16 Search Results for "Edmonds, Jeff"

Document
DOI: 10.4230/LIPIcs.MFCS.2025.6
Catalytic Computing and Register Programs Beyond Log-Depth

Authors: Yaroslav Alekseev, Yuval Filmus, Ian Mertz, Alexander Smal, and Antoine Vinciguerra

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract
In a seminal work, Buhrman et al. (STOC 2014) defined the class CSPACE(s,c) of problems solvable in space s with an additional catalytic tape of size c, which is a tape whose initial content must be restored at the end of the computation. They showed that uniform TC¹ circuits are computable in catalytic logspace, i.e., CL = CSPACE(O(log{n}), 2^{O(log{n})}), thus giving strong evidence that catalytic space gives L strict additional power. Their study focuses on an arithmetic model called register programs, which has been a focal point in development since then. Understanding CL remains a major open problem, as TC¹ remains the most powerful containment to date. In this work, we study the power of catalytic space and register programs to compute circuits of larger depth. Using register programs, we show that for every ε > 0, SAC² ⊆ CSPACE (O((log²n)/(log log n)), 2^{O(log^{1+ε} n)}) . On the other hand, we know that SAC² ⊆ TC² ⊆ CSPACE(O(log²{n}) , 2^{O(log{n})}). Our result thus shows an O(log log n) factor improvement on the free space needed to compute SAC², at the expense of a nearly-polynomial-sized catalytic tape. We also exhibit non-trivial register programs for matrix powering, which is a further step towards showing NC² ⊆ CL.
Cite as

Yaroslav Alekseev, Yuval Filmus, Ian Mertz, Alexander Smal, and Antoine Vinciguerra. Catalytic Computing and Register Programs Beyond Log-Depth. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{alekseev_et_al:LIPIcs.MFCS.2025.6,
  author =	{Alekseev, Yaroslav and Filmus, Yuval and Mertz, Ian and Smal, Alexander and Vinciguerra, Antoine},
  title =	{{Catalytic Computing and Register Programs Beyond Log-Depth}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{6:1--6:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.6},
  URN =		{urn:nbn:de:0030-drops-241136},
  doi =		{10.4230/LIPIcs.MFCS.2025.6},
  annote =	{Keywords: catalytic computing, circuit classes, polynomial method}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2025.23
Which Graph Motif Parameters Count?

Authors: Markus Bläser, Radu Curticapean, Julian Dörfler, and Christian Ikenmeyer

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract
For a fixed graph H, the function #Ind(H → ⋆) maps graphs G to the count of induced H-copies in G; this function obviously "counts something" in that it has a combinatorial interpretation. Linear combinations of such functions are called graph motif parameters and have recently received significant attention in counting complexity after a seminal paper by Curticapean, Dell and Marx (STOC'17). We show that, among linear combinations of functions #Ind(H → ⋆) involving only graphs H without isolated vertices, precisely those with positive integer coefficients maintain a combinatorial interpretation. It is important to note that graph motif parameters can be nonnegative for all inputs G, even when some coefficients are negative. Formally, we show that evaluating any graph motif parameter with a negative coefficient is impossible in an oracle variant of #P, where an implicit graph is accessed by oracle queries. Our proof follows the classification of the relativizing closure properties of #P by Hertrampf, Vollmer, and Wagner (SCT'95) and the framework developed by Ikenmeyer and Pak (STOC'22), but our application of the required Ramsey theorem turns out to be more subtle, as graphs do not have the required Ramsey property. Our techniques generalize from graphs to relational structures, including colored graphs. Vastly generalizing this, we introduce motif parameters over categories that count occurrences of sub-objects in the category. We then prove a general dichotomy theorem that characterizes which such parameters have a combinatorial interpretation. Using known results in Ramsey theory for categories, we obtain a dichotomy for motif parameters of finite vector spaces as well as parameter sets.
Cite as

Markus Bläser, Radu Curticapean, Julian Dörfler, and Christian Ikenmeyer. Which Graph Motif Parameters Count?. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 23:1-23:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{blaser_et_al:LIPIcs.MFCS.2025.23,
  author =	{Bl\"{a}ser, Markus and Curticapean, Radu and D\"{o}rfler, Julian and Ikenmeyer, Christian},
  title =	{{Which Graph Motif Parameters Count?}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{23:1--23:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.23},
  URN =		{urn:nbn:de:0030-drops-241307},
  doi =		{10.4230/LIPIcs.MFCS.2025.23},
  annote =	{Keywords: Graph motif parameters, Combinatorics, Combinatorial Interpretability}
}
Document
DOI: 10.4230/LIPIcs.SAT.2025.6
An Algebraic Approach to MaxCSP

Authors: Ilario Bonacina and Jordi Levy

Published in: LIPIcs, Volume 341, 28th International Conference on Theory and Applications of Satisfiability Testing (SAT 2025)

Abstract
Recently, there have been some attempts to base SAT and MaxSAT solvers on calculi beyond Resolution, even trying to solve SAT using MaxSAT proof systems. One of these directions was to perform MaxSAT sound inferences using polynomials over finite fields, extending the proof system Polynomial Calculus, which is known to be more powerful than Resolution. In this work, we extend the use of the Polynomial Calculus for optimization, showing its completeness over many-valued variables. This allows using a more direct and efficient encoding of CSP problems (e.g., k-Coloring) and solving the maximization version of the problem on such encoding (e.g., Max-k-Coloring).
Cite as

Ilario Bonacina and Jordi Levy. An Algebraic Approach to MaxCSP. In 28th International Conference on Theory and Applications of Satisfiability Testing (SAT 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 341, pp. 6:1-6:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bonacina_et_al:LIPIcs.SAT.2025.6,
  author =	{Bonacina, Ilario and Levy, Jordi},
  title =	{{An Algebraic Approach to MaxCSP}},
  booktitle =	{28th International Conference on Theory and Applications of Satisfiability Testing (SAT 2025)},
  pages =	{6:1--6:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-381-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{341},
  editor =	{Berg, Jeremias and Nordstr\"{o}m, Jakob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAT.2025.6},
  URN =		{urn:nbn:de:0030-drops-237407},
  doi =		{10.4230/LIPIcs.SAT.2025.6},
  annote =	{Keywords: MaxCSP, Polynomial Calculus, MaxSAT}
}
Document
DOI: 10.4230/LIPIcs.CCC.2025.28
Provably Total Functions in the Polynomial Hierarchy

Authors: Noah Fleming, Deniz Imrek, and Christophe Marciot

Published in: LIPIcs, Volume 339, 40th Computational Complexity Conference (CCC 2025)

Abstract
TFNP studies the complexity of total, verifiable search problems, and represents the first layer of the total function polynomial hierarchy (TFPH). Recently, problems in higher levels of the TFPH have gained significant attention, partly due to their close connection to circuit lower bounds. However, very little is known about the relationships between problems in levels of the hierarchy beyond TFNP. Connections to proof complexity have had an outsized impact on our understanding of the relationships between subclasses of TFNP in the black-box model. Subclasses are characterized by provability in certain proof systems, which has allowed for tools from proof complexity to be applied in order to separate TFNP problems. In this work we begin a systematic study of the relationship between subclasses of total search problems in the polynomial hierarchy and proof systems. We show that, akin to TFNP, reductions to a problem in TFΣ_d are equivalent to proofs of the formulas expressing the totality of the problems in some Σ_d-proof system. Having established this general correspondence, we examine important subclasses of TFPH. We show that reductions to the StrongAvoid problem are equivalent to proofs in a Σ₂-variant of the (unary) Sherali-Adams proof system. As well, we explore the TFPH classes which result from well-studied proof systems, introducing a number of new TFΣ₂ classes which characterize variants of DNF resolution, as well as TFΣ_d classes capturing levels of Σ_d-bounded-depth Frege.
Cite as

Noah Fleming, Deniz Imrek, and Christophe Marciot. Provably Total Functions in the Polynomial Hierarchy. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 28:1-28:40, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{fleming_et_al:LIPIcs.CCC.2025.28,
  author =	{Fleming, Noah and Imrek, Deniz and Marciot, Christophe},
  title =	{{Provably Total Functions in the Polynomial Hierarchy}},
  booktitle =	{40th Computational Complexity Conference (CCC 2025)},
  pages =	{28:1--28:40},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-379-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{339},
  editor =	{Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2025.28},
  URN =		{urn:nbn:de:0030-drops-237223},
  doi =		{10.4230/LIPIcs.CCC.2025.28},
  annote =	{Keywords: TFNP, TFPH, Proof Complxity, Characterizations}
}
Document
DOI: 10.4230/LIPIcs.CCC.2025.14
On the Automatability of Tree-Like k-DNF Resolution

Authors: Gaia Carenini and Susanna F. de Rezende

Published in: LIPIcs, Volume 339, 40th Computational Complexity Conference (CCC 2025)

Abstract
A proof system 𝒫 is said to be automatable in time f(N) if there exists an algorithm that given as input an unsatisfiable formula F outputs a refutation of F in the proof system 𝒫 in time f(N), where N is the size of the smallest 𝒫-refutation of F plus the size of F. Atserias and Bonet (ECCC 2002), observed that tree-like k-DNF resolution is automatable in time N^{c⋅klog N} for a universal constant c. We show that, under the randomized exponential-time hypothesis (rETH), this is tight up to a O(log k)-factor in the exponent, i.e., we prove that tree-like k-DNF resolution, for k at most logarithmic in the number of variables of F, is not automatable in time N^o((k/log k)⋅log N) unless rETH is false. Our proof builds on the non-automatability results for resolution by Atserias and Müller (FOCS 2019), for algebraic proof systems by de Rezende, Göös, Nordström, Pitassi, Robere and Sokolov (STOC 2021), and for tree-like resolution by de Rezende (LAGOS 2021).
Cite as

Gaia Carenini and Susanna F. de Rezende. On the Automatability of Tree-Like k-DNF Resolution. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 14:1-14:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{carenini_et_al:LIPIcs.CCC.2025.14,
  author =	{Carenini, Gaia and de Rezende, Susanna F.},
  title =	{{On the Automatability of Tree-Like k-DNF Resolution}},
  booktitle =	{40th Computational Complexity Conference (CCC 2025)},
  pages =	{14:1--14:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-379-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{339},
  editor =	{Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2025.14},
  URN =		{urn:nbn:de:0030-drops-237081},
  doi =		{10.4230/LIPIcs.CCC.2025.14},
  annote =	{Keywords: Proof Complexity, Tree-like k-DNF Resolution, Automatability}
}
Document
DOI: 10.4230/LIPIcs.SEA.2025.27
Sparsity-Driven Aggregation of Mixed Integer Programs

Authors: Liding Xu, Gioni Mexi, and Ksenia Bestuzheva

Published in: LIPIcs, Volume 338, 23rd International Symposium on Experimental Algorithms (SEA 2025)

Abstract
Cutting planes are crucial for the performance of branch-and-cut algorithms for solving mixed-integer programming (MIP) problems, and linear row aggregation has been successfully applied to better leverage the potential of several major families of MIP cutting planes. This paper formulates the problem of finding good quality aggregations as an 𝓁₀-norm minimization problem and employs a combination of the lasso method and iterative reweighting to efficiently find sparse solutions corresponding to good aggregations. A comparative analysis of the proposed algorithm and the state-of-the-art greedy heuristic approach is presented, showing that the greedy heuristic implements a stepwise selection algorithm for the 𝓁₀-norm minimization problem. Further, we present an example where our approach succeeds, whereas the standard heuristic fails to find an aggregation with desired properties. The algorithm is implemented within the constraint integer programming solver SCIP, and computational experiments on the MIPLIB 2017 benchmark show that although the algorithm leads to slowdowns on relatively "easier" instances, our aggregation approach decreases the mean running time on a subset of challenging instances and leads to smaller branch-and-bound trees.
Cite as

Liding Xu, Gioni Mexi, and Ksenia Bestuzheva. Sparsity-Driven Aggregation of Mixed Integer Programs. In 23rd International Symposium on Experimental Algorithms (SEA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 338, pp. 27:1-27:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{xu_et_al:LIPIcs.SEA.2025.27,
  author =	{Xu, Liding and Mexi, Gioni and Bestuzheva, Ksenia},
  title =	{{Sparsity-Driven Aggregation of Mixed Integer Programs}},
  booktitle =	{23rd International Symposium on Experimental Algorithms (SEA 2025)},
  pages =	{27:1--27:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-375-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{338},
  editor =	{Mutzel, Petra and Prezza, Nicola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2025.27},
  URN =		{urn:nbn:de:0030-drops-232652},
  doi =		{10.4230/LIPIcs.SEA.2025.27},
  annote =	{Keywords: mixed integer linear programming, cutting plane, valid inequality, separation, aggregation, projection, sparse optimization}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.49
Submodular Hypergraph Partitioning: Metric Relaxations and Fast Algorithms via an Improved Cut-Matching Game

Authors: Antares Chen, Lorenzo Orecchia, and Erasmo Tani

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
Despite there being significant work on developing spectral- [Chan et al., 2018; Lau et al., 2023; Kwok et al., 2022], and metric-embedding-based [Louis and Makarychev, 2016] approximation algorithms for hypergraph conductance, little is known regarding the approximability of other hypergraph partitioning objectives. This work proposes algorithms for a general model of hypergraph partitioning that unifies both undirected and directed versions of many well-studied partitioning objectives. The first contribution of this paper introduces polymatroidal cut functions, a large class of cut functions amenable to approximation algorithms via metric embeddings and routing multicommodity flows. We demonstrate a simple O(√{log n})-approximation, where n is the number of vertices in the hypergraph, for these problems by rounding relaxations to metrics of negative-type. The second contribution of this paper generalizes the cut-matching game framework of Khandekar et al. [Khandekar et al., 2007] to tackle polymatroidal cut functions. This yields an almost-linear time O(log n)-approximation algorithm for standard versions of undirected and directed hypergraph partitioning [Kwok et al., 2022]. A technical contribution of our construction is a novel cut-matching game, which greatly relaxes the set of allowed actions by the cut player and allows for the use of approximate s-t maximum flows by the matching player. We believe this to be of independent interest.
Cite as

Antares Chen, Lorenzo Orecchia, and Erasmo Tani. Submodular Hypergraph Partitioning: Metric Relaxations and Fast Algorithms via an Improved Cut-Matching Game. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 49:1-49:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chen_et_al:LIPIcs.ICALP.2025.49,
  author =	{Chen, Antares and Orecchia, Lorenzo and Tani, Erasmo},
  title =	{{Submodular Hypergraph Partitioning: Metric Relaxations and Fast Algorithms via an Improved Cut-Matching Game}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{49:1--49:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.49},
  URN =		{urn:nbn:de:0030-drops-234261},
  doi =		{10.4230/LIPIcs.ICALP.2025.49},
  annote =	{Keywords: Hypergraph Partitioning, Cut Improvement, Cut-Matching Game}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.115
On the Complexity of Hazard-Free Formulas

Authors: Leah London Arazi and Amir Shpilka

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
This paper studies the hazard-free formula complexity of Boolean functions. Our first result shows that unate functions are the only Boolean functions for which the monotone formula complexity of the hazard-derivative equals the hazard-free formula complexity of the function itself. Consequently, they are the only functions for which the hazard-derivative approach of Ikenmeyer et al. (J. ACM, 2019) yields optimal bounds. Our second result proves that the hazard-free formula complexity of a uniformly random Boolean function is at most 2^{(1+o(1))n}. Prior to this, no better upper bound than O(3ⁿ) was known. Notably, unlike in the general case of Boolean circuits and formulas, where the typical complexity is derived from that of the multiplexer function with n-bit selector, the hazard-free formula complexity of a random function is smaller than the optimal hazard-free formula for the multiplexer by an exponential factor in n. We provide two proofs of this fact. The first is direct, bounding the number of prime implicants of a random Boolean function and using this bound to construct a DNF of the claimed size. The second introduces a new and independently interesting result: a weak converse to the hazard-derivative lower bound method, which gives an upper bound on the hazard-free complexity of a function in terms of the monotone complexity of a subset of its hazard-derivatives. Additionally, we explore the hazard-free formula complexity of block composition of Boolean functions and obtain a result in the hazard-free setting that is analogous to a result of Karchmer, Raz, and Wigderson (Computational Complexity, 1995) in the monotone setting. We show that our result implies a stronger lower bound on the hazard-free formula depth of the block composition of the set covering function with the multiplexer function than the bound obtained via the hazard-derivative method.
Cite as

Leah London Arazi and Amir Shpilka. On the Complexity of Hazard-Free Formulas. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 115:1-115:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{londonarazi_et_al:LIPIcs.ICALP.2025.115,
  author =	{London Arazi, Leah and Shpilka, Amir},
  title =	{{On the Complexity of Hazard-Free Formulas}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{115:1--115:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.115},
  URN =		{urn:nbn:de:0030-drops-234920},
  doi =		{10.4230/LIPIcs.ICALP.2025.115},
  annote =	{Keywords: Hazard-free computation, Boolean formulas, monotone formulas, Karchmer-Wigderson games, communication complexity, lower bounds}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.34
On the Degree Automatability of Sum-Of-Squares Proofs

Authors: Alex Bortolotti, Monaldo Mastrolilli, and Luis Felipe Vargas

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
The Sum-of-Squares (SoS) hierarchy, also known as Lasserre hierarchy, has emerged as a promising tool in optimization. However, it remains unclear whether fixed-degree SoS proofs can be automated [O'Donnell (2017)]. Indeed, there are examples of polynomial systems with bounded coefficients that admit low-degree SoS proofs, but these proofs necessarily involve numbers with an exponential number of bits, implying that low-degree SoS proofs cannot always be found efficiently. A sufficient condition derived from the Nullstellensatz proof system [Raghavendra and Weitz (2017)] identifies cases where bit complexity issues can be circumvented. One of the main problems left open by Raghavendra and Weitz is proving any result for refutations, as their condition applies only to polynomial systems with a large set of solutions. In this work, we broaden the class of polynomial systems for which degree-d SoS proofs can be automated. To achieve this, we develop a new criterion and we demonstrate how our criterion applies to polynomial systems beyond the scope of Raghavendra and Weitz’s result. In particular, we establish a separation for instances arising from Constraint Satisfaction Problems (CSPs). Moreover, our result extends to refutations, establishing that polynomial-time refutation is possible for broad classes of polynomial time solvable constraint problems, highlighting a first advancement in this area.
Cite as

Alex Bortolotti, Monaldo Mastrolilli, and Luis Felipe Vargas. On the Degree Automatability of Sum-Of-Squares Proofs. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 34:1-34:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bortolotti_et_al:LIPIcs.ICALP.2025.34,
  author =	{Bortolotti, Alex and Mastrolilli, Monaldo and Vargas, Luis Felipe},
  title =	{{On the Degree Automatability of Sum-Of-Squares Proofs}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{34:1--34:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.34},
  URN =		{urn:nbn:de:0030-drops-234110},
  doi =		{10.4230/LIPIcs.ICALP.2025.34},
  annote =	{Keywords: Sum of squares, Polynomial calculus, Polynomial ideal membership, Polymorphisms, Gr\"{o}bner basis theory, Constraint satisfaction problems, Proof complexity}
}
Document
DOI: 10.4230/LIPIcs.STACS.2025.26
Toward Better Depth Lower Bounds: Strong Composition of XOR and a Random Function

Authors: Nikolai Chukhin, Alexander S. Kulikov, and Ivan Mihajlin

Published in: LIPIcs, Volume 327, 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)

Abstract
Proving formula depth lower bounds is a fundamental challenge in complexity theory, with the strongest known bound of (3 - o(1))log n established by Håstad over 25 years ago. The Karchmer-Raz-Wigderson (KRW) conjecture offers a promising approach to advance these bounds and separate P from NC¹. It suggests that the depth complexity of a function composition f ⋄ g approximates the sum of the depth complexities of f and g. The Karchmer-Wigderson (KW) relation framework translates formula depth into communication complexity, restating the KRW conjecture as CC(KW_f ⋄ KW_g) ≈ CC(KW_f) + CC(KW_g). Prior work has confirmed the conjecture under various relaxations, often replacing one or both KW relations with the universal relation or constraining the communication game through strong composition. In this paper, we examine the strong composition KW_XOR ⊛ KW_f of the parity function and a random Boolean function f. We prove that with probability 1-o(1), any protocol solving this composition requires at least n^{3 - o(1)} leaves. This result establishes a depth lower bound of (3 - o(1))log n, matching Håstad’s bound, but is applicable to a broader class of inner functions, even when the outer function is simple. Though bounds for the strong composition do not translate directly to formula depth bounds, they usually help to analyze the standard composition (of the corresponding two functions) which is directly related to formula depth. Our proof utilizes formal complexity measures. First, we apply Khrapchenko’s method to show that numerous instances of f remain unsolved after several communication steps. Subsequently, we transition to a different formal complexity measure to demonstrate that the remaining communication problem is at least as hard as KW_OR ⊛ KW_f. This hybrid approach not only achieves the desired lower bound, but also introduces a novel technique for analyzing formula depth, potentially informing future research in complexity theory.
Cite as

Nikolai Chukhin, Alexander S. Kulikov, and Ivan Mihajlin. Toward Better Depth Lower Bounds: Strong Composition of XOR and a Random Function. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 26:1-26:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chukhin_et_al:LIPIcs.STACS.2025.26,
  author =	{Chukhin, Nikolai and Kulikov, Alexander S. and Mihajlin, Ivan},
  title =	{{Toward Better Depth Lower Bounds: Strong Composition of XOR and a Random Function}},
  booktitle =	{42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)},
  pages =	{26:1--26:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-365-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{327},
  editor =	{Beyersdorff, Olaf and Pilipczuk, Micha{\l} and Pimentel, Elaine and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2025.26},
  URN =		{urn:nbn:de:0030-drops-228513},
  doi =		{10.4230/LIPIcs.STACS.2025.26},
  annote =	{Keywords: complexity, formula complexity, lower bounds, Boolean functions, depth}
}
Document
DOI: 10.4230/LIPIcs.CCC.2018.15
Hardness of Function Composition for Semantic Read once Branching Programs

Authors: Jeff Edmonds, Venkatesh Medabalimi, and Toniann Pitassi

Published in: LIPIcs, Volume 102, 33rd Computational Complexity Conference (CCC 2018)

Abstract
In this work, we study time/space trade-offs for function composition. We prove asymptotically optimal lower bounds for function composition in the setting of nondeterministic read once branching programs, for the syntactic model as well as the stronger semantic model of read-once nondeterministic computation. We prove that such branching programs for solving the tree evaluation problem over an alphabet of size k requires size roughly k^{Omega(h)}, i.e space Omega(h log k). Our lower bound nearly matches the natural upper bound which follows the best strategy for black-white pebbling the underlying tree. While previous super-polynomial lower bounds have been proven for read-once nondeterministic branching programs (for both the syntactic as well as the semantic models), we give the first lower bounds for iterated function composition, and in these models our lower bounds are near optimal.
Cite as

Jeff Edmonds, Venkatesh Medabalimi, and Toniann Pitassi. Hardness of Function Composition for Semantic Read once Branching Programs. In 33rd Computational Complexity Conference (CCC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 102, pp. 15:1-15:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{edmonds_et_al:LIPIcs.CCC.2018.15,
  author =	{Edmonds, Jeff and Medabalimi, Venkatesh and Pitassi, Toniann},
  title =	{{Hardness of Function Composition for Semantic Read once Branching Programs}},
  booktitle =	{33rd Computational Complexity Conference (CCC 2018)},
  pages =	{15:1--15:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-069-9},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{102},
  editor =	{Servedio, Rocco A.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2018.15},
  URN =		{urn:nbn:de:0030-drops-88747},
  doi =		{10.4230/LIPIcs.CCC.2018.15},
  annote =	{Keywords: Branching Programs, Function Composition, Time-Space Tradeoffs, Semantic Read Once, Tree Evaluation Problem}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2016.36
Lower Bounds for Nondeterministic Semantic Read-Once Branching Programs

Authors: Stephen Cook, Jeff Edmonds, Venkatesh Medabalimi, and Toniann Pitassi

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

Abstract
We prove exponential lower bounds on the size of semantic read-once 3-ary nondeterministic branching programs. Prior to our result the best that was known was for D-ary branching programs with |D| >= 2^{13}.
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Stephen Cook, Jeff Edmonds, Venkatesh Medabalimi, and Toniann Pitassi. Lower Bounds for Nondeterministic Semantic Read-Once Branching Programs. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 36:1-36:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{cook_et_al:LIPIcs.ICALP.2016.36,
  author =	{Cook, Stephen and Edmonds, Jeff and Medabalimi, Venkatesh and Pitassi, Toniann},
  title =	{{Lower Bounds for Nondeterministic Semantic Read-Once Branching Programs}},
  booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
  pages =	{36:1--36:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-013-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{55},
  editor =	{Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.36},
  URN =		{urn:nbn:de:0030-drops-63166},
  doi =		{10.4230/LIPIcs.ICALP.2016.36},
  annote =	{Keywords: Branching Programs, Semantic, Non-deterministic, Lower Bounds}
}
Document
DOI: 10.4230/DagSemProc.10071.3
10071 Open Problems – Scheduling

Authors: Jim Anderson, Björn Andersson, Yossi Azar, Nikhil Bansal, Enrico Bini, Marek Chrobak, José Correa, Liliana Cucu-Grosjean, Rob Davis, Arvind Easwaran, Jeff Edmonds, Shelby Funk, Sathish Gopalakrishnan, Han Hoogeveen, Claire Mathieu, Nicole Megow, Seffi Naor, Kirk Pruhs, Maurice Queyranne, Adi Rosén, Nicolas Schabanel, Jiří Sgall, René Sitters, Sebastian Stiller, Marc Uetz, Tjark Vredeveld, and Gerhard J. Woeginger

Published in: Dagstuhl Seminar Proceedings, Volume 10071, Scheduling (2010)

Abstract
Collection of the open problems presented at the scheduling seminar.
Cite as

Jim Anderson, Björn Andersson, Yossi Azar, Nikhil Bansal, Enrico Bini, Marek Chrobak, José Correa, Liliana Cucu-Grosjean, Rob Davis, Arvind Easwaran, Jeff Edmonds, Shelby Funk, Sathish Gopalakrishnan, Han Hoogeveen, Claire Mathieu, Nicole Megow, Seffi Naor, Kirk Pruhs, Maurice Queyranne, Adi Rosén, Nicolas Schabanel, Jiří Sgall, René Sitters, Sebastian Stiller, Marc Uetz, Tjark Vredeveld, and Gerhard J. Woeginger. 10071 Open Problems – Scheduling. In Scheduling. Dagstuhl Seminar Proceedings, Volume 10071, pp. 1-24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{anderson_et_al:DagSemProc.10071.3,
  author =	{Anderson, Jim and Andersson, Bj\"{o}rn and Azar, Yossi and Bansal, Nikhil and Bini, Enrico and Chrobak, Marek and Correa, Jos\'{e} and Cucu-Grosjean, Liliana and Davis, Rob and Easwaran, Arvind and Edmonds, Jeff and Funk, Shelby and Gopalakrishnan, Sathish and Hoogeveen, Han and Mathieu, Claire and Megow, Nicole and Naor, Seffi and Pruhs, Kirk and Queyranne, Maurice and Ros\'{e}n, Adi and Schabanel, Nicolas and Sgall, Ji\v{r}{\'\i} and Sitters, Ren\'{e} and Stiller, Sebastian and Uetz, Marc and Vredeveld, Tjark and Woeginger, Gerhard J.},
  title =	{{10071 Open Problems – Scheduling}},
  booktitle =	{Scheduling},
  pages =	{1--24},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2010},
  volume =	{10071},
  editor =	{Susanne Albers and Sanjoy K. Baruah and Rolf H. M\"{o}hring and Kirk Pruhs},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.10071.3},
  URN =		{urn:nbn:de:0030-drops-25367},
  doi =		{10.4230/DagSemProc.10071.3},
  annote =	{Keywords: Open problems, scheduling}
}
Document
DOI: 10.4230/DagSemProc.10071.6
Every Deterministic Nonclairvoyant Scheduler has a Suboptimal Load Threshold

Authors: Jeff Edmonds

Published in: Dagstuhl Seminar Proceedings, Volume 10071, Scheduling (2010)

Abstract
The goal is to prove a surprising lower bound for resource augmented nonclairvoyant algorithms for scheduling jobs with sublinear nondecreasing speed-up curves on multiple processors with the objective of average response time. Edmonds and Pruhs in SODA09 prove that for every $\e > 0$, there is an algorithm $\alg_{\e}$ that is $(1\!+\!\epsilon)$-speed $O({1 \over \e2})$-competitive. A problem, however, is that this algorithm $\alg_{\e}$ depends on $\e$. The goal is to prove that every fixed deterministic nonclairvoyant algorithm has a suboptimal speed threshold, namely for every (graceful) algorithm $\alg$, there is a threshold $1\!+\!\beta_{\alg}$ that is $\beta_{\alg} > 0$ away from being optimal such that the algorithm is $\Omega({1 \over \e \beta_{\alg}})$ competitive with speed $(1 \!+\! \beta_{\alg}) \!+\! \e$ and is $\omega(1)$ competitive with speed $1 \!+\! \beta_{\alg}$. I have worked very hard on it and have felt that I was close. The proof technique is to use Brouwer's fixed point theorem to break the cycle of needing to know which input will be given before one can know what the algorithm will do and needing to know what the algorithm will do before one can know which input to give. Every thing I have can be found at
Cite as

Jeff Edmonds. Every Deterministic Nonclairvoyant Scheduler has a Suboptimal Load Threshold. In Scheduling. Dagstuhl Seminar Proceedings, Volume 10071, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{edmonds:DagSemProc.10071.6,
  author =	{Edmonds, Jeff},
  title =	{{Every Deterministic Nonclairvoyant Scheduler has a Suboptimal Load Threshold}},
  booktitle =	{Scheduling},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2010},
  volume =	{10071},
  editor =	{Susanne Albers and Sanjoy K. Baruah and Rolf H. M\"{o}hring and Kirk Pruhs},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.10071.6},
  URN =		{urn:nbn:de:0030-drops-25447},
  doi =		{10.4230/DagSemProc.10071.6},
  annote =	{Keywords: Scheduling}
}
Document
DOI: 10.4230/DagSemProc.10071.12
Scalably Scheduling Processes with Arbitrary Speedup Curves

Authors: Jeff Edmonds and Kirk Pruhs

Published in: Dagstuhl Seminar Proceedings, Volume 10071, Scheduling (2010)

Abstract
We give a scalable ((1+\epsilon)-speed O(1)-competitive) nonclairvoyant algorithm for scheduling jobs with sublinear nondecreasing speed-up curves on multiple processors with the objective of average response time.
Cite as

Jeff Edmonds and Kirk Pruhs. Scalably Scheduling Processes with Arbitrary Speedup Curves. In Scheduling. Dagstuhl Seminar Proceedings, Volume 10071, pp. 1-9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{edmonds_et_al:DagSemProc.10071.12,
  author =	{Edmonds, Jeff and Pruhs, Kirk},
  title =	{{Scalably Scheduling Processes with Arbitrary Speedup Curves}},
  booktitle =	{Scheduling},
  pages =	{1--9},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2010},
  volume =	{10071},
  editor =	{Susanne Albers and Sanjoy K. Baruah and Rolf H. M\"{o}hring and Kirk Pruhs},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.10071.12},
  URN =		{urn:nbn:de:0030-drops-25463},
  doi =		{10.4230/DagSemProc.10071.12},
  annote =	{Keywords: Scheduling}
}
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