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Documents authored by Rudolph, Dorian

Document
DOI: 10.4230/LIPIcs.ITCS.2026.7
An Unholy Trinity: TFNP, Polynomial Systems, and the Quantum Satisfiability Problem

Authors: Marco Aldi, Sevag Gharibian, and Dorian Rudolph

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
The theory of Total Function NP (TFNP) and its subclasses says that, even if one is promised an efficiently verifiable proof exists for a problem, finding this proof can be intractable. Despite the success of the theory at showing intractability of problems such as computing Brouwer fixed points and Nash equilibria, subclasses of TFNP remain arguably few and far between. In this work, we define two new subclasses of TFNP borne of the study of complex polynomial systems: Multi-homogeneous Systems (MHS) and Sparse Fundamental Theorem of Algebra (SFTA). The first of these is based on Bézout’s theorem from algebraic geometry, marking the first TFNP subclass based on an algebraic geometric principle. At the heart of our study is the computational problem known as Quantum SAT (QSAT) with a System of Distinct Representatives (SDR), first studied by [Laumann, Läuchli, Moessner, Scardicchio, and Sondhi 2010]. Among other results, we show that QSAT with SDR is MHS-complete, thus giving not only the first link between quantum complexity theory and TFNP, but also the first TFNP problem whose classical variant (SAT with SDR) is easy but whose quantum variant is hard. We also show how to embed the roots of a sparse, high-degree, univariate polynomial into QSAT with SDR, obtaining that SFTA is contained in a zero-error version of MHS. We conjecture this construction also works in the low-error setting, which would imply SFTA ⊆ MHS.
Cite as

Marco Aldi, Sevag Gharibian, and Dorian Rudolph. An Unholy Trinity: TFNP, Polynomial Systems, and the Quantum Satisfiability Problem. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 7:1-7:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{aldi_et_al:LIPIcs.ITCS.2026.7,
  author =	{Aldi, Marco and Gharibian, Sevag and Rudolph, Dorian},
  title =	{{An Unholy Trinity: TFNP, Polynomial Systems, and the Quantum Satisfiability Problem}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{7:1--7:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.7},
  URN =		{urn:nbn:de:0030-drops-252946},
  doi =		{10.4230/LIPIcs.ITCS.2026.7},
  annote =	{Keywords: quantum complexity theory, Quantum Merlin Arthur (QMA), Quantum Satisfiability Problem (QSAT), total function NP (TFNP)}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.83
The Pure-State Consistency of Local Density Matrices Problem: In PSPACE and Complete for a Class Between QMA and QMA(2)

Authors: Jonas Kamminga and Dorian Rudolph

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
In this work we investigate the computational complexity of the pure consistency of local density matrices (PureCLDM) and pure N-representability (Pure-N-Representability; analog of PureCLDM for bosonic or fermionic systems) problems. In these problems the input is a set of reduced density matrices and the task is to determine whether there exists a global pure state consistent with these reduced density matrices. While mixed CLDM, i.e. where the global state can be mixed, was proven to be QMA-complete by Broadbent and Grilo [JoC 2022], almost nothing was known about the complexity of the pure version. Before our work the best upper and lower bounds were QMA(2) and QMA. Our contribution to the understanding of these problems is twofold. Firstly, we define a pure state analogue of the complexity class QMA^+ of Aharanov and Regev [FOCS 2003], which we call PureSuperQMA. We prove that both pure-N-Representability and PureCLDM are complete for this new class. Along the way we supplement Broadbent and Grilo by proving hardness for 2-qubit reduced density matrices and showing that mixed N-Representability is QMA-complete. Secondly, we improve the upper bound on PureCLDM. Using methods from algebraic geometry, we prove that PureSuperQMA ⊆ PSPACE. Our methods, and the PSPACE upper bound, are also valid for PureCLDM with exponential or even perfect precision, hence precisePureCLDM is not preciseQMA(2) = NEXP-complete, unless PSPACE = NEXP. We view this as evidence for a negative answer to the longstanding open question whether PureCLDM is QMA(2)-complete. The techniques we develop for our PSPACE upper bound are quite general. We are able to use them for various applications: from proving PSPACE upper bounds on other quantum problems to giving an efficient parallel (NC) algorithm for (non-convex) quadratically constrained quadratic programs with few constraints.
Cite as

Jonas Kamminga and Dorian Rudolph. The Pure-State Consistency of Local Density Matrices Problem: In PSPACE and Complete for a Class Between QMA and QMA(2). In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 83:1-83:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{kamminga_et_al:LIPIcs.ITCS.2026.83,
  author =	{Kamminga, Jonas and Rudolph, Dorian},
  title =	{{The Pure-State Consistency of Local Density Matrices Problem: In PSPACE and Complete for a Class Between QMA and QMA(2)}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{83:1--83:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.83},
  URN =		{urn:nbn:de:0030-drops-253701},
  doi =		{10.4230/LIPIcs.ITCS.2026.83},
  annote =	{Keywords: Quantum Complexity Theory, PSPACE, QMA(2), Consistency of Local Density Matrices, Polynomial Optimization}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2025.85
Quantum 2-SAT on Low Dimensional Systems Is QMAsubscript{1}-Complete: Direct Embeddings and Black-Box Simulation

Authors: Dorian Rudolph, Sevag Gharibian, and Daniel Nagaj

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract
Despite the fundamental role the Quantum Satisfiability (QSAT) problem has played in quantum complexity theory, a central question remains open: At which local dimension does the complexity of QSAT transition from "easy" to "hard"? Here, we study QSAT with each constraint acting on a d_A-dimensional and d_B-dimensional qudit pair, denoted (d_A×d_B)-QSAT. Our first main result shows that, surprisingly, QSAT on qubits can remain QMA_1-hard, in that (2×5)-QSAT is QMA_1-complete. (QMA_1 is a quantum analogue of MA with perfect completeness.) In contrast, (2×2)-QSAT (i.e. Quantum 2-SAT on qubits) is well-known to be poly-time solvable [Bravyi, 2006]. Our second main result proves that (3×d)-QSAT on the 1D line with d ∈ O(1) is also QMA_1-hard. Finally, we initiate the study of (2×d)-QSAT on the 1D line by giving a frustration-free 1D Hamiltonian with a unique, entangled ground state. As implied by our title, our first result uses a direct embedding: We combine a novel clock construction with the 2D circuit-to-Hamiltonian construction of [Gosset and Nagaj, 2013]. Of note is a new simplified and analytic proof for the latter (as opposed to a partially numeric proof in [GN13]). This exploits Unitary Labelled Graphs [Bausch, Cubitt, Ozols, 2017] together with a new "Nullspace Connection Lemma", allowing us to break low energy analyses into small patches of projectors, and to improve the soundness analysis of [GN13] from Ω(1/T⁶) to Ω(1/T²), for T the number of gates. Our second result goes via black-box reduction: Given an arbitrary 1D Hamiltonian H on d'-dimensional qudits, we show how to embed it into an effective 1D (3×d)-QSAT instance, for d ∈ O(1). Our approach may be viewed as a weaker notion of "analog simulation" (à la [Bravyi, Hastings 2017], [Cubitt, Montanaro, Piddock 2018]). As far as we are aware, this gives the first "black-box simulation"-based QMA_1-hardness result.
Cite as

Dorian Rudolph, Sevag Gharibian, and Daniel Nagaj. Quantum 2-SAT on Low Dimensional Systems Is QMAsubscript{1}-Complete: Direct Embeddings and Black-Box Simulation. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 85:1-85:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{rudolph_et_al:LIPIcs.ITCS.2025.85,
  author =	{Rudolph, Dorian and Gharibian, Sevag and Nagaj, Daniel},
  title =	{{Quantum 2-SAT on Low Dimensional Systems Is QMAsubscript\{1\}-Complete: Direct Embeddings and Black-Box Simulation}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{85:1--85:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.85},
  URN =		{urn:nbn:de:0030-drops-227139},
  doi =		{10.4230/LIPIcs.ITCS.2025.85},
  annote =	{Keywords: quantum complexity theory, Quantum Merlin Arthur (QMA), Quantum Satisfiability Problem (QSAT), Hamiltonian simulation}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2024.7
Quantum Polynomial Hierarchies: Karp-Lipton, Error Reduction, and Lower Bounds

Authors: Avantika Agarwal, Sevag Gharibian, Venkata Koppula, and Dorian Rudolph

Published in: LIPIcs, Volume 306, 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)

Abstract
The Polynomial-Time Hierarchy (PH) is a staple of classical complexity theory, with applications spanning randomized computation to circuit lower bounds to "quantum advantage" analyses for near-term quantum computers. Quantumly, however, despite the fact that at least four definitions of quantum PH exist, it has been challenging to prove analogues for these of even basic facts from PH. This work studies three quantum-verifier based generalizations of PH, two of which are from [Gharibian, Santha, Sikora, Sundaram, Yirka, 2022] and use classical strings (QCPH) and quantum mixed states (QPH) as proofs, and one of which is new to this work, utilizing quantum pure states (QPHpure) as proofs. We first resolve several open problems from [GSSSY22], including a collapse theorem and a Karp-Lipton theorem for QCPH. Then, for our new class QPHpure, we show one-sided error reduction QPHpure, as well as the first bounds relating these quantum variants of PH, namely QCPH ⊆ QPHpure ⊆ EXP^PP.
Cite as

Avantika Agarwal, Sevag Gharibian, Venkata Koppula, and Dorian Rudolph. Quantum Polynomial Hierarchies: Karp-Lipton, Error Reduction, and Lower Bounds. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 7:1-7:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{agarwal_et_al:LIPIcs.MFCS.2024.7,
  author =	{Agarwal, Avantika and Gharibian, Sevag and Koppula, Venkata and Rudolph, Dorian},
  title =	{{Quantum Polynomial Hierarchies: Karp-Lipton, Error Reduction, and Lower Bounds}},
  booktitle =	{49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024)},
  pages =	{7:1--7:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-335-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{306},
  editor =	{Kr\'{a}lovi\v{c}, Rastislav and Ku\v{c}era, Anton{\'\i}n},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2024.7},
  URN =		{urn:nbn:de:0030-drops-205632},
  doi =		{10.4230/LIPIcs.MFCS.2024.7},
  annote =	{Keywords: Quantum complexity, polynomial hierarchy}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2023.53
Quantum Space, Ground Space Traversal, and How to Embed Multi-Prover Interactive Proofs into Unentanglement

Authors: Sevag Gharibian and Dorian Rudolph

Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Abstract
A celebrated result in classical complexity theory is Savitch’s theorem, which states that non-deterministic polynomial-space computations (NPSPACE) can be simulated by deterministic poly-space computations (PSPACE). In this work, we initiate the study of a quantum analogue of NPSPACE, denoted Streaming-QCMASPACE (SQCMASPACE), in which an exponentially long classical proof is streamed to a poly-space quantum verifier. We first show that a quantum analogue of Savitch’s theorem is unlikely to hold, in that SQCMASPACE = NEXP. For completeness, we also introduce the companion class Streaming-QMASPACE (SQMASPACE) with an exponentially long streamed quantum proof, and show SQMASPACE = QMAEXP (the quantum analogue of NEXP). Our primary focus, however, is on the study of exponentially long streaming classical proofs, where we next show the following two main results. The first result shows that, in strong contrast to the classical setting, the solution space of a quantum constraint satisfaction problem (i.e. a local Hamiltonian) is always connected when exponentially long proofs are permitted. For this, we show how to simulate any Lipschitz continuous path on the unit hypersphere via a sequence of local unitary gates, at the expense of blowing up the circuit size. This shows that quantum error-correcting codes can be unable to detect one codeword erroneously evolving to another if the evolution happens sufficiently slowly, and answers an open question of [Gharibian, Sikora, ICALP 2015] regarding the Ground State Connectivity problem. Our second main result is that any SQCMASPACE computation can be embedded into "unentanglement", i.e. into a quantum constraint satisfaction problem with unentangled provers. Formally, we show how to embed SQCMASPACE into the Sparse Separable Hamiltonian problem of [Chailloux, Sattath, CCC 2012] (QMA(2)-complete for 1/poly promise gap), at the expense of scaling the promise gap with the streamed proof size. As a corollary, we obtain the first systematic construction for obtaining QMA(2)-type upper bounds on arbitrary multi-prover interactive proof systems, where the QMA(2) promise gap scales exponentially with the number of bits of communication in the interactive proof. Our construction uses a new technique for exploiting unentanglement to simulate quadratic Boolean functions, which in some sense allows history states to encode the future.
Cite as

Sevag Gharibian and Dorian Rudolph. Quantum Space, Ground Space Traversal, and How to Embed Multi-Prover Interactive Proofs into Unentanglement. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 53:1-53:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{gharibian_et_al:LIPIcs.ITCS.2023.53,
  author =	{Gharibian, Sevag and Rudolph, Dorian},
  title =	{{Quantum Space, Ground Space Traversal, and How to Embed Multi-Prover Interactive Proofs into Unentanglement}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{53:1--53:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.53},
  URN =		{urn:nbn:de:0030-drops-175564},
  doi =		{10.4230/LIPIcs.ITCS.2023.53},
  annote =	{Keywords: quantum complexity theory, Quantum Merlin Arthur (QMA), QMA(2), ground state connectivity (GSCON), quantum error correction}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2022.75
On Polynomially Many Queries to NP or QMA Oracles

Authors: Sevag Gharibian and Dorian Rudolph

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Abstract
We study the complexity of problems solvable in deterministic polynomial time with access to an NP or Quantum Merlin-Arthur (QMA)-oracle, such as P^NP and P^QMA, respectively. The former allows one to classify problems more finely than the Polynomial-Time Hierarchy (PH), whereas the latter characterizes physically motivated problems such as Approximate Simulation (APX-SIM) [Ambainis, CCC 2014]. In this area, a central role has been played by the classes P^NP[log] and P^QMA[log], defined identically to P^NP and P^QMA, except that only logarithmically many oracle queries are allowed. Here, [Gottlob, FOCS 1993] showed that if the adaptive queries made by a P^NP machine have a "query graph" which is a tree, then this computation can be simulated in P^NP[log]. In this work, we first show that for any verification class C ∈ {NP, MA, QCMA, QMA, QMA(2), NEXP, QMA_exp}, any P^C machine with a query graph of "separator number" s can be simulated using deterministic time exp(slog n) and slog n queries to a C-oracle. When s ∈ O(1) (which includes the case of O(1)-treewidth, and thus also of trees), this gives an upper bound of P^C[log], and when s ∈ O(log^k(n)), this yields bound QP^{C[log^{k+1}]} (QP meaning quasi-polynomial time). We next show how to combine Gottlob’s "admissible-weighting function" framework with the "flag-qubit" framework of [Watson, Bausch, Gharibian, 2020], obtaining a unified approach for embedding P^C computations directly into APX-SIM instances in a black-box fashion. Finally, we formalize a simple no-go statement about polynomials (c.f. [Krentel, STOC 1986]): Given a multi-linear polynomial p specified via an arithmetic circuit, if one can "weakly compress" p so that its optimal value requires m bits to represent, then P^NP can be decided with only m queries to an NP-oracle.
Cite as

Sevag Gharibian and Dorian Rudolph. On Polynomially Many Queries to NP or QMA Oracles. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 75:1-75:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{gharibian_et_al:LIPIcs.ITCS.2022.75,
  author =	{Gharibian, Sevag and Rudolph, Dorian},
  title =	{{On Polynomially Many Queries to NP or QMA Oracles}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{75:1--75:27},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.75},
  URN =		{urn:nbn:de:0030-drops-156717},
  doi =		{10.4230/LIPIcs.ITCS.2022.75},
  annote =	{Keywords: admissible weighting function, oracle complexity class, quantum complexity theory, Quantum Merlin Arthur (QMA), simulation of local measurement}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2018.52
Shape Recognition by a Finite Automaton Robot

Authors: Robert Gmyr, Kristian Hinnenthal, Irina Kostitsyna, Fabian Kuhn, Dorian Rudolph, and Christian Scheideler

Published in: LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

Abstract
Motivated by the problem of shape recognition by nanoscale computing agents, we investigate the problem of detecting the geometric shape of a structure composed of hexagonal tiles by a finite-state automaton robot. In particular, in this paper we consider the question of recognizing whether the tiles are assembled into a parallelogram whose longer side has length l = f(h), for a given function f(*), where h is the length of the shorter side. To determine the computational power of the finite-state automaton robot, we identify functions that can or cannot be decided when the robot is given a certain number of pebbles. We show that the robot can decide whether l = ah+b for constant integers a and b without any pebbles, but cannot detect whether l = f(h) for any function f(x) = omega(x). For a robot with a single pebble, we present an algorithm to decide whether l = p(h) for a given polynomial p(*) of constant degree. We contrast this result by showing that, for any constant k, any function f(x) = omega(x^(6k + 2)) cannot be decided by a robot with k states and a single pebble. We further present exponential functions that can be decided using two pebbles. Finally, we present a family of functions f_n(*) such that the robot needs more than n pebbles to decide whether l = f_n(h).
Cite as

Robert Gmyr, Kristian Hinnenthal, Irina Kostitsyna, Fabian Kuhn, Dorian Rudolph, and Christian Scheideler. Shape Recognition by a Finite Automaton Robot. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 52:1-52:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{gmyr_et_al:LIPIcs.MFCS.2018.52,
  author =	{Gmyr, Robert and Hinnenthal, Kristian and Kostitsyna, Irina and Kuhn, Fabian and Rudolph, Dorian and Scheideler, Christian},
  title =	{{Shape Recognition by a Finite Automaton Robot}},
  booktitle =	{43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)},
  pages =	{52:1--52:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-086-6},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{117},
  editor =	{Potapov, Igor and Spirakis, Paul and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.52},
  URN =		{urn:nbn:de:0030-drops-96347},
  doi =		{10.4230/LIPIcs.MFCS.2018.52},
  annote =	{Keywords: finite automata, shape recognition, computational geometry}
}
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