs and Monotone Penalties. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 18, pp. 185-198, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{khandekar_et_al:LIPIcs.FSTTCS.2012.185, author = {Khandekar, Rohit and Hildrum, Kirsten and Rajan, Deepak and Wolf, Joel}, title = {{Scheduling with Setup Costs and Monotone Penalties}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)}, pages = {185--198}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-47-7}, ISSN = {1868-8969}, year = {2012}, volume = {18}, editor = {D'Souza, Deepak and Radhakrishnan, Jaikumar and Telikepalli, Kavitha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2012.185}, URN = {urn:nbn:de:0030-drops-38576}, doi = {10.4230/LIPIcs.FSTTCS.2012.185}, annote = {Keywords: Scheduling, resource augmentation, approximation algorithm, preemption, setup times} }

Document

**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} }

Document

**Published in:** LIPIcs, Volume 4, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (2009)

In this paper, we initiate the study of designing approximation algorithms for
{\sf Fault-Tolerant Group-Steiner} ({\sf FTGS}) problems. The motivation is to protect
the well-studied group-Steiner networks from edge or vertex failures.
In {\sf Fault-Tolerant Group-Steiner} problems, we are given a graph with edge- (or vertex-) costs,
a root vertex, and a collection of subsets of vertices called groups. The objective is to find a
minimum-cost subgraph that has two edge- (or vertex-) disjoint paths from each group to the root.
We present approximation algorithms and hardness results for several variants of this basic problem, e.g.,
edge-costs vs. vertex-costs, edge-connectivity vs. vertex-connectivity,
and $2$-connecting from each group a single vertex vs. many vertices.
Main contributions of our paper include the introduction
of very general structural lemmas on connectivity and a charging scheme that may find more applications in the future.
Our algorithmic results are supplemented by inapproximability results, which are tight in some cases.
Our algorithms employ a variety of techniques.
For the edge-connectivity variant, we use a primal-dual based
algorithm for covering an {\em uncros\-sable} set-family, while for the vertex-connectivity version,
we prove a new graph-theoretic lemma that shows equivalence between obtaining two vertex-disjoint paths
from two vertices and $2$-connecting a carefully chosen single vertex. To handle large group-sizes,
we use a $p$-Steiner tree algorithm to identify the ``correct'' pair of terminals from each group to be
connected to the root. We also use a non-trivial charging scheme
to improve the approximation ratio for the most general problem we consider.

Rohit Khandekar, Guy Kortsarz, and Zeev Nutov. Approximating Fault-Tolerant Group-Steiner Problems. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 4, pp. 263-274, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{khandekar_et_al:LIPIcs.FSTTCS.2009.2324, author = {Khandekar, Rohit and Kortsarz, Guy and Nutov, Zeev}, title = {{Approximating Fault-Tolerant Group-Steiner Problems}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science}, pages = {263--274}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-13-2}, ISSN = {1868-8969}, year = {2009}, volume = {4}, editor = {Kannan, Ravi and Narayan Kumar, K.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2009.2324}, URN = {urn:nbn:de:0030-drops-23243}, doi = {10.4230/LIPIcs.FSTTCS.2009.2324}, annote = {Keywords: Fault-tolerance, group Steiner problem, edge-disjointness, vertex-disjointness, approximation, connectivity} }

Document

**Published in:** LIPIcs, Volume 4, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (2009)

We introduce a graph clustering problem motivated by a stream processing application. Input to our problem is an undirected graph with vertex and edge weights. A cluster is a subset of the vertices. The {\em size} of a cluster is
defined as the total vertex weight in the subset plus the total edge weight at the boundary of the cluster. The bounded size graph clustering problem ($\GC$) is to partition the vertices into clusters of size at most a given budget and minimize the total edge-weight across the clusters. In the {\em multiway cut} version of the problem, we are also given a subset of vertices called {\em terminals}. No cluster is allowed to contain more than one terminal. Our problem differs from most of the previously studied clustering problems in that the number of clusters is not specified. We first show that the feasibility version of the multiway cut $\GC$ problem,
i.e., determining if there exists a clustering with bounded-size clusters satisfying the multiway cut constraint, can be solved in polynomial time. Our algorithm is based on the min-cut subroutine and an uncrossing argument. This result is in contrast with the NP-hardness of the min-max multiway cut problem, considered by Svitkina and Tardos (2004), in which the number of clusters must equal the number of terminals. Our results for the feasibility version also generalize to any symmetric submodular function. We next show that the optimization version of $\GC$ is NP-hard by showing an
approximation-preserving reduction from the $\frac 13$-balanced cut problem.
Our main result is an $O(\log^2 n)$-approximation to the optimization version
of the multiway cut $\GC$ problem violating the budget by an $O(\log n)$
factor, where $n$ denotes the number of vertices. Our algorithm is based on a
set-cover-like greedy approach which iteratively computes bounded-size clusters
to maximize the number of new vertices covered.

Rohit Khandekar, Kirsten Hildrum, Sujay Parekh, Deepak Rajan, Jay Sethuraman, and Joel Wolf. Bounded Size Graph Clustering with Applications to Stream Processing. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 4, pp. 275-286, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{khandekar_et_al:LIPIcs.FSTTCS.2009.2325, author = {Khandekar, Rohit and Hildrum, Kirsten and Parekh, Sujay and Rajan, Deepak and Sethuraman, Jay and Wolf, Joel}, title = {{Bounded Size Graph Clustering with Applications to Stream Processing}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science}, pages = {275--286}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-13-2}, ISSN = {1868-8969}, year = {2009}, volume = {4}, editor = {Kannan, Ravi and Narayan Kumar, K.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2009.2325}, URN = {urn:nbn:de:0030-drops-23250}, doi = {10.4230/LIPIcs.FSTTCS.2009.2325}, annote = {Keywords: Graph partitioning, uncrossing, Gomory-Hu trees, symmetric submodular functions} }

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