13 Search Results for "Xu, Chao"

Document
DOI: 10.4230/LIPIcs.ISAAC.2024.12
Partitioning Problems with Splittings and Interval Targets

Authors: Samuel Bismuth, Vladislav Makarov, Erel Segal-Halevi, and Dana Shapira

Published in: LIPIcs, Volume 322, 35th International Symposium on Algorithms and Computation (ISAAC 2024)

Abstract
The n-way number partitioning problem is a classic problem in combinatorial optimization, with applications to diverse settings such as fair allocation and machine scheduling. All these problems are NP-hard, but various approximation algorithms are known. We consider three closely related kinds of approximations. The first two variants optimize the partition such that: in the first variant some fixed number s of items can be split between two or more bins and in the second variant we allow at most a fixed number t of splittings. The third variant is a decision problem: the largest bin sum must be within a pre-specified interval, parameterized by a fixed rational number u times the largest item size. When the number of bins n is unbounded, we show that every variant is strongly NP-complete. When the number of bins n is fixed, the running time depends on the fixed parameters s,t,u. For each variant, we give a complete picture of its running time. For n = 2, the running time is easy to identify. Our main results consider any fixed integer n ≥ 3. Using a two-way polynomial-time reduction between the first and the third variant, we show that n-way number-partitioning with s split items can be solved in polynomial time if s ≥ n-2, and it is NP-complete otherwise. Also, n-way number-partitioning with t splittings can be solved in polynomial time if t ≥ n-1, and it is NP-complete otherwise. Finally, we show that the third variant can be solved in polynomial time if u ≥ (n-2)/n, and it is NP-complete otherwise. Our positive results for the optimization problems consider both min-max and max-min versions. Using the same reduction, we provide a fully polynomial-time approximation scheme for the case where the number of split items is lower than n-2.
Cite as

Samuel Bismuth, Vladislav Makarov, Erel Segal-Halevi, and Dana Shapira. Partitioning Problems with Splittings and Interval Targets. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 12:1-12:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bismuth_et_al:LIPIcs.ISAAC.2024.12,
  author =	{Bismuth, Samuel and Makarov, Vladislav and Segal-Halevi, Erel and Shapira, Dana},
  title =	{{Partitioning Problems with Splittings and Interval Targets}},
  booktitle =	{35th International Symposium on Algorithms and Computation (ISAAC 2024)},
  pages =	{12:1--12:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-354-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{322},
  editor =	{Mestre, Juli\'{a}n and Wirth, Anthony},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2024.12},
  URN =		{urn:nbn:de:0030-drops-221394},
  doi =		{10.4230/LIPIcs.ISAAC.2024.12},
  annote =	{Keywords: Number Partitioning, Fair Division, Identical Machine Scheduling}
}
Document
DOI: 10.4230/OASIcs.ATMOS.2024.2
Indexing Graphs for Shortest Beer Path Queries

Authors: David Coudert, Andrea D'Ascenzo, and Mattia D'Emidio

Published in: OASIcs, Volume 123, 24th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2024)

Abstract
A beer graph is an edge-weighted graph G = (V,E,ω) with beer vertices B ⊆ V. A beer path between two vertices s and t of a beer graph is a path that connects s and t and visits at least one vertex in B. The beer distance between two vertices is the weight of a shortest beer path, i.e. a beer path having minimum total weight. A graph indexing scheme is a two-phase method that constructs an index data structure by a one-time preprocessing of an input graph and then exploits it to compute (or accelerate the computation of) answers to queries on structures of the graph dataset. In the last decade, such indexing schemes have been designed to perform, effectively, many relevant types of queries, e.g. on reachability, and have gained significant popularity in essentially all data-intensive application domains where large number of queries have to be routinely answered (e.g. journey planners), since they have been shown, through many experimental studies, to offer extremely low query times at the price of limited preprocessing time and space overheads. In this paper, we showcase that an indexing scheme, to efficiently execute queries on beer distances or shortest beer paths for pairs of vertices of a beer graph, can be obtained by adapting the highway labeling, a recently introduced indexing method to accelerate the computation of classical shortest paths. We design a preprocessing algorithm to build a whl index, i.e. a weighted highway labeling of a beer graph, and show how it can be queried to compute beer distances and shortest beer paths. Through extensive experimentation on real networks, we empirically demonstrate its practical effectiveness and superiority, in terms of offered trade-off between preprocessing time, space overhead and query time, with respect to the state-of-the-art.
Cite as

David Coudert, Andrea D'Ascenzo, and Mattia D'Emidio. Indexing Graphs for Shortest Beer Path Queries. In 24th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2024). Open Access Series in Informatics (OASIcs), Volume 123, pp. 2:1-2:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{coudert_et_al:OASIcs.ATMOS.2024.2,
  author =	{Coudert, David and D'Ascenzo, Andrea and D'Emidio, Mattia},
  title =	{{Indexing Graphs for Shortest Beer Path Queries}},
  booktitle =	{24th Symposium on Algorithmic Approaches for Transportation Modelling, Optimization, and Systems (ATMOS 2024)},
  pages =	{2:1--2:18},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-350-8},
  ISSN =	{2190-6807},
  year =	{2024},
  volume =	{123},
  editor =	{Bouman, Paul C. and Kontogiannis, Spyros C.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.ATMOS.2024.2},
  URN =		{urn:nbn:de:0030-drops-211907},
  doi =		{10.4230/OASIcs.ATMOS.2024.2},
  annote =	{Keywords: Graph Algorithms, Indexing Schemes, Beer Distances, Algorithms Engineering}
}
Document
DOI: 10.4230/LIPIcs.AFT.2024.7
DeFiAligner: Leveraging Symbolic Analysis and Large Language Models for Inconsistency Detection in Decentralized Finance

Authors: Rundong Gan, Liyi Zhou, Le Wang, Kaihua Qin, and Xiaodong Lin

Published in: LIPIcs, Volume 316, 6th Conference on Advances in Financial Technologies (AFT 2024)

Abstract
Decentralized Finance (DeFi) has witnessed a monumental surge, reaching 53.039 billion USD in total value locked. As this sector continues to expand, ensuring the reliability of DeFi smart contracts becomes increasingly crucial. While some users are adept at reading code or the compiled bytecode to understand smart contracts, many rely on documentation. Therefore, discrepancies between the documentation and the deployed code can pose significant risks, whether these discrepancies are due to errors or intentional fraud. To tackle these challenges, we developed DeFiAligner, an end-to-end system to identify inconsistencies between documentation and smart contracts. DeFiAligner incorporates a symbolic execution tool, SEVM, which explores execution paths of on-chain binary code, recording memory and stack states. It automatically generates symbolic expressions for token balance changes and branch conditions, which, along with related project documents, are processed by LLMs. Using structured prompts, the LLMs evaluate the alignment between the symbolic expressions and the documentation. Our tests across three distinct scenarios demonstrate DeFiAligner’s capability to automate inconsistency detection in DeFi, achieving recall rates of 92% and 90% on two public datasets respectively.
Cite as

Rundong Gan, Liyi Zhou, Le Wang, Kaihua Qin, and Xiaodong Lin. DeFiAligner: Leveraging Symbolic Analysis and Large Language Models for Inconsistency Detection in Decentralized Finance. In 6th Conference on Advances in Financial Technologies (AFT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 316, pp. 7:1-7:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{gan_et_al:LIPIcs.AFT.2024.7,
  author =	{Gan, Rundong and Zhou, Liyi and Wang, Le and Qin, Kaihua and Lin, Xiaodong},
  title =	{{DeFiAligner: Leveraging Symbolic Analysis and Large Language Models for Inconsistency Detection in Decentralized Finance}},
  booktitle =	{6th Conference on Advances in Financial Technologies (AFT 2024)},
  pages =	{7:1--7:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-345-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{316},
  editor =	{B\"{o}hme, Rainer and Kiffer, Lucianna},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AFT.2024.7},
  URN =		{urn:nbn:de:0030-drops-209431},
  doi =		{10.4230/LIPIcs.AFT.2024.7},
  annote =	{Keywords: Decentralized Finance Security, Large Language Models, Project Review, Symbolic Analysis, Smart Contracts}
}
Document
DOI: 10.4230/LIPIcs.ECOOP.2024.7
HOBBIT: Hashed OBject Based InTegrity

Authors: Matthias Bernad and Stefan Brunthaler

Published in: LIPIcs, Volume 313, 38th European Conference on Object-Oriented Programming (ECOOP 2024)

Abstract
C vulnerabilities usually hold verbatim for C++ programs. The counterfeit-object-oriented programming attack demonstrated that this relation is asymmetric, i.e., it only applies to C++. The problem pinpointed by this COOP attack is that C++ does not validate the integrity of its objects. By injecting malicious objects with manipulated virtual function table pointers, attackers can hijack control-flow of programs. The software security community addressed the COOP-problem in the years following its discovery, but together with the emergence of transient-execution attacks, such as Spectre, researchers also shifted their attention. We present Hobbit, a software-only solution to prevent COOP attacks by validating object integrity for virtual function pointer tables. Hobbit does not require any hardware specific features, scales to multi-million lines of C++ source code, and our LLVM-based implementation offers a configurable performance impact between 121.63% and 2.80% on compute-intensive SPEC CPU C++ benchmarks. Hobbit’s security analysis indicates strong resistance to brute forcing attacks and demonstrates additional benefits of using execute-only memory.
Cite as

Matthias Bernad and Stefan Brunthaler. HOBBIT: Hashed OBject Based InTegrity. In 38th European Conference on Object-Oriented Programming (ECOOP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 313, pp. 7:1-7:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bernad_et_al:LIPIcs.ECOOP.2024.7,
  author =	{Bernad, Matthias and Brunthaler, Stefan},
  title =	{{HOBBIT: Hashed OBject Based InTegrity}},
  booktitle =	{38th European Conference on Object-Oriented Programming (ECOOP 2024)},
  pages =	{7:1--7:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-341-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{313},
  editor =	{Aldrich, Jonathan and Salvaneschi, Guido},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ECOOP.2024.7},
  URN =		{urn:nbn:de:0030-drops-208566},
  doi =		{10.4230/LIPIcs.ECOOP.2024.7},
  annote =	{Keywords: software security, code-reuse attacks, language-based security, counterfeit-object-oriented programming, object integrity, compiler security}
}
Document
DOI: 10.4230/LIPIcs.SAT.2024.26
Optimal Layout Synthesis for Deep Quantum Circuits on NISQ Processors with 100+ Qubits

Authors: Irfansha Shaik and Jaco van de Pol

Published in: LIPIcs, Volume 305, 27th International Conference on Theory and Applications of Satisfiability Testing (SAT 2024)

Abstract
Layout synthesis is mapping a quantum circuit to a quantum processor. SWAP gate insertions are needed for scheduling 2-qubit gates only on connected physical qubits. With the ever-increasing number of qubits in NISQ processors, scalable layout synthesis is of utmost importance. With large optimality gaps observed in heuristic approaches, scalable exact methods are needed. While recent exact and near-optimal approaches scale to moderate circuits, large deep circuits are still out of scope. In this work, we propose a SAT encoding based on parallel plans that apply 1 SWAP and a group of CNOTs at each time step. Using domain-specific information, we maintain optimality in parallel plans while scaling to large and deep circuits. From our results, we show the scalability of our approach which significantly outperforms leading exact and near-optimal approaches (up to 100x). For the first time, we can optimally map several 8, 14, and 16 qubit circuits onto 54, 80, and 127 qubit platforms with up to 17 SWAPs. While adding optimal SWAPs, we also report near-optimal depth in our mapped circuits.
Cite as

Irfansha Shaik and Jaco van de Pol. Optimal Layout Synthesis for Deep Quantum Circuits on NISQ Processors with 100+ Qubits. In 27th International Conference on Theory and Applications of Satisfiability Testing (SAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 305, pp. 26:1-26:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{shaik_et_al:LIPIcs.SAT.2024.26,
  author =	{Shaik, Irfansha and van de Pol, Jaco},
  title =	{{Optimal Layout Synthesis for Deep Quantum Circuits on NISQ Processors with 100+ Qubits}},
  booktitle =	{27th International Conference on Theory and Applications of Satisfiability Testing (SAT 2024)},
  pages =	{26:1--26:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-334-8},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{305},
  editor =	{Chakraborty, Supratik and Jiang, Jie-Hong Roland},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAT.2024.26},
  URN =		{urn:nbn:de:0030-drops-205487},
  doi =		{10.4230/LIPIcs.SAT.2024.26},
  annote =	{Keywords: Layout Synthesis, Transpiling, Qubit Mapping and Routing, Quantum Circuits, Propositional Satisfiability, Parallel Plans}
}
Document
PACE Solver Description
DOI: 10.4230/LIPIcs.IPEC.2022.29
PACE Solver Description: Hust-Solver - A Heuristic Algorithm of Directed Feedback Vertex Set Problem

Authors: YuMing Du, QingYun Zhang, JunZhou Xu, ShunGen Zhang, Chao Liao, ZhiHuai Chen, ZhiBo Sun, ZhouXing Su, JunWen Ding, Chen Wu, PinYan Lu, and ZhiPeng Lv

Published in: LIPIcs, Volume 249, 17th International Symposium on Parameterized and Exact Computation (IPEC 2022)

Abstract
A directed graph is formed by vertices and arcs from one vertex to another. The feedback vertex set problem (FVSP) consists in making a given directed graph acyclic by removing as few vertices as possible. In this write-up, we outline the core techniques used in the heuristic feedback vertex set algorithm, submitted to the heuristic track of the 2022 PACE challenge.
Cite as

YuMing Du, QingYun Zhang, JunZhou Xu, ShunGen Zhang, Chao Liao, ZhiHuai Chen, ZhiBo Sun, ZhouXing Su, JunWen Ding, Chen Wu, PinYan Lu, and ZhiPeng Lv. PACE Solver Description: Hust-Solver - A Heuristic Algorithm of Directed Feedback Vertex Set Problem. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 29:1-29:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{du_et_al:LIPIcs.IPEC.2022.29,
  author =	{Du, YuMing and Zhang, QingYun and Xu, JunZhou and Zhang, ShunGen and Liao, Chao and Chen, ZhiHuai and Sun, ZhiBo and Su, ZhouXing and Ding, JunWen and Wu, Chen and Lu, PinYan and Lv, ZhiPeng},
  title =	{{PACE Solver Description: Hust-Solver - A Heuristic Algorithm of Directed Feedback Vertex Set Problem}},
  booktitle =	{17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
  pages =	{29:1--29:3},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-260-0},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{249},
  editor =	{Dell, Holger and Nederlof, Jesper},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.29},
  URN =		{urn:nbn:de:0030-drops-173855},
  doi =		{10.4230/LIPIcs.IPEC.2022.29},
  annote =	{Keywords: directed feedback vertex set, local search, simulated annealing, set covering}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2022.6
Approximate Representation of Symmetric Submodular Functions via Hypergraph Cut Functions

Authors: Calvin Beideman, Karthekeyan Chandrasekaran, Chandra Chekuri, and Chao Xu

Published in: LIPIcs, Volume 250, 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)

Abstract
Submodular functions are fundamental to combinatorial optimization. Many interesting problems can be formulated as special cases of problems involving submodular functions. In this work, we consider the problem of approximating symmetric submodular functions everywhere using hypergraph cut functions. Devanur, Dughmi, Schwartz, Sharma, and Singh [Devanur et al., 2013] showed that symmetric submodular functions over n-element ground sets cannot be approximated within (n/8)-factor using a graph cut function and raised the question of approximating them using hypergraph cut functions. Our main result is that there exist symmetric submodular functions over n-element ground sets that cannot be approximated within a o(n^{1/3}/log² n)-factor using a hypergraph cut function. On the positive side, we show that symmetrized concave linear functions and symmetrized rank functions of uniform matroids and partition matroids can be constant-approximated using hypergraph cut functions.
Cite as

Calvin Beideman, Karthekeyan Chandrasekaran, Chandra Chekuri, and Chao Xu. Approximate Representation of Symmetric Submodular Functions via Hypergraph Cut Functions. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{beideman_et_al:LIPIcs.FSTTCS.2022.6,
  author =	{Beideman, Calvin and Chandrasekaran, Karthekeyan and Chekuri, Chandra and Xu, Chao},
  title =	{{Approximate Representation of Symmetric Submodular Functions via Hypergraph Cut Functions}},
  booktitle =	{42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
  pages =	{6:1--6:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-261-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{250},
  editor =	{Dawar, Anuj and Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.6},
  URN =		{urn:nbn:de:0030-drops-173986},
  doi =		{10.4230/LIPIcs.FSTTCS.2022.6},
  annote =	{Keywords: Submodular Functions, Hypergraphs, Approximation, Representation}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2022.83
Improved Approximation Algorithms for the Traveling Tournament Problem

Authors: Jingyang Zhao, Mingyu Xiao, and Chao Xu

Published in: LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

Abstract
The Traveling Tournament Problem (TTP) is a well-known benchmark problem in the field of tournament timetabling, which asks us to design a double round-robin schedule such that each pair of teams plays one game in each other’s home venue, minimizing the total distance traveled by all n teams (n is even). TTP-k is the problem with one more constraint that each team can have at most k consecutive home games or away games. The case where k = 3, TTP-3, is one of the most investigated cases. In this paper, we improve the approximation ratio of TTP-3 from (1.667+ε) to (1.598+ε), for any ε > 0. Previous schedules were constructed based on a Hamiltonian cycle of the graph. We propose a novel construction based on triangle packing. Then, by combining our triangle packing schedule with the Hamiltonian cycle schedule, we obtain the improved approximation ratio. The idea of our construction can also be extended to k ≥ 4. We demonstrate that the approximation ratio of TTP-4 can be improved from (1.750+ε) to (1.700+ε) by the same method. As an additional product, we also improve the approximation ratio of LDTTP-3 (TTP-3 where all teams are allocated on a straight line) from 4/3 to (6/5+ε).
Cite as

Jingyang Zhao, Mingyu Xiao, and Chao Xu. Improved Approximation Algorithms for the Traveling Tournament Problem. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 83:1-83:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{zhao_et_al:LIPIcs.MFCS.2022.83,
  author =	{Zhao, Jingyang and Xiao, Mingyu and Xu, Chao},
  title =	{{Improved Approximation Algorithms for the Traveling Tournament Problem}},
  booktitle =	{47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
  pages =	{83:1--83:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-256-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{241},
  editor =	{Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.83},
  URN =		{urn:nbn:de:0030-drops-168813},
  doi =		{10.4230/LIPIcs.MFCS.2022.83},
  annote =	{Keywords: Sports scheduling, Traveling Tournament Problem, Approximation algorithm}
}
Document
DOI: 10.4230/LIPIcs.ESA.2020.29
More on Change-Making and Related Problems

Authors: Timothy M. Chan and Qizheng He

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

Abstract
Given a set of n integer-valued coin types and a target value t, the well-known change-making problem asks for the minimum number of coins that sum to t, assuming an unlimited number of coins in each type. In the more general all-targets version of the problem, we want the minimum number of coins summing to j, for every j = 0,…,t. For example, the textbook dynamic programming algorithms can solve the all-targets problem in O(nt) time. Recently, Chan and He (SOSA'20) described a number of O(t polylog t)-time algorithms for the original (single-target) version of the change-making problem, but not the all-targets version. In this paper, we obtain a number of new results on change-making and related problems: - We present a new algorithm for the all-targets change-making problem with running time Õ(t^{4/3}), improving a previous Õ(t^{3/2})-time algorithm. - We present a very simple Õ(u²+t)-time algorithm for the all-targets change-making problem, where u denotes the maximum coin value. The analysis of the algorithm uses a theorem of Erdős and Graham (1972) on the Frobenius problem. This algorithm can be extended to solve the all-capacities version of the unbounded knapsack problem (for integer item weights bounded by u). - For the original (single-target) coin changing problem, we describe a simple modification of one of Chan and He’s algorithms that runs in Õ(u) time (instead of Õ(t)). - For the original (single-capacity) unbounded knapsack problem, we describe a simple algorithm that runs in Õ(nu) time, improving previous near-u²-time algorithms. - We also observe how one of our ideas implies a new result on the minimum word break problem, an optimization version of a string problem studied by Bringmann et al. (FOCS'17), generalizing change-making (which corresponds to the unary special case).
Cite as

Timothy M. Chan and Qizheng He. More on Change-Making and Related Problems. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 29:1-29:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chan_et_al:LIPIcs.ESA.2020.29,
  author =	{Chan, Timothy M. and He, Qizheng},
  title =	{{More on Change-Making and Related Problems}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{29:1--29:14},
  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.2020.29},
  URN =		{urn:nbn:de:0030-drops-128958},
  doi =		{10.4230/LIPIcs.ESA.2020.29},
  annote =	{Keywords: Coin changing, knapsack, dynamic programming, Frobenius problem, fine-grained complexity}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2020.17
Multicriteria Cuts and Size-Constrained k-Cuts in Hypergraphs

Authors: Calvin Beideman, Karthekeyan Chandrasekaran, and Chao Xu

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

Abstract
We address counting and optimization variants of multicriteria global min-cut and size-constrained min-k-cut in hypergraphs. 1) For an r-rank n-vertex hypergraph endowed with t hyperedge-cost functions, we show that the number of multiobjective min-cuts is O(r2^{tr}n^{3t-1}). In particular, this shows that the number of parametric min-cuts in constant rank hypergraphs for a constant number of criteria is strongly polynomial, thus resolving an open question by Aissi, Mahjoub, McCormick, and Queyranne [Aissi et al., 2015]. In addition, we give randomized algorithms to enumerate all multiobjective min-cuts and all pareto-optimal cuts in strongly polynomial-time. 2) We also address node-budgeted multiobjective min-cuts: For an n-vertex hypergraph endowed with t vertex-weight functions, we show that the number of node-budgeted multiobjective min-cuts is O(r2^{r}n^{t+2}), where r is the rank of the hypergraph, and the number of node-budgeted b-multiobjective min-cuts for a fixed budget-vector b ∈ ℝ^t_+ is O(n²). 3) We show that min-k-cut in hypergraphs subject to constant lower bounds on part sizes is solvable in polynomial-time for constant k, thus resolving an open problem posed by Queyranne [Guinez and Queyranne, 2012]. Our technique also shows that the number of optimal solutions is polynomial. All of our results build on the random contraction approach of Karger [Karger, 1993]. Our techniques illustrate the versatility of the random contraction approach to address counting and algorithmic problems concerning multiobjective min-cuts and size-constrained k-cuts in hypergraphs.
Cite as

Calvin Beideman, Karthekeyan Chandrasekaran, and Chao Xu. Multicriteria Cuts and Size-Constrained k-Cuts in Hypergraphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 17:1-17:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{beideman_et_al:LIPIcs.APPROX/RANDOM.2020.17,
  author =	{Beideman, Calvin and Chandrasekaran, Karthekeyan and Xu, Chao},
  title =	{{Multicriteria Cuts and Size-Constrained k-Cuts in Hypergraphs}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)},
  pages =	{17:1--17:21},
  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.2020.17},
  URN =		{urn:nbn:de:0030-drops-126203},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2020.17},
  annote =	{Keywords: Multiobjective Optimization, Hypergraph min-cut, Hypergraph-k-cut}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2019.23
Fast and Deterministic Approximations for k-Cut

Authors: Kent Quanrud

Published in: LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

Abstract
In an undirected graph, a k-cut is a set of edges whose removal breaks the graph into at least k connected components. The minimum weight k-cut can be computed in n^O(k) time, but when k is treated as part of the input, computing the minimum weight k-cut is NP-Hard [Goldschmidt and Hochbaum, 1994]. For poly(m,n,k)-time algorithms, the best possible approximation factor is essentially 2 under the small set expansion hypothesis [Manurangsi, 2017]. Saran and Vazirani [1995] showed that a (2 - 2/k)-approximately minimum weight k-cut can be computed via O(k) minimum cuts, which implies a O~(km) randomized running time via the nearly linear time randomized min-cut algorithm of Karger [2000]. Nagamochi and Kamidoi [2007] showed that a (2 - 2/k)-approximately minimum weight k-cut can be computed deterministically in O(mn + n^2 log n) time. These results prompt two basic questions. The first concerns the role of randomization. Is there a deterministic algorithm for 2-approximate k-cuts matching the randomized running time of O~(km)? The second question qualitatively compares minimum cut to 2-approximate minimum k-cut. Can 2-approximate k-cuts be computed as fast as the minimum cut - in O~(m) randomized time? We give a deterministic approximation algorithm that computes (2 + eps)-minimum k-cuts in O(m log^3 n / eps^2) time, via a (1 + eps)-approximation for an LP relaxation of k-cut.
Cite as

Kent Quanrud. Fast and Deterministic Approximations for k-Cut. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 23:1-23:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{quanrud:LIPIcs.APPROX-RANDOM.2019.23,
  author =	{Quanrud, Kent},
  title =	{{Fast and Deterministic Approximations for k-Cut}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
  pages =	{23:1--23:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-125-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{145},
  editor =	{Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.23},
  URN =		{urn:nbn:de:0030-drops-112388},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2019.23},
  annote =	{Keywords: k-cut, multiplicative weight updates}
}
Document
DOI: 10.4230/OASIcs.SOSA.2019.7
LP Relaxation and Tree Packing for Minimum k-cuts

Authors: Chandra Chekuri, Kent Quanrud, and Chao Xu

Published in: OASIcs, Volume 69, 2nd Symposium on Simplicity in Algorithms (SOSA 2019)

Abstract
Karger used spanning tree packings [Karger, 2000] to derive a near linear-time randomized algorithm for the global minimum cut problem as well as a bound on the number of approximate minimum cuts. This is a different approach from his well-known random contraction algorithm [Karger, 1995; Karger and Stein, 1996]. Thorup developed a fast deterministic algorithm for the minimum k-cut problem via greedy recursive tree packings [Thorup, 2008]. In this paper we revisit properties of an LP relaxation for k-cut proposed by Naor and Rabani [Naor and Rabani, 2001], and analyzed in [Chekuri et al., 2006]. We show that the dual of the LP yields a tree packing, that when combined with an upper bound on the integrality gap for the LP, easily and transparently extends Karger's analysis for mincut to the k-cut problem. In addition to the simplicity of the algorithm and its analysis, this allows us to improve the running time of Thorup's algorithm by a factor of n. We also improve the bound on the number of alpha-approximate k-cuts. Second, we give a simple proof that the integrality gap of the LP is 2(1-1/n). Third, we show that an optimum solution to the LP relaxation, for all values of k, is fully determined by the principal sequence of partitions of the input graph. This allows us to relate the LP relaxation to the Lagrangean relaxation approach of Barahona [Barahona, 2000] and Ravi and Sinha [Ravi and Sinha, 2008]; it also shows that the idealized recursive tree packing considered by Thorup gives an optimum dual solution to the LP. This work arose from an effort to understand and simplify the results of Thorup [Thorup, 2008].
Cite as

Chandra Chekuri, Kent Quanrud, and Chao Xu. LP Relaxation and Tree Packing for Minimum k-cuts. In 2nd Symposium on Simplicity in Algorithms (SOSA 2019). Open Access Series in Informatics (OASIcs), Volume 69, pp. 7:1-7:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chekuri_et_al:OASIcs.SOSA.2019.7,
  author =	{Chekuri, Chandra and Quanrud, Kent and Xu, Chao},
  title =	{{LP Relaxation and Tree Packing for Minimum k-cuts}},
  booktitle =	{2nd Symposium on Simplicity in Algorithms (SOSA 2019)},
  pages =	{7:1--7:18},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-099-6},
  ISSN =	{2190-6807},
  year =	{2019},
  volume =	{69},
  editor =	{Fineman, Jeremy T. and Mitzenmacher, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2019.7},
  URN =		{urn:nbn:de:0030-drops-100335},
  doi =		{10.4230/OASIcs.SOSA.2019.7},
  annote =	{Keywords: k-cut, LP relaxation, tree packing}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2017.2
Global and Fixed-Terminal Cuts in Digraphs

Authors: Kristóf Bérczi, Karthekeyan Chandrasekaran, Tamás Király, Euiwoong Lee, and Chao Xu

Published in: LIPIcs, Volume 81, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)

Abstract
The computational complexity of multicut-like problems may vary significantly depending on whether the terminals are fixed or not. In this work we present a comprehensive study of this phenomenon in two types of cut problems in directed graphs: double cut and bicut. 1. Fixed-terminal edge-weighted double cut is known to be solvable efficiently. We show that fixed-terminal node-weighted double cut cannot be approximated to a factor smaller than 2 under the Unique Games Conjecture (UGC), and we also give a 2-approximation algorithm. For the global version of the problem, we prove an inapproximability bound of 3/2 under UGC. 2. Fixed-terminal edge-weighted bicut is known to have an approximability factor of 2 that is tight under UGC. We show that the global edge-weighted bicut is approximable to a factor strictly better than 2, and that the global node-weighted bicut cannot be approximated to a factor smaller than 3/2 under UGC. 3. In relation to these investigations, we also prove two results on undirected graphs which are of independent interest. First, we show NP-completeness and a tight inapproximability bound of 4/3 for the node-weighted 3-cut problem under UGC. Second, we show that for constant k, there exists an efficient algorithm to solve the minimum {s,t}-separating k-cut problem. Our techniques for the algorithms are combinatorial, based on LPs and based on the enumeration of approximate min-cuts. Our hardness results are based on combinatorial reductions and integrality gap instances.
Cite as

Kristóf Bérczi, Karthekeyan Chandrasekaran, Tamás Király, Euiwoong Lee, and Chao Xu. Global and Fixed-Terminal Cuts in Digraphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 2:1-2:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{berczi_et_al:LIPIcs.APPROX-RANDOM.2017.2,
  author =	{B\'{e}rczi, Krist\'{o}f and Chandrasekaran, Karthekeyan and Kir\'{a}ly, Tam\'{a}s and Lee, Euiwoong and Xu, Chao},
  title =	{{Global and Fixed-Terminal Cuts in Digraphs}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
  pages =	{2:1--2:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-044-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{81},
  editor =	{Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.2},
  URN =		{urn:nbn:de:0030-drops-75511},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2017.2},
  annote =	{Keywords: Directed Graphs, Arborescence, Graph Cuts, Hardness of Approximation}
}
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