13 Search Results for "Garg, Vijay K."


Document
Parallel and Distributed Algorithms for the Housing Allocation Problem

Authors: Xiong Zheng and Vijay K. Garg

Published in: LIPIcs, Volume 153, 23rd International Conference on Principles of Distributed Systems (OPODIS 2019)


Abstract
We propose parallel and distributed algorithms for the housing allocation problem. In this problem, there is a set of agents and a set of houses. Each agent has a strict preference list for a subset of houses. We need to find a matching for agents to houses such that some criterion is optimized. One such criterion which has attracted much attention is Pareto optimality. A matching is Pareto optimal if no coalition of agents can be strictly better off by exchanging houses among themselves. We also study the housing market problem, a variant of the housing allocation problem, where each agent initially owns a house. In addition to Pareto optimality, we are also interested in finding the a matching in the core of a housing market. A matching is in the core if there is no coalition of agents that can be better off by breaking away from other agents and switching houses only among themselves in the initial allocation. In the first part of this work, we show that computing a Pareto optimal matching of a house allocation is in CC and computing a matching in the core of a housing market is CC-hard, where CC is the class of problems logspace reducible to the comparator circuit value problem. Given a matching of agents to houses, we show that verifying whether it is Pareto optimal is in NC. We also show that verifying whether it is in the core can be done in NC. We then give an algorithm to show that computing a maximum cardinality Pareto optimal matching for the housing allocation problem is in RNC² and quasi-NC². In the second part of this work, we present a distributed version of the top trading cycle algorithm for finding a matching in the core of a housing market. To that end, we first present two algorithms for finding all the disjoint cycles in a functional graph. The first algorithm is a Las Vegas algorithm which terminates in O(log l) rounds with high probability, where l is the length of the longest cycle. The second algorithm is a deterministic algorithm which terminates in O(log* n log l) rounds, where n is the number of nodes in the graph. Both algorithms work in the synchronous distributed model and use messages of size O(log n). By applying these two algorithms for finding cycles in a functional graph, we give the distributed top trading cycle algorithm which terminates in O(n) rounds and requires O(n²) messages.

Cite as

Xiong Zheng and Vijay K. Garg. Parallel and Distributed Algorithms for the Housing Allocation Problem. In 23rd International Conference on Principles of Distributed Systems (OPODIS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 153, pp. 23:1-23:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


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@InProceedings{zheng_et_al:LIPIcs.OPODIS.2019.23,
  author =	{Zheng, Xiong and Garg, Vijay K.},
  title =	{{Parallel and Distributed Algorithms for the Housing Allocation Problem}},
  booktitle =	{23rd International Conference on Principles of Distributed Systems (OPODIS 2019)},
  pages =	{23:1--23:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-133-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{153},
  editor =	{Felber, Pascal and Friedman, Roy and Gilbert, Seth and Miller, Avery},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2019.23},
  URN =		{urn:nbn:de:0030-drops-118090},
  doi =		{10.4230/LIPIcs.OPODIS.2019.23},
  annote =	{Keywords: Parallel Algorithm, Distributed Algorithm, Housing Allocation, Housing Markets, Pareto optimality}
}
Document
Linearizable Replicated State Machines With Lattice Agreement

Authors: Xiong Zheng, Vijay K. Garg, and John Kaippallimalil

Published in: LIPIcs, Volume 153, 23rd International Conference on Principles of Distributed Systems (OPODIS 2019)


Abstract
This paper studies the lattice agreement problem in asynchronous systems and explores its application to building a linearizable replicated state machine (RSM). First, we propose an algorithm to solve the lattice agreement problem in O(log f) asynchronous rounds, where f is the number of crash failures that the system can tolerate. This is an exponential improvement over the previous best upper bound of O(f). Second, Faleiro et al have shown in [Faleiro et al. PODC, 2012] that combination of conflict-free data types and lattice agreement protocols can be applied to implement a linearizable RSM. They give a Paxos style lattice agreement protocol, which can be adapted to implement a linearizable RSM and guarantee that a command by a client can be learned in at most O(n) message delays, where n is the number of proposers. Later, Xiong et al in [Xiong et al. DISC, 2018] gave a lattice agreement protocol which improves the O(n) message delay guarantee to O(f). However, neither of the protocols is practical for building a linearizable RSM. Thus, in the second part of the paper, we first give an improved protocol based on the one proposed by Xiong et al. Then, we implement a simple linearizable RSM using our improved protocol and compare our implementation with an open source Java implementation of Paxos. Results show that better performance can be obtained by using lattice agreement based protocols to implement a linearizable RSM compared to traditional consensus based protocols.

Cite as

Xiong Zheng, Vijay K. Garg, and John Kaippallimalil. Linearizable Replicated State Machines With Lattice Agreement. In 23rd International Conference on Principles of Distributed Systems (OPODIS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 153, pp. 29:1-29:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


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@InProceedings{zheng_et_al:LIPIcs.OPODIS.2019.29,
  author =	{Zheng, Xiong and Garg, Vijay K. and Kaippallimalil, John},
  title =	{{Linearizable Replicated State Machines With Lattice Agreement}},
  booktitle =	{23rd International Conference on Principles of Distributed Systems (OPODIS 2019)},
  pages =	{29:1--29:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-133-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{153},
  editor =	{Felber, Pascal and Friedman, Roy and Gilbert, Seth and Miller, Avery},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2019.29},
  URN =		{urn:nbn:de:0030-drops-118158},
  doi =		{10.4230/LIPIcs.OPODIS.2019.29},
  annote =	{Keywords: Lattice Agreement, Generalized Lattice Agreement, Replicated State Machine, Consensus}
}
Document
Lattice Agreement in Message Passing Systems

Authors: Xiong Zheng, Changyong Hu, and Vijay K. Garg

Published in: LIPIcs, Volume 121, 32nd International Symposium on Distributed Computing (DISC 2018)


Abstract
This paper studies the lattice agreement problem and the generalized lattice agreement problem in distributed message passing systems. In the lattice agreement problem, given input values from a lattice, processes have to non-trivially decide output values that lie on a chain. We consider the lattice agreement problem in both synchronous and asynchronous systems. For synchronous lattice agreement, we present two algorithms which run in log(f) and min{O(log^2 h(L)), O(log^2 f)} rounds, respectively, where h(L) denotes the height of the input sublattice L, f < n is the number of crash failures the system can tolerate, and n is the number of processes in the system. These algorithms have significant better round complexity than previously known algorithms. The algorithm by Attiya et al. [Attiya et al. DISC, 1995] takes log(n) synchronous rounds, and the algorithm by Mavronicolasa [Mavronicolasa, 2018] takes min{O(h(L)), O(sqrt(f))} rounds. For asynchronous lattice agreement, we propose an algorithm which has time complexity of 2*min{h(L), f + 1} message delays which improves on the previously known time complexity of O(n) message delays. The generalized lattice agreement problem defined by Faleiro et al in [Faleiro et al. PODC, 2012] is a generalization of the lattice agreement problem where it is applied for the replicated state machine. We propose an algorithm which guarantees liveness when a majority of the processes are correct in asynchronous systems. Our algorithm requires min{O(h(L)), O(f)} units of time in the worst case which is better than O(n) units of time required by the algorithm in [Faleiro et al. PODC, 2012].

Cite as

Xiong Zheng, Changyong Hu, and Vijay K. Garg. Lattice Agreement in Message Passing Systems. In 32nd International Symposium on Distributed Computing (DISC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 121, pp. 41:1-41:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)


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@InProceedings{zheng_et_al:LIPIcs.DISC.2018.41,
  author =	{Zheng, Xiong and Hu, Changyong and Garg, Vijay K.},
  title =	{{Lattice Agreement in Message Passing Systems}},
  booktitle =	{32nd International Symposium on Distributed Computing (DISC 2018)},
  pages =	{41:1--41:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-092-7},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{121},
  editor =	{Schmid, Ulrich and Widder, Josef},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2018.41},
  URN =		{urn:nbn:de:0030-drops-98301},
  doi =		{10.4230/LIPIcs.DISC.2018.41},
  annote =	{Keywords: Lattice Agreement, Replicated State Machine, Consensus}
}
Document
Fast Detection of Stable and Count Predicates in Parallel Computations

Authors: Himanshu Chauhan and Vijay K. Garg

Published in: LIPIcs, Volume 95, 21st International Conference on Principles of Distributed Systems (OPODIS 2017)


Abstract
Enumerating all consistent states of a parallel computation that satisfy a given predicate is an important problem in debugging and verification of parallel programs. We give a fast algorithm to enumerate all consistent states of a parallel computation that satisfy a stable predicate. In addi- tion, we define a new category of global predicates called count predicates and give an algorithm to enumerate all consistent states (of the computation) that satisfy it. All existing predicate detection algorithms, such as BFS, DFS and Lex algorithms, do not exploit the knowledge about the nature of the predicates, and thus may visit all global states of the computation in the worst case. In comparison, our algorithms only visit the states that satisfy the given predicate, and thus take time and space that is a polynomial function of the number of states of interest. In doing so, they provide a significant reduction — exponential in many cases — in time complexities in comparison to existing algorithms.

Cite as

Himanshu Chauhan and Vijay K. Garg. Fast Detection of Stable and Count Predicates in Parallel Computations. In 21st International Conference on Principles of Distributed Systems (OPODIS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 95, pp. 20:1-20:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)


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@InProceedings{chauhan_et_al:LIPIcs.OPODIS.2017.20,
  author =	{Chauhan, Himanshu and Garg, Vijay K.},
  title =	{{Fast Detection of Stable and Count Predicates in Parallel Computations}},
  booktitle =	{21st International Conference on Principles of Distributed Systems (OPODIS 2017)},
  pages =	{20:1--20:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-061-3},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{95},
  editor =	{Aspnes, James and Bessani, Alysson and Felber, Pascal and Leit\~{a}o, Jo\~{a}o},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2017.20},
  URN =		{urn:nbn:de:0030-drops-86496},
  doi =		{10.4230/LIPIcs.OPODIS.2017.20},
  annote =	{Keywords: Algorithms, Theory, Predicate Detection, Parallel Programs}
}
Document
Brief Announcement
Brief Announcement: Applying Predicate Detection to the Stable Marriage Problem

Authors: Vijay K. Garg

Published in: LIPIcs, Volume 91, 31st International Symposium on Distributed Computing (DISC 2017)


Abstract
We show that many techniques developed in the context of predicate detection are applicable to the stable marriage problem. The standard Gale-Shapley algorithm can be derived as a special case of detecting linear predicates. We also show that techniques in computation slicing can be used to represent the set of all constrained stable matchings.

Cite as

Vijay K. Garg. Brief Announcement: Applying Predicate Detection to the Stable Marriage Problem. In 31st International Symposium on Distributed Computing (DISC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 91, pp. 52:1-52:3, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)


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@InProceedings{garg:LIPIcs.DISC.2017.52,
  author =	{Garg, Vijay K.},
  title =	{{Brief Announcement: Applying Predicate Detection to the Stable Marriage Problem}},
  booktitle =	{31st International Symposium on Distributed Computing (DISC 2017)},
  pages =	{52:1--52:3},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-053-8},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{91},
  editor =	{Richa, Andr\'{e}a},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2017.52},
  URN =		{urn:nbn:de:0030-drops-79719},
  doi =		{10.4230/LIPIcs.DISC.2017.52},
  annote =	{Keywords: Stable Matching, Linear Predicates, Distributive Lattices}
}
Document
Predicate Detection for Parallel Computations with Locking Constraints

Authors: Yen-Jung Chang and Vijay K. Garg

Published in: LIPIcs, Volume 70, 20th International Conference on Principles of Distributed Systems (OPODIS 2016)


Abstract
The happened-before model (or the poset model) has been widely used for modeling the computations (execution traces) of parallel programs and detecting predicates (user-specified conditions). This model captures potential causality as well as locking constraints among the executed events of computations using Lamport's happened-before relation. The detection of a predicate in a computation is performed by checking if the predicate could become true in any reachable global state of the computation. In this paper, we argue that locking constraints are fundamentally different from potential causality. Hence, a poset is not an appropriate model for debugging purposes when the computations contain locking constraints. We present a model called Locking Poset, or a Loset, that generalizes the poset model for locking constraints. Just as a poset captures possibly an exponential number of total orders, a loset captures possibly an exponential number of posets. Therefore, detecting a predicate in a loset is equivalent to detecting the predicate in all corresponding posets. Since determining if a global state is reachable in a computation is a fundamental problem for detecting predicates, this paper first studies the reachability problem in the loset model. We show that the problem is NP-complete. Afterwards, we introduce a subset of reachable global states called lock-free feasible global states such that we can check whether a global state is lock-free feasible in polynomial time. Moreover, we show that lock-free feasible global states can act as "reset" points for reachability and be used to drastically reduce the time for determining the reachability of other global states. We also introduce strongly feasible global states that contain all reachable global states and show that the strong feasibility of a global state can be checked in polynomial time. We show that strong feasibility provides an effective approximation of reachability for many practical applications.

Cite as

Yen-Jung Chang and Vijay K. Garg. Predicate Detection for Parallel Computations with Locking Constraints. In 20th International Conference on Principles of Distributed Systems (OPODIS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 70, pp. 17:1-17:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)


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@InProceedings{chang_et_al:LIPIcs.OPODIS.2016.17,
  author =	{Chang, Yen-Jung and Garg, Vijay K.},
  title =	{{Predicate Detection for Parallel Computations with Locking Constraints}},
  booktitle =	{20th International Conference on Principles of Distributed Systems (OPODIS 2016)},
  pages =	{17:1--17:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-031-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{70},
  editor =	{Fatourou, Panagiota and Jim\'{e}nez, Ernesto and Pedone, Fernando},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2016.17},
  URN =		{urn:nbn:de:0030-drops-70867},
  doi =		{10.4230/LIPIcs.OPODIS.2016.17},
  annote =	{Keywords: predicate detection, parallel computations, reachable global states, locking constraints, happened-before relation}
}
Document
QuickLex: A Fast Algorithm for Consistent Global States Enumeration of Distributed Computations

Authors: Yen-Jung Chang and Vijay K. Garg

Published in: LIPIcs, Volume 46, 19th International Conference on Principles of Distributed Systems (OPODIS 2015)


Abstract
Verifying the correctness of executions of concurrent and distributed programs is difficult because they show nondeterministic behavior due to different process scheduling order. Predicate detection can alleviate this problem by predicting whether the user-specified condition (predicate) could have become true in any global state of the given concurrent or distributed computation. The method is predictive because it generates inferred global states from the observed execution path and then checks if those global states satisfy the predicate. An important part of the predicate detection method is global states enumeration, which generates the consistent global states, including the inferred ones, of the given computation. Cooper and Marzullo gave the first enumeration algorithm based on a breadth first strategy (BFS). Later, many algorithms have been proposed to improve the space and time complexity. Among the existing algorithms, the Tree algorithm due to Jegou et al. has the smallest time complexity and requires O(|P|) space, which is linear to the size of the computation P. In this paper, we present a fast algorithm, QuickLex, to enumerate global states in the lexical order. QuickLex requires much smaller space than O(|P|). From our experiments, the Tree algorithm requires 2-10 times more memory space than QuickLex. Moreover, QuickLex is 4 times faster than Tree even though the asymptotic time complexity of QuickLex is higher than that of Tree. The reason is that the worst case time complexity of QuickLex happens only in computations that are not common in practice. Moreover, Tree is built on linked-lists and QuickLex can be implemented using integer arrays. In comparison with the existing lexical algorithm (Lex), QuickLex is 7 times faster and uses almost the same amount of memory as Lex. Finally, we implement a parallel-and-online predicate detector for concurrent programs using QuickLex, which can detect data races and violation of invariants in the programs.

Cite as

Yen-Jung Chang and Vijay K. Garg. QuickLex: A Fast Algorithm for Consistent Global States Enumeration of Distributed Computations. In 19th International Conference on Principles of Distributed Systems (OPODIS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 46, pp. 25:1-25:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)


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@InProceedings{chang_et_al:LIPIcs.OPODIS.2015.25,
  author =	{Chang, Yen-Jung and Garg, Vijay K.},
  title =	{{QuickLex: A Fast Algorithm for Consistent Global States Enumeration of Distributed Computations}},
  booktitle =	{19th International Conference on Principles of Distributed Systems (OPODIS 2015)},
  pages =	{25:1--25:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-98-9},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{46},
  editor =	{Anceaume, Emmanuelle and Cachin, Christian and Potop-Butucaru, Maria},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2015.25},
  URN =		{urn:nbn:de:0030-drops-66142},
  doi =		{10.4230/LIPIcs.OPODIS.2015.25},
  annote =	{Keywords: consistent global state, algorithm, computation}
}
Document
ActiveMonitor: Asynchronous Monitor Framework for Scalability and Multi-Object Synchronization

Authors: Wei-Lun Hung, Himanshu Chauhan, and Vijay K. Garg

Published in: LIPIcs, Volume 46, 19th International Conference on Principles of Distributed Systems (OPODIS 2015)


Abstract
Monitor objects are used extensively for thread-safety and synchronization in shared memory parallel programs. They provide ease of use, and enable straightforward correctness analysis. However, they inhibit parallelism by enforcing serial executions of critical sections, and thus the performance of parallel programs with monitors scales poorly with number of processes. Their current design and implementation is also ill-suited for thread synchronization across multiple thread-safe objects. We present ActiveMonitor - a framework that allows multi-object synchronization without global locks, and improves parallelism by exploiting asynchronous execution of critical sections. We evaluate the performance of Java based implementation of ActiveMonitor on micro-benchmarks involving light and heavy critical sections, as well as on single-source-shortest-path problem in directed graphs. Our results show that on most of these problems, ActiveMonitor based programs outperform programs implemented using Java's reentrant-lock and condition constructs.

Cite as

Wei-Lun Hung, Himanshu Chauhan, and Vijay K. Garg. ActiveMonitor: Asynchronous Monitor Framework for Scalability and Multi-Object Synchronization. In 19th International Conference on Principles of Distributed Systems (OPODIS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 46, pp. 29:1-29:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)


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@InProceedings{hung_et_al:LIPIcs.OPODIS.2015.29,
  author =	{Hung, Wei-Lun and Chauhan, Himanshu and Garg, Vijay K.},
  title =	{{ActiveMonitor: Asynchronous Monitor Framework for Scalability and Multi-Object Synchronization}},
  booktitle =	{19th International Conference on Principles of Distributed Systems (OPODIS 2015)},
  pages =	{29:1--29:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-98-9},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{46},
  editor =	{Anceaume, Emmanuelle and Cachin, Christian and Potop-Butucaru, Maria},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2015.29},
  URN =		{urn:nbn:de:0030-drops-66188},
  doi =		{10.4230/LIPIcs.OPODIS.2015.29},
  annote =	{Keywords: concurrent/parallel programming, monitors, concurrency}
}
Document
Approximate Hypergraph Coloring under Low-discrepancy and Related Promises

Authors: Vijay V. S. P. Bhattiprolu, Venkatesan Guruswami, and Euiwoong Lee

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


Abstract
A hypergraph is said to be X-colorable if its vertices can be colored with X colors so that no hyperedge is monochromatic. 2-colorability is a fundamental property (called Property B) of hypergraphs and is extensively studied in combinatorics. Algorithmically, however, given a 2-colorable k-uniform hypergraph, it is NP-hard to find a 2-coloring miscoloring fewer than a fraction 2^(-k+1) of hyperedges (which is trivially achieved by a random 2-coloring), and the best algorithms to color the hypergraph properly require about n^(1-1/k) colors, approaching the trivial bound of n as k increases. In this work, we study the complexity of approximate hypergraph coloring, for both the maximization (finding a 2-coloring with fewest miscolored edges) and minimization (finding a proper coloring using fewest number of colors) versions, when the input hypergraph is promised to have the following stronger properties than 2-colorability: (A) Low-discrepancy: If the hypergraph has a 2-coloring of discrepancy l << sqrt(k), we give an algorithm to color the hypergraph with about n^(O(l^2/k)) colors. However, for the maximization version, we prove NP-hardness of finding a 2-coloring miscoloring a smaller than 2^(-O(k)) (resp. k^(-O(k))) fraction of the hyperedges when l = O(log k) (resp. l=2). Assuming the Unique Games conjecture, we improve the latter hardness factor to 2^(-O(k)) for almost discrepancy-1 hypergraphs. (B) Rainbow colorability: If the hypergraph has a (k-l)-coloring such that each hyperedge is polychromatic with all these colors (this is stronger than a (l+1)-discrepancy 2-coloring), we give a 2-coloring algorithm that miscolors at most k^(-Omega(k)) of the hyperedges when l << sqrt(k), and complement this with a matching Unique Games hardness result showing that when l = sqrt(k), it is hard to even beat the 2^(-k+1) bound achieved by a random coloring. (C) Strong Colorability: We obtain similar (stronger) Min- and Max-2-Coloring algorithmic results in the case of (k+l)-strong colorability.

Cite as

Vijay V. S. P. Bhattiprolu, Venkatesan Guruswami, and Euiwoong Lee. Approximate Hypergraph Coloring under Low-discrepancy and Related Promises. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 152-174, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{bhattiprolu_et_al:LIPIcs.APPROX-RANDOM.2015.152,
  author =	{Bhattiprolu, Vijay V. S. P. and Guruswami, Venkatesan and Lee, Euiwoong},
  title =	{{Approximate Hypergraph Coloring under Low-discrepancy and Related Promises}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
  pages =	{152--174},
  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.2015.152},
  URN =		{urn:nbn:de:0030-drops-53011},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2015.152},
  annote =	{Keywords: Hypergraph Coloring, Discrepancy, Rainbow Coloring, Stong Coloring, Algorithms, Semidefinite Programming, Hardness of Approximation}
}
Document
Inapproximability of H-Transversal/Packing

Authors: Venkatesan Guruswami and Euiwoong Lee

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


Abstract
Given an undirected graph G=(V,E) and a fixed pattern graph H with k vertices, we consider the H-Transversal and H-Packing problems. The former asks to find the smallest subset S of vertices such that the subgraph induced by V - S does not have H as a subgraph, and the latter asks to find the maximum number of pairwise disjoint k-subsets S1, ..., Sm such that the subgraph induced by each Si has H as a subgraph. We prove that if H is 2-connected, H-Transversal and H-Packing are almost as hard to approximate as general k-Hypergraph Vertex Cover and k-Set Packing, so it is NP-hard to approximate them within a factor of Omega(k) and Omega(k / polylog(k)) respectively. We also show that there is a 1-connected H where H-Transversal admits an O(log k)-approximation algorithm, so that the connectivity requirement cannot be relaxed from 2 to 1. For a special case of H-Transversal where H is a (family of) cycles, we mention the implication of our result to the related Feedback Vertex Set problem, and give a different hardness proof for directed graphs.

Cite as

Venkatesan Guruswami and Euiwoong Lee. Inapproximability of H-Transversal/Packing. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 284-304, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{guruswami_et_al:LIPIcs.APPROX-RANDOM.2015.284,
  author =	{Guruswami, Venkatesan and Lee, Euiwoong},
  title =	{{Inapproximability of H-Transversal/Packing}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
  pages =	{284--304},
  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.2015.284},
  URN =		{urn:nbn:de:0030-drops-53085},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2015.284},
  annote =	{Keywords: Constraint Satisfaction Problems, Approximation resistance}
}
Document
The Hardness of Approximation of Euclidean k-Means

Authors: Pranjal Awasthi, Moses Charikar, Ravishankar Krishnaswamy, and Ali Kemal Sinop

Published in: LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)


Abstract
The Euclidean k-means problem is a classical problem that has been extensively studied in the theoretical computer science, machine learning and the computational geometry communities. In this problem, we are given a set of n points in Euclidean space R^d, and the goal is to choose k center points in R^d so that the sum of squared distances of each point to its nearest center is minimized. The best approximation algorithms for this problem include a polynomial time constant factor approximation for general k and a (1+c)-approximation which runs in time poly(n) exp(k/c). At the other extreme, the only known computational complexity result for this problem is NP-hardness [Aloise et al.'09]. The main difficulty in obtaining hardness results stems from the Euclidean nature of the problem, and the fact that any point in R^d can be a potential center. This gap in understanding left open the intriguing possibility that the problem might admit a PTAS for all k, d. In this paper we provide the first hardness of approximation for the Euclidean k-means problem. Concretely, we show that there exists a constant c > 0 such that it is NP-hard to approximate the k-means objective to within a factor of (1+c). We show this via an efficient reduction from the vertex cover problem on triangle-free graphs: given a triangle-free graph, the goal is to choose the fewest number of vertices which are incident on all the edges. Additionally, we give a proof that the current best hardness results for vertex cover can be carried over to triangle-free graphs. To show this we transform G, a known hard vertex cover instance, by taking a graph product with a suitably chosen graph H, and showing that the size of the (normalized) maximum independent set is almost exactly preserved in the product graph using a spectral analysis, which might be of independent interest.

Cite as

Pranjal Awasthi, Moses Charikar, Ravishankar Krishnaswamy, and Ali Kemal Sinop. The Hardness of Approximation of Euclidean k-Means. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 754-767, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{awasthi_et_al:LIPIcs.SOCG.2015.754,
  author =	{Awasthi, Pranjal and Charikar, Moses and Krishnaswamy, Ravishankar and Sinop, Ali Kemal},
  title =	{{The Hardness of Approximation of Euclidean k-Means}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{754--767},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Arge, Lars and Pach, J\'{a}nos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.754},
  URN =		{urn:nbn:de:0030-drops-51178},
  doi =		{10.4230/LIPIcs.SOCG.2015.754},
  annote =	{Keywords: Euclidean k-means, Hardness of Approximation, Vertex Cover}
}
Document
Parameterized Complexity Dichotomy for Steiner Multicut

Authors: Karl Bringmann, Danny Hermelin, Matthias Mnich, and Erik Jan van Leeuwen

Published in: LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)


Abstract
We consider the Steiner Multicut problem, which asks, given an undirected graph G, a collection T = \{T_{1},...,T_{t}}, T_i \subseteq V(G), of terminal sets of size at most p, and an integer k, whether there is a set S of at most k edges or nodes such that of each set T_{i} at least one pair of terminals is in different connected components of G \ S. This problem generalizes several well-studied graph cut problems, in particular the Multicut problem, which corresponds to the case p = 2. The Multicut problem was recently shown to be fixed-parameter tractable for parameter k [Marx and Razgon, Bousquet et al., STOC 2011]. The question whether this result generalizes to Steiner Multicut motivates the present work. We answer the question that motivated this work, and in fact provide a dichotomy of the parameterized complexity of Steiner Multicut on general graphs. That is, for any combination of k, t, p, and the treewidth tw(G) as constant, parameter, or unbounded, and for all versions of the problem (edge deletion and node deletion with and without deletable terminals), we prove either that the problem is fixed-parameter tractable or that the problem is hard (W[1]-hard or even (para-)NP-complete). Among the many results in the paper, we highlight that: - The edge deletion version of Steiner Multicut is fixed-parameter tractable for parameter k+t on general graphs (but has no polynomial kernel, even on trees). - In contrast, both node deletion versions of Steiner Multicut are W[1]-hard for the parameter k+t on general graphs. - All versions of Steiner Multicut are W[1]-hard for the parameter k, even when p=3 and the graph is a tree plus one node. Since we allow k, t, p, and tw(G) to be any constants, our characterization includes a dichotomy for Steiner Multicut on trees (for tw(G) = 1) as well as a polynomial time versus NP-hardness dichotomy (by restricting k,t,p,tw(G) to constant or unbounded).

Cite as

Karl Bringmann, Danny Hermelin, Matthias Mnich, and Erik Jan van Leeuwen. Parameterized Complexity Dichotomy for Steiner Multicut. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 157-170, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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@InProceedings{bringmann_et_al:LIPIcs.STACS.2015.157,
  author =	{Bringmann, Karl and Hermelin, Danny and Mnich, Matthias and van Leeuwen, Erik Jan},
  title =	{{Parameterized Complexity Dichotomy for Steiner Multicut}},
  booktitle =	{32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)},
  pages =	{157--170},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-78-1},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{30},
  editor =	{Mayr, Ernst W. and Ollinger, Nicolas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.157},
  URN =		{urn:nbn:de:0030-drops-49115},
  doi =		{10.4230/LIPIcs.STACS.2015.157},
  annote =	{Keywords: graph cut problems, Steiner cut, fixed-parameter tractability}
}
Document
An Improved Approximation Algorithm for the Hard Uniform Capacitated k-median Problem

Authors: Shanfei Li

Published in: LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)


Abstract
In the k-median problem, given a set of locations, the goal is to select a subset of at most k centers so as to minimize the total cost of connecting each location to its nearest center. We study the uniform hard capacitated version of the k-median problem, in which each selected center can only serve a limited number of locations. Inspired by the algorithm of Charikar, Guha, Tardos and Shmoys, we give an improved approximation algorithm for this problem with increasing the capacities by a constant factor, which improves the previous best approximation algorithm proposed by Byrka, Fleszar, Rybicki and Spoerhase.

Cite as

Shanfei Li. An Improved Approximation Algorithm for the Hard Uniform Capacitated k-median Problem. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 325-338, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)


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@InProceedings{li:LIPIcs.APPROX-RANDOM.2014.325,
  author =	{Li, Shanfei},
  title =	{{An Improved Approximation Algorithm for the Hard Uniform Capacitated k-median Problem}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)},
  pages =	{325--338},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-74-3},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{28},
  editor =	{Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.325},
  URN =		{urn:nbn:de:0030-drops-47062},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2014.325},
  annote =	{Keywords: Approximation algorithm; k-median problem; LP-rounding; Hard capacities}
}
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