18 Search Results for "Mucha, Marcin"


Document
Even Faster Knapsack via Rectangular Monotone Min-Plus Convolution and Balancing

Authors: Karl Bringmann, Anita Dürr, and Adam Polak

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)


Abstract
We present a pseudopolynomial-time algorithm for the Knapsack problem that has running time Õ(n + t√{p_{max}}), where n is the number of items, t is the knapsack capacity, and p_{max} is the maximum item profit. This improves over the Õ(n + t p_{max})-time algorithm based on the convolution and prediction technique by Bateni et al. (STOC 2018). Moreover, we give some evidence, based on a strengthening of the Min-Plus Convolution Hypothesis, that our running time might be optimal. Our algorithm uses two new technical tools, which might be of independent interest. First, we generalize the Õ(n^{1.5})-time algorithm for bounded monotone min-plus convolution by Chi et al. (STOC 2022) to the rectangular case where the range of entries can be different from the sequence length. Second, we give a reduction from general knapsack instances to balanced instances, where all items have nearly the same profit-to-weight ratio, up to a constant factor. Using these techniques, we can also obtain algorithms that run in time Õ(n + OPT√{w_{max}}), Õ(n + (nw_{max}p_{max})^{1/3}t^{2/3}), and Õ(n + (nw_{max}p_{max})^{1/3} OPT^{2/3}), where OPT is the optimal total profit and w_{max} is the maximum item weight.

Cite as

Karl Bringmann, Anita Dürr, and Adam Polak. Even Faster Knapsack via Rectangular Monotone Min-Plus Convolution and Balancing. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 33:1-33:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{bringmann_et_al:LIPIcs.ESA.2024.33,
  author =	{Bringmann, Karl and D\"{u}rr, Anita and Polak, Adam},
  title =	{{Even Faster Knapsack via Rectangular Monotone Min-Plus Convolution and Balancing}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{33:1--33:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.33},
  URN =		{urn:nbn:de:0030-drops-211047},
  doi =		{10.4230/LIPIcs.ESA.2024.33},
  annote =	{Keywords: 0-1-Knapsack problem, bounded monotone min-plus convolution, fine-grained complexity}
}
Document
Exploring the Approximability Landscape of 3SUM

Authors: Karl Bringmann, Ahmed Ghazy, and Marvin Künnemann

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)


Abstract
Since an increasing number of problems in P have conditional lower bounds against exact algorithms, it is natural to study which of these problems can be efficiently approximated. Often, however, there are many potential ways to formulate an approximate version of a problem. We ask: How sensitive is the (in-)approximability of a problem in P to its precise formulation? To this end, we perform a case study using the popular 3SUM problem. Its many equivalent formulations give rise to a wide range of potential approximate relaxations. Specifically, to obtain an approximate relaxation in our framework, one can choose among the options: (a) 3SUM or Convolution 3SUM, (b) monochromatic or trichromatic, (c) allowing under-approximation, over-approximation, or both, (d) approximate decision or approximate optimization, (e) single output or multiple outputs and (f) implicit or explicit target (given as input). We show general reduction principles between some variants and find that we can classify the remaining problems (over polynomially bounded positive integers) into three regimes: 1) (1+ε)-approximable in near-linear time Õ(n + 1/ε), 2) (1+ε)-approximable in near-quadratic time Õ(n/ε) or Õ(n+1/ε²), or 3) non-approximable, i.e., requiring time n^{2± o(1)} even for any approximation factor. In each of these three regimes, we provide matching upper and conditional lower bounds. To prove our results, we establish two results that may be of independent interest: Over polynomially bounded integers, we show subquadratic equivalence of (min,+)-convolution and polyhedral 3SUM, and we prove equivalence of the Strong 3SUM conjecture and the Strong Convolution 3SUM conjecture.

Cite as

Karl Bringmann, Ahmed Ghazy, and Marvin Künnemann. Exploring the Approximability Landscape of 3SUM. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 34:1-34:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{bringmann_et_al:LIPIcs.ESA.2024.34,
  author =	{Bringmann, Karl and Ghazy, Ahmed and K\"{u}nnemann, Marvin},
  title =	{{Exploring the Approximability Landscape of 3SUM}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{34:1--34:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.34},
  URN =		{urn:nbn:de:0030-drops-211057},
  doi =		{10.4230/LIPIcs.ESA.2024.34},
  annote =	{Keywords: Fine-grained Complexity, Conditional Lower Bounds, Approximation Schemes, Min-Plus Convolution}
}
Document
Finding Perfect Matchings in Bridgeless Cubic Multigraphs Without Dynamic (2-)connectivity

Authors: Paweł Gawrychowski and Mateusz Wasylkiewicz

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)


Abstract
Petersen’s theorem, one of the earliest results in graph theory, states that every bridgeless cubic multigraph contains a perfect matching. While the original proof was neither constructive nor algorithmic, Biedl, Bose, Demaine, and Lubiw [J. Algorithms 38(1)] showed how to implement a later constructive proof by Frink in 𝒪(nlog⁴n) time using a fully dynamic 2-edge-connectivity structure. Then, Diks and Stańczyk [SOFSEM 2010] described a faster approach that only needs a fully dynamic connectivity structure and works in 𝒪(nlog²n) time. Both algorithms, while reasonable simple, utilize non-trivial (2-edge-)connectivity structures. We show that this is not necessary, and in fact a structure for maintaining a dynamic tree, e.g. link-cut trees, suffices to obtain a simple 𝒪(nlog n) time algorithm.

Cite as

Paweł Gawrychowski and Mateusz Wasylkiewicz. Finding Perfect Matchings in Bridgeless Cubic Multigraphs Without Dynamic (2-)connectivity. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 59:1-59:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{gawrychowski_et_al:LIPIcs.ESA.2024.59,
  author =	{Gawrychowski, Pawe{\l} and Wasylkiewicz, Mateusz},
  title =	{{Finding Perfect Matchings in Bridgeless Cubic Multigraphs Without Dynamic (2-)connectivity}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{59:1--59:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.59},
  URN =		{urn:nbn:de:0030-drops-211301},
  doi =		{10.4230/LIPIcs.ESA.2024.59},
  annote =	{Keywords: perfect matching, cubic graphs, bridgeless graphs, link-cut tree}
}
Document
Minimizing the Weighted Number of Tardy Jobs Is W[1]-Hard

Authors: Klaus Heeger and Danny Hermelin

Published in: LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)


Abstract
We consider the 1∣∣∑ w_jU_j problem, the problem of minimizing the weighted number of tardy jobs on a single machine. This problem is one of the most basic and fundamental problems in scheduling theory, with several different applications both in theory and practice. Using a reduction from the Multicolored Clique problem, we prove that 1∣∣∑ w_jU_j is W[1]-hard with respect to the number p_# of different processing times in the input, as well as with respect to the number w_# of different weights in the input. This, along with previous work, provides a complete picture for 1∣∣∑ w_jU_j from the perspective of parameterized complexity, as well as almost tight complexity bounds for the problem under the Exponential Time Hypothesis (ETH).

Cite as

Klaus Heeger and Danny Hermelin. Minimizing the Weighted Number of Tardy Jobs Is W[1]-Hard. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 68:1-68:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{heeger_et_al:LIPIcs.ESA.2024.68,
  author =	{Heeger, Klaus and Hermelin, Danny},
  title =	{{Minimizing the Weighted Number of Tardy Jobs Is W\lbrack1\rbrack-Hard}},
  booktitle =	{32nd Annual European Symposium on Algorithms (ESA 2024)},
  pages =	{68:1--68:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-338-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{308},
  editor =	{Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.68},
  URN =		{urn:nbn:de:0030-drops-211392},
  doi =		{10.4230/LIPIcs.ESA.2024.68},
  annote =	{Keywords: single-machine scheduling, number of different weights, number of different processing times}
}
Document
Track A: Algorithms, Complexity and Games
Sublinear Algorithms for TSP via Path Covers

Authors: Soheil Behnezhad, Mohammad Roghani, Aviad Rubinstein, and Amin Saberi

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)


Abstract
We study sublinear time algorithms for the traveling salesman problem (TSP). First, we focus on the closely related maximum path cover problem, which asks for a collection of vertex disjoint paths that include the maximum number of edges. We show that for any fixed ε > 0, there is an algorithm that (1/2 - ε)-approximates the maximum path cover size of an n-vertex graph in Õ(n) time. This improves upon a (3/8-ε)-approximate Õ(n √n)-time algorithm of Chen, Kannan, and Khanna [ICALP'20]. Equipped with our path cover algorithm, we give an Õ(n) time algorithm that estimates the cost of (1,2)-TSP within a factor of (1.5+ε) which is an improvement over a folklore (1.75 + ε)-approximate Õ(n)-time algorithm, as well as a (1.625+ε)-approximate Õ(n√n)-time algorithm of [CHK ICALP'20]. For graphic TSP, we present an Õ(n) algorithm that estimates the cost of graphic TSP within a factor of 1.83 which is an improvement over a 1.92-approximate Õ(n) time algorithm due to [CHK ICALP'20, Behnezhad FOCS'21]. We show that the approximation can be further improved to 1.66 using n^{2-Ω(1)} time. All of our Õ(n) time algorithms are information-theoretically time-optimal up to polylog n factors. Additionally, we show that our approximation guarantees for path cover and (1,2)-TSP hit a natural barrier: We show better approximations require better sublinear time algorithms for the well-studied maximum matching problem.

Cite as

Soheil Behnezhad, Mohammad Roghani, Aviad Rubinstein, and Amin Saberi. Sublinear Algorithms for TSP via Path Covers. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 19:1-19:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{behnezhad_et_al:LIPIcs.ICALP.2024.19,
  author =	{Behnezhad, Soheil and Roghani, Mohammad and Rubinstein, Aviad and Saberi, Amin},
  title =	{{Sublinear Algorithms for TSP via Path Covers}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{19:1--19:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.19},
  URN =		{urn:nbn:de:0030-drops-201623},
  doi =		{10.4230/LIPIcs.ICALP.2024.19},
  annote =	{Keywords: Sublinear Algorithms, Traveling Salesman Problem, Approximation Algorithm, (1, 2)-TSP, Graphic TSP}
}
Document
Track A: Algorithms, Complexity and Games
Lower Bounds for Matroid Optimization Problems with a Linear Constraint

Authors: Ilan Doron-Arad, Ariel Kulik, and Hadas Shachnai

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)


Abstract
We study a family of matroid optimization problems with a linear constraint (MOL). In these problems, we seek a subset of elements which optimizes (i.e., maximizes or minimizes) a linear objective function subject to (i) a matroid independent set, or a matroid basis constraint, (ii) additional linear constraint. A notable member in this family is budgeted matroid independent set (BM), which can be viewed as classic 0/1-Knapsack with a matroid constraint. While special cases of BM, such as knapsack with cardinality constraint and multiple-choice knapsack, admit a fully polynomial-time approximation scheme (Fully PTAS), the best known result for BM on a general matroid is an Efficient PTAS. Prior to this work, the existence of a Fully PTAS for BM, and more generally, for any problem in the family of MOL problems, has been open. In this paper, we answer this question negatively by showing that none of the (non-trivial) problems in this family admits a Fully PTAS. This resolves the complexity status of several well studied problems. Our main result is obtained by showing first that exact weight matroid basis (EMB) does not admit a pseudo-polynomial time algorithm. This distinguishes EMB from the special cases of k-subset sum and EMB on a linear matroid, which are solvable in pseudo-polynomial time. We then obtain unconditional hardness results for the family of MOL problems in the oracle model (even if randomization is allowed), and show that the same results hold when the matroids are encoded as part of the input, assuming P ≠ NP. For the hardness proof of EMB, we introduce the Π-matroid family. This intricate subclass of matroids, which exploits the interaction between a weight function and the matroid constraint, may find use in tackling other matroid optimization problems.

Cite as

Ilan Doron-Arad, Ariel Kulik, and Hadas Shachnai. Lower Bounds for Matroid Optimization Problems with a Linear Constraint. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 56:1-56:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{doronarad_et_al:LIPIcs.ICALP.2024.56,
  author =	{Doron-Arad, Ilan and Kulik, Ariel and Shachnai, Hadas},
  title =	{{Lower Bounds for Matroid Optimization Problems with a Linear Constraint}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{56:1--56:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.56},
  URN =		{urn:nbn:de:0030-drops-201990},
  doi =		{10.4230/LIPIcs.ICALP.2024.56},
  annote =	{Keywords: matroid optimization, budgeted problems, knapsack, approximation schemes}
}
Document
Track A: Algorithms, Complexity and Games
Minimizing Tardy Processing Time on a Single Machine in Near-Linear Time

Authors: Nick Fischer and Leo Wennmann

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)


Abstract
In this work we revisit the elementary scheduling problem 1||∑ p_j U_j. The goal is to select, among n jobs with processing times and due dates, a subset of jobs with maximum total processing time that can be scheduled in sequence without violating their due dates. This problem is NP-hard, but a classical algorithm by Lawler and Moore from the 60s solves this problem in pseudo-polynomial time O(nP), where P is the total processing time of all jobs. With the aim to develop best-possible pseudo-polynomial-time algorithms, a recent wave of results has improved Lawler and Moore’s algorithm for 1||∑ p_j U_j: First to time Õ(P^{7/4}) [Bringmann, Fischer, Hermelin, Shabtay, Wellnitz; ICALP'20], then to time Õ(P^{5/3}) [Klein, Polak, Rohwedder; SODA'23], and finally to time Õ(P^{7/5}) [Schieber, Sitaraman; WADS'23]. It remained an exciting open question whether these works can be improved further. In this work we develop an algorithm in near-linear time Õ(P) for the 1||∑ p_j U_j problem. This running time not only significantly improves upon the previous results, but also matches conditional lower bounds based on the Strong Exponential Time Hypothesis or the Set Cover Hypothesis and is therefore likely optimal (up to subpolynomial factors). Our new algorithm also extends to the case of m machines in time Õ(P^m). In contrast to the previous improvements, we take a different, more direct approach inspired by the recent reductions from Modular Subset Sum to dynamic string problems. We thereby arrive at a satisfyingly simple algorithm.

Cite as

Nick Fischer and Leo Wennmann. Minimizing Tardy Processing Time on a Single Machine in Near-Linear Time. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 64:1-64:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{fischer_et_al:LIPIcs.ICALP.2024.64,
  author =	{Fischer, Nick and Wennmann, Leo},
  title =	{{Minimizing Tardy Processing Time on a Single Machine in Near-Linear Time}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{64:1--64:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.64},
  URN =		{urn:nbn:de:0030-drops-202079},
  doi =		{10.4230/LIPIcs.ICALP.2024.64},
  annote =	{Keywords: Scheduling, Fine-Grained Complexity, Dynamic Strings}
}
Document
Track A: Algorithms, Complexity and Games
A Faster Algorithm for Pigeonhole Equal Sums

Authors: Ce Jin and Hongxun Wu

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)


Abstract
An important area of research in exact algorithms is to solve Subset-Sum-type problems faster than meet-in-middle. In this paper we study Pigeonhole Equal Sums, a total search problem proposed by Papadimitriou (1994): given n positive integers w₁,… ,w_n of total sum ∑_{i = 1}ⁿ w_i < 2ⁿ-1, the task is to find two distinct subsets A, B ⊆ [n] such that ∑_{i ∈ A}w_i = ∑_{i ∈ B}w_i. Similar to the status of the Subset Sum problem, the best known algorithm for Pigeonhole Equal Sums runs in O^*(2^{n/2}) time, via either meet-in-middle or dynamic programming (Allcock, Hamoudi, Joux, Klingelhöfer, and Santha, 2022). Our main result is an improved algorithm for Pigeonhole Equal Sums in O^*(2^{0.4n}) time. We also give a polynomial-space algorithm in O^*(2^{0.75n}) time. Unlike many previous works in this area, our approach does not use the representation method, but rather exploits a simple structural characterization of input instances with few solutions.

Cite as

Ce Jin and Hongxun Wu. A Faster Algorithm for Pigeonhole Equal Sums. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 94:1-94:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{jin_et_al:LIPIcs.ICALP.2024.94,
  author =	{Jin, Ce and Wu, Hongxun},
  title =	{{A Faster Algorithm for Pigeonhole Equal Sums}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{94:1--94:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.94},
  URN =		{urn:nbn:de:0030-drops-202375},
  doi =		{10.4230/LIPIcs.ICALP.2024.94},
  annote =	{Keywords: Subset Sum, Pigeonhole, PPP}
}
Document
Track A: Algorithms, Complexity and Games
No Polynomial Kernels for Knapsack

Authors: Klaus Heeger, Danny Hermelin, Matthias Mnich, and Dvir Shabtay

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)


Abstract
This paper focuses on kernelization algorithms for the fundamental Knapsack problem. A kernelization algorithm (or kernel) is a polynomial-time reduction from a problem onto itself, where the output size is bounded by a function of some problem-specific parameter. Such algorithms provide a theoretical model for data reduction and preprocessing and are central in the area of parameterized complexity. In this way, a kernel for Knapsack for some parameter k reduces any instance of Knapsack to an equivalent instance of size at most f(k) in polynomial time, for some computable function f. When f(k) = k^{O(1)} then we call such a reduction a polynomial kernel. Our study focuses on two natural parameters for Knapsack: The number w_# of different item weights, and the number p_# of different item profits. Our main technical contribution is a proof showing that Knapsack does not admit a polynomial kernel for any of these two parameters under standard complexity-theoretic assumptions. Our proof discovers an elaborate application of the standard kernelization lower bound framework, and develops along the way novel ideas that should be useful for other problems as well. We complement our lower bounds by showing that Knapsack admits a polynomial kernel for the combined parameter w_# ⋅ p_#.

Cite as

Klaus Heeger, Danny Hermelin, Matthias Mnich, and Dvir Shabtay. No Polynomial Kernels for Knapsack. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 83:1-83:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{heeger_et_al:LIPIcs.ICALP.2024.83,
  author =	{Heeger, Klaus and Hermelin, Danny and Mnich, Matthias and Shabtay, Dvir},
  title =	{{No Polynomial Kernels for Knapsack}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{83:1--83:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.83},
  URN =		{urn:nbn:de:0030-drops-202261},
  doi =		{10.4230/LIPIcs.ICALP.2024.83},
  annote =	{Keywords: Knapsack, polynomial kernels, compositions, number of different weights, number of different profits}
}
Document
Computing Treedepth in Polynomial Space and Linear FPT Time

Authors: Wojciech Nadara, Michał Pilipczuk, and Marcin Smulewicz

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)


Abstract
The treedepth of a graph G is the least possible depth of an elimination forest of G: a rooted forest on the same vertex set where every pair of vertices adjacent in G is bound by the ancestor/descendant relation. We propose an algorithm that given a graph G and an integer d, either finds an elimination forest of G of depth at most d or concludes that no such forest exists; thus the algorithm decides whether the treedepth of G is at most d. The running time is 2^𝒪(d²)⋅n^𝒪(1) and the space usage is polynomial in n. Further, by allowing randomization, the time and space complexities can be improved to 2^𝒪(d²)⋅n and d^𝒪(1)⋅n, respectively. This improves upon the algorithm of Reidl et al. [ICALP 2014], which also has time complexity 2^𝒪(d²)⋅n, but uses exponential space.

Cite as

Wojciech Nadara, Michał Pilipczuk, and Marcin Smulewicz. Computing Treedepth in Polynomial Space and Linear FPT Time. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 79:1-79:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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@InProceedings{nadara_et_al:LIPIcs.ESA.2022.79,
  author =	{Nadara, Wojciech and Pilipczuk, Micha{\l} and Smulewicz, Marcin},
  title =	{{Computing Treedepth in Polynomial Space and Linear FPT Time}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{79:1--79:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.79},
  URN =		{urn:nbn:de:0030-drops-170175},
  doi =		{10.4230/LIPIcs.ESA.2022.79},
  annote =	{Keywords: treedepth, FPT, polynomial space}
}
Document
The PACE 2020 Parameterized Algorithms and Computational Experiments Challenge: Treedepth

Authors: Łukasz Kowalik, Marcin Mucha, Wojciech Nadara, Marcin Pilipczuk, Manuel Sorge, and Piotr Wygocki

Published in: LIPIcs, Volume 180, 15th International Symposium on Parameterized and Exact Computation (IPEC 2020)


Abstract
This year’s Parameterized Algorithms and Computational Experiments challenge (PACE 2020) was devoted to the problem of computing the treedepth of a given graph. Altogether 51 participants from 20 teams, 12 countries and 3 continents submitted their implementations to the competition. In this report, we describe the setup of the challenge, the selection of benchmark instances and the ranking of the participating teams. We also briefly discuss the approaches used in the submitted solvers and the differences in their performance on our benchmark dataset.

Cite as

Łukasz Kowalik, Marcin Mucha, Wojciech Nadara, Marcin Pilipczuk, Manuel Sorge, and Piotr Wygocki. The PACE 2020 Parameterized Algorithms and Computational Experiments Challenge: Treedepth. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 37:1-37:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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@InProceedings{kowalik_et_al:LIPIcs.IPEC.2020.37,
  author =	{Kowalik, {\L}ukasz and Mucha, Marcin and Nadara, Wojciech and Pilipczuk, Marcin and Sorge, Manuel and Wygocki, Piotr},
  title =	{{The PACE 2020 Parameterized Algorithms and Computational Experiments Challenge: Treedepth}},
  booktitle =	{15th International Symposium on Parameterized and Exact Computation (IPEC 2020)},
  pages =	{37:1--37:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-172-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{180},
  editor =	{Cao, Yixin and Pilipczuk, Marcin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2020.37},
  URN =		{urn:nbn:de:0030-drops-133404},
  doi =		{10.4230/LIPIcs.IPEC.2020.37},
  annote =	{Keywords: computing treedepth, contest, implementation challenge, FPT}
}
Document
Equal-Subset-Sum Faster Than the Meet-in-the-Middle

Authors: Marcin Mucha, Jesper Nederlof, Jakub Pawlewicz, and Karol Węgrzycki

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


Abstract
In the Equal-Subset-Sum problem, we are given a set S of n integers and the problem is to decide if there exist two disjoint nonempty subsets A,B subseteq S, whose elements sum up to the same value. The problem is NP-complete. The state-of-the-art algorithm runs in O^*(3^(n/2)) <= O^*(1.7321^n) time and is based on the meet-in-the-middle technique. In this paper, we improve upon this algorithm and give O^*(1.7088^n) worst case Monte Carlo algorithm. This answers a question suggested by Woeginger in his inspirational survey. Additionally, we analyse the polynomial space algorithm for Equal-Subset-Sum. A naive polynomial space algorithm for Equal-Subset-Sum runs in O^*(3^n) time. With read-only access to the exponentially many random bits, we show a randomized algorithm running in O^*(2.6817^n) time and polynomial space.

Cite as

Marcin Mucha, Jesper Nederlof, Jakub Pawlewicz, and Karol Węgrzycki. Equal-Subset-Sum Faster Than the Meet-in-the-Middle. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 73:1-73:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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@InProceedings{mucha_et_al:LIPIcs.ESA.2019.73,
  author =	{Mucha, Marcin and Nederlof, Jesper and Pawlewicz, Jakub and W\k{e}grzycki, Karol},
  title =	{{Equal-Subset-Sum Faster Than the Meet-in-the-Middle}},
  booktitle =	{27th Annual European Symposium on Algorithms (ESA 2019)},
  pages =	{73:1--73:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-124-5},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{144},
  editor =	{Bender, Michael A. and Svensson, Ola and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.73},
  URN =		{urn:nbn:de:0030-drops-111946},
  doi =		{10.4230/LIPIcs.ESA.2019.73},
  annote =	{Keywords: Equal-Subset-Sum, Subset-Sum, meet-in-the-middle, enumeration technique, randomized algorithm}
}
Document
Online Facility Location with Deletions

Authors: Marek Cygan, Artur Czumaj, Marcin Mucha, and Piotr Sankowski

Published in: LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)


Abstract
In this paper we study three previously unstudied variants of the online Facility Location problem, considering an intrinsic scenario when the clients and facilities are not only allowed to arrive to the system, but they can also depart at any moment. We begin with the study of a natural fully-dynamic online uncapacitated model where clients can be both added and removed. When a client arrives, then it has to be assigned either to an existing facility or to a new facility opened at the client's location. However, when a client who has been also one of the open facilities is to be removed, then our model has to allow to reconnect all clients that have been connected to that removed facility. In this model, we present an optimal O(log(n_{act}) / log log(n_{act}))-competitive algorithm, where n_{act} is the number of active clients at the end of the input sequence. Next, we turn our attention to the capacitated Facility Location problem. We first note that if no deletions are allowed, then one can achieve an optimal competitive ratio of O(log(n) / log(log n)), where n is the length of the sequence. However, when deletions are allowed, the capacitated version of the problem is significantly more challenging than the uncapacitated one. We show that still, using a more sophisticated algorithmic approach, one can obtain an online O(log N + log c log n)-competitive algorithm for the capacitated Facility Location problem in the fully dynamic model, where N is number of points in the input metric and c is the capacity of any open facility.

Cite as

Marek Cygan, Artur Czumaj, Marcin Mucha, and Piotr Sankowski. Online Facility Location with Deletions. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 21:1-21:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


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@InProceedings{cygan_et_al:LIPIcs.ESA.2018.21,
  author =	{Cygan, Marek and Czumaj, Artur and Mucha, Marcin and Sankowski, Piotr},
  title =	{{Online Facility Location with Deletions}},
  booktitle =	{26th Annual European Symposium on Algorithms (ESA 2018)},
  pages =	{21:1--21:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-081-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{112},
  editor =	{Azar, Yossi and Bast, Hannah and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.21},
  URN =		{urn:nbn:de:0030-drops-94843},
  doi =		{10.4230/LIPIcs.ESA.2018.21},
  annote =	{Keywords: online algorithms, facility location, fully-dynamic online algorithms}
}
Document
Dynamic Beats Fixed: On Phase-Based Algorithms for File Migration

Authors: Marcin Bienkowski, Jaroslaw Byrka, and Marcin Mucha

Published in: LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)


Abstract
In this paper, we construct a deterministic 4-competitive algorithm for the online file migration problem, beating the currently best 20-year old, 4.086-competitive MTLM algorithm by Bartal et al. (SODA 1997). Like MTLM, our algorithm also operates in phases, but it adapts their lengths dynamically depending on the geometry of requests seen so far. The improvement was obtained by carefully analyzing a linear model (factor-revealing LP) of a single phase of the algorithm. We also show that if an online algorithm operates in phases of fixed length and the adversary is able to modify the graph between phases, no algorithm can beat the competitive ratio of 4.086.

Cite as

Marcin Bienkowski, Jaroslaw Byrka, and Marcin Mucha. Dynamic Beats Fixed: On Phase-Based Algorithms for File Migration. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 13:1-13:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


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@InProceedings{bienkowski_et_al:LIPIcs.ICALP.2017.13,
  author =	{Bienkowski, Marcin and Byrka, Jaroslaw and Mucha, Marcin},
  title =	{{Dynamic Beats Fixed: On Phase-Based Algorithms for File Migration}},
  booktitle =	{44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
  pages =	{13:1--13:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-041-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{80},
  editor =	{Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.13},
  URN =		{urn:nbn:de:0030-drops-73942},
  doi =		{10.4230/LIPIcs.ICALP.2017.13},
  annote =	{Keywords: file migration, factor-revealing linear programs, online algorithms, competitive analysis}
}
Document
On Problems Equivalent to (min,+)-Convolution

Authors: Marek Cygan, Marcin Mucha, Karol Wegrzycki, and Michal Wlodarczyk

Published in: LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)


Abstract
In the recent years, significant progress has been made in explaining apparent hardness of improving over naive solutions for many fundamental polynomially solvable problems. This came in the form of conditional lower bounds -- reductions from a problem assumed to be hard. These include 3SUM, All-Pairs Shortest Paths, SAT and Orthogonal Vectors, and others. In the (min,+)-convolution problem, the goal is to compute a sequence c, where c[k] = min_i a[i]+b[k-i], given sequences a and b. This can easily be done in O(n^2) time, but no O(n^{2-eps}) algorithm is known for eps > 0. In this paper we undertake a systematic study of the (min,+)-convolution problem as a hardness assumption. As the first step, we establish equivalence of this problem to a group of other problems, including variants of the classic knapsack problem and problems related to subadditive sequences. The (min,+)-convolution has been used as a building block in algorithms for many problems, notably problems in stringology. It has also already appeared as an ad hoc hardness assumption. We investigate some of these connections and provide new reductions and other results.

Cite as

Marek Cygan, Marcin Mucha, Karol Wegrzycki, and Michal Wlodarczyk. On Problems Equivalent to (min,+)-Convolution. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 22:1-22:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


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@InProceedings{cygan_et_al:LIPIcs.ICALP.2017.22,
  author =	{Cygan, Marek and Mucha, Marcin and Wegrzycki, Karol and Wlodarczyk, Michal},
  title =	{{On Problems Equivalent to (min,+)-Convolution}},
  booktitle =	{44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
  pages =	{22:1--22:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-041-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{80},
  editor =	{Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.22},
  URN =		{urn:nbn:de:0030-drops-74216},
  doi =		{10.4230/LIPIcs.ICALP.2017.22},
  annote =	{Keywords: fine-grained complexity, knapsack, conditional lower bounds, (min,+)-convolution, subquadratic equivalence}
}
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