Jakob Nordström / MIAO seminars

MIAO Seminars

The MIAO seminars are what we call the series of seminars arranged by the Mathematical Insights into Algorithms for Optimization (MIAO) research group at the University of Copenhagen and Lund University.

Our seminars often (but far from always) consist of two parts: first a regular talk, and after this a ca-1-hour informal, technical presentation with discussions. The intention is that all listeners will first get an overview of some exciting research results, and then people who find the topic particularly interesting will get an opportunity to study the same material more in-depth.

To elaborate a bit, for seminars conducted in this format the first part is just a standard, listener-friendly seminar of 50-55 minutes (including time for questions). The audience is assumed to have a general knowledge of computer science and/or mathematics, but is not expected to have any detailed insights into the specific area covered by the talk. This first part is intended to be self-contained, since most attendees tend to leave after the first part.

After a ca-10-minute break it is time for the second part, when those who are interested reconvene for a more in-depth technical presentation. During this second part the goal is to "open up the hood", so that for a theory talk we look at the formal definitions and prove at least some of the key ingredients in the results including all (or at least some of) the gory technical details glossed over in a polished seminar presentation. For a more applied talk, we might go into questions of implementation or discuss detailed experimental results. Questions and discussions are strongly encouraged. However, for those who feel that the first 50-55-minute regular seminar was enough for today, it is perfectly fine to just discretely drop out during the break. No excuses needed; no questions asked.

These seminars typically have a low-key, informal, atmosphere, but they are open to the public and are announced beforehand on our seminar mailing lists (with two different lists for more theoretically oriented or more applied seminars, respectively). In case you want to receive the announcements, please just let us know by sending an e-mail message to jn@di.ku.dk specifying which seminar mailing list you wish to be added to (theoretical, applied, or both). We also add all public seminars to the MIAO seminar calendar.

In response to multiple requests, since the spring of 2021 we have started recording most seminars. The recordings are posted on the MIAO Research YouTube channel.

All seminar times below are in the Central European Time Zone (with daylight saving time observed from the last Sunday in March to the last Sunday in October).

Upcoming seminars

  • Friday Sep 30 at 14:00 on Zoom
    Towards an algorithmic theory of proof complexity
    (Albert Atserias, Universitat Politècnica de Catalunya)

    A possibly unexpected by-product of the mathematical study of the lengths of proofs, as is done in the field of propositional proof complexity, is, I claim, that it may lead to new algorithmic insights. To explain this, I will first recall the origins of proof complexity as a field. Then I will explain why our current understanding of certain proof systems could lead to new algorithms. The key to this is that, for several proof systems of practical use, the class of tautologies with low-complexity proofs comes with a tight semantic characterization. Concretely, the characterization states that a tautology fails to have a low-complexity proof precisely when it is locally falsifiable, in a precise technical sense of the term. One immediate derivative of this is that, for those proof systems that admit such semantic characterizations, the statement that a formula has a low-complexity proofs has: (1) low-complexity certificates when true, and (2) low-complexity refutations when false. We illustrate the point by discussing the recently discovered subexponential-time algorithm that distinguishes the graphs that are 3-colorable from those that are not even n^eps-colorable in time exp(n^{1-2eps}), which beats all previously known approaches.

  • Tuesday Oct 4 at 14:00 at University of Copenhagen and on Zoom
    Matrix multiplication and polynomial identity testing
    (Robert Andrews, University of Illinois Urbana-Champaign)

    Determining the complexity of matrix multiplication is a fundamental problem of theoretical computer science. It is popularly conjectured that ω, the matrix multiplication exponent, equals 2. If true, this conjecture would yield fast algorithms for a wide array of problems in linear algebra and beyond. But what if ω > 2? In this talk, I will describe how lower bounds on ω can be used to make progress on derandomizing polynomial identity testing.

  • Wednesday Oct 5 at 14:00 in seminar room E:2116, Ole Römers väg 3, Lund University, and on Zoom
    Theoretical barriers for efficient proof search
    (Susanna F. de Rezende, Lund University)

    The proof search problem is a central question in automated theorem proving and SAT solving. Clearly, if a propositional tautology F does not have a short (polynomial size) proof in a proof system P, any algorithm that searches for P-proofs of F will necessarily take super-polynomial time. But can proofs of "easy" formulas, i.e., those that have polynomial size proofs, be found in polynomial time? This question motivates the study of automatability of proof systems. In this talk, we give an overview of known non-automatability results, focusing on the more recent ones, and present some of the main ideas used to obtain them.

  • Tuesday Oct 18 at 14:00 at University of Copenhagen and on Zoom
    Exponential separations using guarded extension variables
    (Emre Yolcu, Carnegie Mellon University)

    I will talk about the complexity of proof systems augmenting resolution with inference rules that allow, given a formula F in conjunctive normal form, deriving clauses that are not necessarily logically implied by F but whose addition to F preserves satisfiability. When the derived clauses are allowed to introduce variables not occurring in F, the systems we will consider become equivalent to extended resolution. We are concerned with the versions of these systems "without new variables." They are called BC-, RAT-, SBC-, and GER-, denoting respectively blocked clauses, resolution asymmetric tautologies, set-blocked clauses, and generalized extended resolution. Each of these systems formalizes some restricted version of the ability to make assumptions that hold "without loss of generality," which is commonly used informally to simplify or shorten proofs. Except for SBC-, these systems are known to be exponentially weaker than extended resolution. They are, however, all equivalent to it under a relaxed notion of simulation that allows the translation of the formula along with the proof when moving between proof systems. I will show how to take advantage of this fact to construct formulas that separate RAT- from GER- and vice versa. With a similar strategy, we can also separate SBC- from RAT-. Additionally, I will briefly describe polynomial-size SBC- proofs of the pigeonhole principle, which separates SBC- from GER- by a previously known lower bound. These results also separate the three systems from BC- since they all simulate it. We thus obtain an almost complete picture of their relative strengths.

  • Monday Nov 7 at 14:00 in Lund and on Zoom
    The shortest even cycle problem is tractable
    (Thore Husfeldt, IT University of Copenhagen and Lund University)

    Given a directed graph as input, we show how to efficiently find a shortest (directed, simple) cycle on an even number of vertices. As far as we know, no polynomial-time algorithm was previously known for this problem. In fact, finding any even cycle in a directed graph in polynomial time was open for more than two decades until Robertson, Seymour, and Thomas (Ann. of Math. (2) 1999) and, independently, McCuaig (Electron. J. Combin. 2004; announced jointly at STOC 1997) gave an efficiently testable structural characterisation of even-cycle-free directed graphs.

    Almost no knowledge of graph algorithms is necessary to appreciate the result. The second hour will be more about algebra and combinatorics than algorithms — polynomials, matrix functions, and generating functions.

    This is joint work with Andreas Björklund (Lund) and Petteri Kaski (Helsinki).

  • Friday Nov 18 at 14:00
    Title TBD
    (Srikanth Srinivasan, Aarhus University)

  • Tuesday Dec 6 at 14:00
    Title TBD
    (Matti Järvisalo, University of Helsinki)

MIAO seminars autumn 2022

  • Monday Sep 19 at 14:00 on Zoom
    On vanishing sums of roots of unity in polynomial calculus and sum-of-squares
    (Ilario Bonacina, Universitat Politècnica de Catalunya)

    Vanishing sums of roots of unity can be seen as a natural generalization of knapsack from Boolean variables to variables taking values over the roots of unity. We show that these sums are hard to prove for polynomial calculus and for sum-of-squares, both in terms of degree and size. This talk is based on a joint work with M. Lauria and N. Galesi.

MIAO seminars spring 2022

  • Wednesday Jun 8 at 14:00 in seminar room E:2116, Ole Römers väg 3, Lund University
    Short proofs in strong proof systems
    (Marijn Heule, Carnegie Mellon University)

    The success of satisfiability solving presents us with an interesting peculiarity: modern solvers can frequently handle gigantic formulas while failing miserably on supposedly easy problems. Their poor performance is typically caused by the weakness of their underlying proof system—resolution. To overcome this obstacle, we need solvers that are based on stronger proof systems. Unfortunately, existing strong proof systems—such as extended resolution or Frege systems—do not seem to lend themselves to mechanization.

    In this talk, we discuss some recently proposed strong proof systems that are more suitable for mechanization. These proof systems are surprisingly strong, even without the introduction of new variables—a key feature of short proofs presented in the proof-complexity literature. We demonstrate the strength of these proof systems on two famous problems: the pigeonhole principle and mutilated chessboards. For both problems, we present proofs of linear size without new variables. We also present a novel technique to find such proofs automatically and show its usefulness on SAT Competition benchmarks. We conclude with some theoretical and practical challenges.

  • Wednesday Jun 1 at 14:00 in Auditorium No. 2, Hans Christian Ørsted Building, Universitetsparken 5, University of Copenhagen, and on Zoom
    Certifying correctness for combinatorial algorithms by using pseudo-Boolean reasoning
    (Stephan Gocht, Lund University and University of Copenhagen)

    Although solving NP-complete problems is widely believed to require exponential time in the worst case, state-of-the-art algorithms are amazingly efficient for many NP-hard optimisation problems. However, this is achieved through highly sophisticated algorithms that are prone to implementation errors which can cause incorrect results, even for the best commercial tools. A promising approach to address this problem is to use certifying algorithms that produce not only the desired output but also a simple, machine-verifiable certificate or proof of correctness of the output. By verifying this proof with an external tool, we can guarantee that the given answer is valid.

    This talk will give an introduction to a new proof system that can certify answers of algorithms for various combinatorial problems, such as Boolean satisfiability (SAT) solving and optimisation, constraint programming, and graph solving. This proof system operates on 0-1 integer linear constraints, generalising the cutting planes proof system. As a running example, we will show how to certify the correctness of a maximum matching algorithm.

    After the official seminar, there will be an optional second part with more technical discussions. This second part will demonstrate how to generate such machine-verifiable proofs in practice and how to check them using the verifier VeriPB, which I developed during my PhD project.

  • Monday May 30 at 14:00 in seminar room E:2116, Ole Römers väg 3, Lund University, and on Zoom
    CDCL versus resolution
    (Marc Vinyals)

    The effectiveness of the CDCL algorithm, the one most used to solve SAT in practice, is complicated to understand, and so far one of the most successful tools for that has been proof complexity. CDCL is easily seen to be limited by the resolution proof system, and furthermore it has been shown to be equivalent to resolution, in the sense that CDCL can reproduce a given resolution proof with only polynomial overhead.

    But the question of the power of CDCL with respect to resolution is far from closed. To begin with, the previous equivalence is subject to some assumptions, only some of which are reasonable in practice. In addition, in a world where algorithms are expected to run in linear time, a polynomial overhead is too coarse a measure.

    In this talk we will discuss what breaks when we try to make the assumptions more realistic, and how much of an overhead CDCL needs in order to simulate resolution.

  • Tuesday Mar 22 at 14:00 in seminar room A107, Hans Christian Ørsted Building, Universitetsparken 5, University of Copenhagen, and on Zoom
    Solving large scale instances of a geometric set cover problem with coloring conflicts
    (Allan Sapucaia, University of Campinas)

    When designing wireless networks, one wants to install antennas ensuring that all important regions are covered. There are many different secondary goals to consider such as minimizing the number of antennas, or ensuring that the network can still work even when some of the antennas are down.

    In this talk, we will discuss a geometric wireless network design problem where we want to minimize the number of antennas, while constrained by frequency assignment conflicts. We will study a Integer Linear Program model for this problem, and discuss how it can be improved using both its geometric and graph-theoretical aspects. We show that the problem can be efficiently solved in practice using decomposition and conclude by providing theoretical explanations for its performance.

MIAO seminars autumn 2021

  • Monday Dec 6 at 17:00 (note the time!)
    Core-guided minimal correction set and core enumeration (video)
    (Nina Narodytska, VMware Research)

    A set of constraints is unsatisfiable if there is no solution that satisfies these constraints. To analyze unsatisfiable problems, the user needs to understand where inconsistencies come from and how they can be repaired. Minimal unsatisfiable cores and correction sets are important subsets of constraints that enable such analysis. In the first part of the talk, we analyze core-guided MaxSAT algorithms and discuss how the cores found by these algorithms are related to the cores of the original unsatisfiable formula. Based on these results, in the second part, we discuss a new algorithm for extracting minimal unsatisfiable cores and correction sets. Our new solver significantly outperforms several state of the art algorithms on common benchmarks when it comes to extracting correction sets and compares favorably on core extraction.

  • Monday Nov 22 at 14:00
    TFNP: Collapses, separations, and characterisations (video)
    (Mika Göös, EPFL)

    We discuss a range of results for total NP search problem classes (TFNP) that includes new collapses of classes, black-box separations, and characterisations using propositional proof systems. We will focus on a surprising collapse result, EoPL = PLS ∩ PPAD, and see its full proof.

    Joint work with Alexandros Hollender, Siddhartha Jain, Gilbert Maystre, William Pires, Robert Robere, and Ran Tao.

  • Wednesday Nov 17 at 14:00 in E:1408, Ole Römers väg 3, Lund University and on Zoom
    Decomposition approaches for a large-scale scheduling problem (video of first half of seminar)
    (Elina Rönnberg, Linköping University)

    Generic state-of-the-art solvers for discrete optimisation are exceptionally powerful tools that efficiently solve a variety of problems. However, the scale and complexity of practically relevant problems can sometimes render these solvers incapable of even finding feasible solutions. In such cases, one possibility is to develop solution approaches that exploit problem structure to decompose the problem and then use the generic solvers for addressing the resulting subproblems. The success of such approaches relies on that the subproblems can be efficiently solved and that the information gained from that is sufficient to solve the original problem.

    In an industry-academia collaboration between Linköping University and Saab Aeronautics, we address the scheduling of a kind of electronic systems in future aircraft. Briefly, this problem can be described as a rich multiprocessor scheduling problem that also includes the scheduling of a communication network. To find feasible solutions to practically relevant instances of this problem is challenging.

    In this talk, I will present two different ways of decomposing the problem. The first relies on making a strong relaxation of the problem and then applying a constraint generation procedure. In this approach, both the relaxed problem and the subproblem are solved by a mixed-integer programming solver. The performance of the method is enhanced by an integration with adaptive large neighbourhood search. The second decomposition strategy is based on logic-based Benders decomposition. In this case, the master problem is solved as a mixed-integer program and the subproblem is formulated as a constraint program. The computational results from the two approaches are compared. To conclude the talk, the different ways of exploiting the problem structure and efficiency of solvers will be discussed and we will compare the properties of the resulting decomposition approaches.

  • Joint BARC/MIAO seminar:
    Friday Nov 12 at 14:00 in Auditorium No. 10, Hans Christian Ørsted Building, Universitetsparken 5, University of Copenhagen, and on Zoom
    Superpolynomial lower bounds against low-depth algebraic circuits (video)
    (Nutan Limaye, IT University of Copenhagen)

    Every multivariate polynomial P(X) can be written as a sum of monomials, i.e. a sum of products of variables and field constants. In general, the size of such an expression is the number of monomials that have a non-zero coefficient in P.

    What happens if we add another layer of complexity, and consider sums of products of sums (of variables and field constants) expressions? Now, it becomes unclear how to prove that a given polynomial P(X) does not have small expressions. In this result, we solve exactly this problem.

    More precisely, we prove that certain explicit polynomials have no polynomial-sized "Sigma-Pi-Sigma" (sums of products of sums) representations. We can also show similar results for Sigma-Pi-Sigma-Pi, Sigma-Pi-Sigma-Pi-Sigma and so on for all "constant-depth" expressions.

    In the first part of this two-part talk, I will present the statements of the main results and some background. In the second part of the talk, I will give some proof details.

    This is a joint work with Srikanth Srinivasan and Sébastien Tavenas.

  • Monday Nov 8 at 14:00
    Lifting with inner-product and other low discrepancy gadgets (video)
    (Sajin Koroth, Simon Fraser University)

    Lifting theorems are a powerful technique that exports our understanding of a weak model of computation (usually query complexity) to a powerful model of computation (usually communication complexity). They have been instrumental in solving many open problems in diverse areas of theoretical computer science like communication complexity, circuit complexity, proof complexity, linear programming, game theory, etc. A typical lifting theorem connects the complexity of computing any function f in the weak model to computing an associated function f ∘ g in the powerful model. The associated function f ∘ g is obtained by composing f with a specific function g known as the gadget. A key aspect of broad applicability and success of the lifting theorems is that they work for any f. In a typical lifting theorem, this universality is achieved by designing the gadget g with specific properties. The applicability of a lifting theorem is usually limited by the choice of gadget g and the size of the gadget. Thus, it is crucial to understand what functions g can be used as a gadget and how small g can be. In this talk, we present a lifting theorem that connects query complexity to communication complexity and works for almost all gadgets g of "smallish" size.

    This talk is based on joint work with Arkadev Chattopadhyay, Yuval Filmus, Or Meir, and Toniann Pitassi.

  • Monday Oct 25 at 14:00
    Proof complexity lower bounds by composition (video)
    (Robert Robere, McGill University)

    In recent years, there has been an explosion in the development of so-called "lifting theorems" in proof complexity and other closely related areas which have led to the resolution of many long-standing open problems. The basic idea of a lifting theorem is simple: in order to prove a lower bound in a "complicated" system that we do not understand (for example, Cutting Planes refutations), we will "escalate" the hardness from a "simple" system that we do understand (for example, Resolution refutations) by taking a hard formula F for the simple system and composing it with a specially chosen "gadget function" that removes the power of the complicated system. In many cases, it is possible to prove that the "best" possible strategy in the complicated system for refuting the composed formula is to simulate the proof of the uncomposed formula in the simple system (and thus lower bounds for the uncomposed formula are "escalated" or "lifted" to lower bounds for the composed formula). While lifting theorems in circuit complexity can be traced back to the work of Raz and McKenzie [RM99], recent developments have pushed these techniques to many new areas, and have shown how lifting is a flexible tool that allows quite finely tuned tradeoffs between different parameters for a wide variety of proof systems.

    In this talk, we give an introduction to and survey of lifting results in proof complexity and related fields, like communication complexity and circuit complexity, and outline some of the ways that lifting can be used to control various "complexity parameters" of unsatisfiable CNF formulas (like proof length, proof space, proof depth, etc.). We will highlight several recent developments in the area and indicate some future directions.

  • Friday Oct 22 at 14:00
    Verified proof checkers (video)
    (Magnus Myreen, Chalmers University of Technology)

    Solvers, such as e.g. SAT solvers, are often complicated pieces of software. How can we trust their results? For highly optimised state-of-the-art solvers, testing is insufficient and proving the functional correctness of the solver's implementation is not practically possible. Fortunately, solvers can often be augmented to produce proof certificates that can be checked by separate simpler proof checkers. In recent years, there has been growing interest in formally proving that the proof checkers will never accept a certificate that contains flaws.

    In this talk, I will describe work that makes it possible to prove functional correctness of proof checkers down to the machine code that runs them. I have worked on (and have supervised work on) several checkers. In this talk, I'll focus on (1) my work on proving end-to-end correctness of Jared Davis's Milawa prover, and (2) recent work on a LPR/LRAT checker for UNSAT proofs. My talk will include a description of the CakeML project, which was the context of (2). I will use demos to show what the tools look like when running.

  • Friday Oct 15 at 10:00
    Logic-based explainable AI (video)
    (Alexey Ignatiev, Monash University)

    Explainable artificial intelligence (XAI) represents arguably one of the most crucial challenges being faced by the area of AI these days. Although the majority of approaches to XAI are of heuristic nature, recent work proposed the use of abductive reasoning to computing provably correct explanations for machine learning (ML) predictions. The proposed rigorous approach was shown to be useful not only for computing trustable explanations but also for reasoning about explanations computed heuristically. It was also applied to uncover a close relationship between XAI and verification of ML models. This talk will overview the advances of the rigorous logic-based approach to XAI as well as the use of reasoning in devising interpretable rule-based ML models including decision trees, decision sets, and decision lists.

  • Monday Oct 4 at 14:00
    Monotone arithmetic lower bounds via communication complexity (video)
    (Arkadev Chattopadhyay, Tata Institute of Fundamental Research, Mumbai)

    How much power does negation or cancellation provide to computation? This is a fundamental question in theoretical computer science that appears in various parts: in Boolean circuits, arithmetic circuits and also in communication complexity. I will talk about some new connections between the latter two fields and their applications to extend two classical results from four decades ago:

    1. Valiant (1979) showed that monotone arithmetic circuits are exponentially weaker than general circuits for computing monotone polynomials. Our first result gives a qualitatively more powerful separation by showing an exponential separation between general monotone circuits and constant-depth multi-linear formulas. Neither such a separation between general formulas and monotone circuits, nor a separation between multi-linear circuits and monotone circuits were known before. Our result uses the recent counter-example to the Log-Approximate-Rank Conjecture in communication complexity.
    2. Jerrum and Snir (1982) also obtained a separation between the powers of general circuits and monotone ones via a different polynomial, i.e. the spanning tree polynomial (STP), a polynomial that is well known to be in VP, using non-multi-linear cancellations of determinantal computation. We provide the first extension of this result to show that the STP remains "robustly hard" for monotone circuits in the sense of Hrubes' recent notion of epsilon-sensitivity. The latter result is proved via formulating a discrepancy method for monotone arithmetic circuits that seems independently interesting.

    We will discuss several open problems arising from these results. (The seminar is based on joint works with Rajit Datta, Utsab Ghosal and Partha Mukhopadhyay.)

  • Monday Sep 20 at 14:00
    KRW composition theorems via lifting (video)
    (Or Meir, University of Haifa)

    One of the major open problems in complexity theory is proving super-logarithmic lower bounds on the depth of circuits (i.e., separating P from NC0). Karchmer, Raz, and Wigderson (Computational Complexity 5(3/4), 1995) suggested approaching this problem by proving that depth complexity behaves "as expected" with respect to composition of functions. They showed that the validity of this conjecture would separate P from NC0. Several works have made progress toward resolving this conjecture by proving special cases. In particular, these works proved the KRW conjecture for every outer function, but only for few inner functions. Thus, it is an important challenge to prove the KRW conjecture for a wider range of inner functions.

    In this work, we extend significantly the range of inner functions that can be handled. First, we consider the monotone version of the KRW conjecture. We prove it for every monotone inner function whose depth complexity can be lower bounded via a query-to-communication lifting theorem. This allows us to handle several new and well-studied functions such as the s-t-connectivity, clique, and generation functions.

    In order to carry this progress back to the non-monotone setting,we introduce a new notion of semi-monotone composition, which combines the non-monotone complexity of the outer function with the monotone complexity of the inner function. In this setting, we prove the KRW conjecture for a similar selection of inner functions, but only for a specific choice of the outer function.

  • Friday Sep 17 at 14:15
    Estimating the size of unions of sets in streaming model (video)
    (Kuldeep S. Meel, National University of Singapore)

    Feeling tired of complicated theorems with complicated proofs? Fret not; this week will be different: A simple theorem with a simpler proof for a general problem. I will make a case for our PODS-21 paper to be the simplest paper ever accepted at PODS.

    A set is Delphic if it supports efficient membership, sampling, and counting calls. The notion of Delphic sets captures several well-known problems, such as Klee's measure, distinct elements, and model counting of DNF formulas. We will discuss estimation of the size of the union of Delphic sets wherein each set is implicitly (and succinctly) represented, and comes in a streaming fashion.

    The primary contribution of our work is a simple, elegant, and efficient sampling-based algorithm for the estimation of the union in the streaming setting. Our algorithm has worst case space complexity and update time that is logarithmic in the size of the universe and stream size. Consequently, our algorithm provides the first algorithm with a linear dependence on d for Klee's measure problem in streaming setting for d>1, thereby settling the open problem of Woodruff and Tirthpura (PODS-12). The Klee's measure problem corresponds to computation of volume of multi-dimension axis-aligned rectangles, i.e., every d-dimension axis-aligned rectangle can be defined as [a_1,b_1] x [a_2,b_2] x … x [a_d, b_d].

    (Joint work with N.V. Vinodchandran and Sourav Chakraborty.)

  • Monday Sep 6 at 14:00
    Lifting with sunflowers (video)
    (Ian Mertz, University of Toronto)

    Lifting theorems are an important class of techniques which can transform functions which are hard for weak computation models into functions which are hard for strong computation models. Starting with Raz and McKenzie (1999), there has been a flurry of work proving query-to-communication lifting theorems, which translate lower bounds on query complexity to lower bounds for the corresponding communication model. In this talk I will present a simplified proof of deterministic query-to-communication lifting. Our proof uses elementary counting together with a novel connection to the sunflower lemma.

  • Friday Aug 27 at 10:00
    Who is Isabelle, and why is she so picky about my proofs?
    (Dmitriy Traytel, University of Copenhagen)

    Proof assistants are tools that mechanically check human-written formal proofs, e.g., of an algorithm's correctness, and support their users in developing these proofs. Landmark achievements in this area include realistic verified compilers, operating system kernels, and distributed systems as well as formal proofs of deep mathematical results such as the four color theorem, the Feit–Thompson theorem, and the Kepler conjecture. In this talk, I will give a brief introduction of the Isabelle/HOL proof assistant, of which I am both a user and a developer. I will explain how Isabelle's foundational approach achieves the highest level of trustworthiness and will examplify the benefits of using a proof assistant via three case studies: the formal verification of security properties of authenticated data structures, the completeness of ordered resolution, and the correctness of an efficient runtime monitoring algorithm.

MIAO seminars spring 2021

During the spring semester of 2021, we had online virtual seminars running every weekday at 17:30 CET from early February to mid-May as part of the Satisfiability: Theory, Practice, and Beyond and Theoretical Foundations of Computer Systems programs at the Simons Institute for the Theory of Computing. On top of this, we had some additional seminars as listed below.

  • Monday Jun 7 at 14:00
    Proof complexity meets finite model theory (video)
    (Joanna Ochremiak, LaBRI, Université de Bordeaux and CNRS)

    Finite model theory studies the power of logics on the class of finite structures. One of its goals is to characterize symmetric computation, that is, computation that abstracts away details which are not essential for the given task, by respecting the symmetries of the input.

    In this talk I will discuss connections between proof complexity and finite model theory, focussing on lower bounds. For certain proof systems, the existence of a succinct refutation can be decided in a symmetry-preserving way. This allows us to transfer lower bounds from finite model theory to proof complexity. I will introduce this approach and explain its key technical ideas such as the method of folding for dealing with symmetries in linear programs.

  • Friday Jun 4 at 14:00
    Abstract cores in implicit hitting set based MaxSAT solving (video)
    (Jeremias Berg, University of Helsinki)

    Maximum satisfiability (MaxSat) solving is an active area of research motivated by numerous successful applications to solving NP-hard combinatorial optimization problems. One of the most successful approaches for solving MaxSat instances from real world domains are the so called implicit hitting set (IHS) solvers. IHS solvers decouple MaxSat solving into separate core-extraction and optimization steps which are tackled by an Boolean satisfiability (SAT) and an integer linear programming (IP) solver, respectively. While the approach shows state-of-the-art performance on many industrial instances, it is known that there exists instances on which IHS solvers need to extract an exponential number of cores before terminating. In this talk I will present the simplest of these problematic instances and talk about how abstract cores, a compact representation of a large number of regular cores that we recently proposed, addresses perhaps the main bottleneck of IHS solvers. I will show how to incorporate abstract core reasoning into the IHS algorithm and demonstrate that that including abstract cores into a state-of-the- art IHS solver improves its performance enough to surpass the best performing solvers of the recent MaxSat Evaluations.

  • Friday May 28 at 14:00
    Feasible interpolation for algebraic proof systems (video)
    (Tuomas Hakoniemi, Universitat Politècnica de Catalunya)

    In this talk we present a form of feasible interpolation for two algebraic proof systems, Polynomial Calculus and Sums-of-Squares. We show that for both systems there is a poly-time algorithm that given two sets P(x,z) and Q(y,z) of polynomial equalities, a refutation of the union of P(x,z) and Q(y,z) and an assignment a to the z-variables outputs either a refutation of P(x,a) or a refutation of Q(y,a). Our proofs are fairly logical in nature, in that they rely heavily on semantic characterizations of resource-bounded refutations to prove the existence of suitable refutations. These semantic existence proofs narrow down the search space for the refutations we are after so that we can actually find them efficiently.

  • Monday May 17 at 14:00
    Average-case perfect matching lower bounds from hardness of Tseitin formulas (video)
    (Kilian Risse, KTH Royal Institute of Technology)

    We study the complexity of proving that a sparse random regular graph on an odd number of vertices does not have a perfect matching, and related problems involving each vertex being matched some pre-specified number of times. We show that this requires proofs of degree Ω(n / log n) in the Polynomial Calculus (over fields of characteristic ≠ 2) and Sum-of-Squares proof systems, and exponential size in the bounded-depth Frege proof system. This resolves a question by Razborov asking whether the Lovász-Schrijver proof system requires nδ rounds to refute these formulas for some δ > 0. The results are obtained by a worst-case to average-case reduction of these formulas relying on a topological embedding theorem which may be of independent interest.

    Joint work with with Per Austrin.

  • Monday May 10 at 14:00
    On the complexity of branch and cut (video)
    (Noah Fleming, University of Toronto)

    The Stabbing Planes proof system was introduced to model practical branch-and-cut integer programming solvers. As a proof system, Stabbing Planes can be viewed as a simple generalization of DPLL to reason about linear inequalities. It is powerful enough to simulate Cutting Planes and produce short refutations of the Tseitin formulas — certain unsatisfiable systems of linear equations mod 2 — which are canonical hard examples for many algebraic proof systems. In a surprising recent result, Dadush and Tiwari showed that these short Stabbing Planes refutations of the Tseitin formulas could be translated into Cutting Planes proofs. This raises the question of whether all Stabbing Planes can be efficiently translated into Cutting Planes. In recent work, we give a partial answer to this question.

    In this talk I will introduce and motivate the Stabbing Planes proof system. I will then show how Stabbing Planes proofs with quasi-polynomially bounded coefficients can be quasi-polynomially translated into Cutting Planes. As a consequence of this translation, we can show that Cutting Planes has quasi-polynomial size refutations of any unsatisfiable system of linear equations over a finite field. A remarkable property of our translation for systems of linear equations over finite fields, and the translation of Dadush and Tiwari for the Tseitin formulas, is that the resulting proofs are also quasi-polynomially deep. A natural question is thus whether these depth bounds can be improved. In the last part of the talk, I will discuss progress towards answering this question in the form of a new depth lower bound technique for Cutting Planes, which works also for the stronger semantic Cutting Planes system, and which allows us to establish the first linear lower bounds on the depth of semantic Cutting Planes refutations of the Tseitin formulas.

  • Monday May 3 at 14:00
    SAT-encodings and scalability (video)
    (André Schidler, Technische Universität Wien)

    Boolean satisfiability (SAT) solvers have reached stunning performance over the last decade. This performance can be harnessed for other problems by means of SAT encodings: translating the problem instance into a SAT instance. SAT encodings have been used successfully for many combinatorial problems and are established in industry. In this talk, I discuss the general idea behind SAT encodings and two specific application areas: graph decompositions and machine learning.

    Graph decompositions allow for specialized algorithms that can solve hard problems efficiently. SAT encodings provide not only the means to optimally compute these decompositions, but can be easily extended by extra constraints specific to the hard problem to be solved.

    For machine learning, SAT encodings gained increased attention in recent years, as they allow the induction of very small AI models. Learning small models has become more important in the context of explainable AI: smaller models are usually easier to understand for humans. We will focus specifically on decision tree induction.

    Scalability is one of the main issues when using SAT encodings. We can overcome this problem by embedding SAT-based solving into a heuristic framework. This allows us to trade off a slightly worse result for large applicability. In the last part of our talk, we discuss our SAT-based local improvement framework that implements this idea.

  • Wednesday Apr 28 at 15:00
    Slicing the hypercube is not easy (video)
    (Amir Yehudayoff, Technion – Israel Institute of Technology)

    How many hyperplanes are needed to slice all edges of the hypercube? This question has been studied in machine learning, geometry and computational complexity since the 1970s. We shall describe (most of) an argument showing that more than n0.57 hyperplanes are needed, for large n. We shall also see a couple of applications. This is joint work with Gal Yehuda.

  • Monday Apr 26 at 14:00
    Recent lower bounds in algebraic complexity theory (video)
    (Ben Lee Volk, University of Texas at Austin)

    Algebraic complexity theory studies the complexity of solving algebraic computational tasks using algebraic models of computation. One major problem in this area is to prove lower bounds on the number of arithmetic operations required for computing explicit polynomials. This natural mathematical problem is the algebraic analog of the famous P vs. NP problem. It also has tight connections to other classical mathematical areas and to fundamental questions in complexity theory.

    In this talk I will provide background and then present some recent progress on proving lower bounds for models of algebraic computation, such as the algebraic analogs of NC1 and NL.

    Based on joint works with Prerona Chatterjee, Mrinal Kumar, and Adrian She.

  • Friday Apr 23 at 14:00
    On the complexity of branching proofs (video)
    (Daniel Dadush, CWI)

    We consider the task of proving integer infeasibility of a bounded convex K in Rn using a general branching proof system. In a general branching proof, one constructs a branching tree by adding an integer disjunction ax ⩽ b or ax ⩾ b+1, for an integer vector a and an integer b, at each node, such that the leaves of the tree correspond to empty sets (i.e., K together with the inequalities picked up from the root to leaf is empty). Recently, Beame et al (ITCS 2018), asked whether the bit size of the coefficients in a branching proof, which they named stabbing planes (SP) refutations, for the case of polytopes derived from SAT formulas, can be assumed to be polynomial in n. We resolve this question by showing that any branching proof can be recompiled so that the integer disjunctions have coefficients of size at most (nR)O(n2), where R is the radius of an l1 ball containing K, while increasing the number of nodes in the branching tree by at most a factor O(n). As our second contribution, we show that Tseitin formulas, an important class of infeasible SAT instances, have quasi-polynomial sized cutting plane (CP) refutations, disproving the conjecture that Tseitin formulas are (exponentially) hard for CP. As our final contribution, we give a simple family of polytopes in [0,1]n requiring branching proofs of length 2n/n.

    Joint work with Samarth Tiwari.

  • Monday Apr 19 at 14:00
    Kidney exchange programmes — saving lives with optimisation algorithms (video)
    (William Pettersson, University of Glasgow)

    Kidney Exchange Programmes (KEPs) increase the rate of living donor kidney transplantation, in turn saving lives. In this talk, I will explain exactly what KEPs are, how KEPs achieve this goal, and the role of optimisation algorithms within KEPs. This will include brief explanations of some of the models used for kidney exchange programmes, some of the more generic optimisation techniques that we use, as well as recent advances and specific research directions for the field into the future. This talk will very much be an overview of kidney exchange, without going into complex details of optimisation algorithms, making it suitable for a wider audience.

  • Thursday Apr 8 at 14:00
    (Semi)Algebraic proofs over {±1} variables (video)
    (Dmitry Sokolov, St.Petersburg State University)

    One of the major open problems in proof complexity is to prove lower bounds on AC0[p]-Frege proof systems. As a step toward this goal Impagliazzo, Mouli and Pitassi in a recent paper suggested proving lower bounds on the size for Polynomial Calculus over the {±1} basis. In this talk we show a technique for proving such lower bounds and moreover, we also give lower bounds on the size for Sum-of-Squares over the {±1} basis.

    We discuss the difference between {0, 1} and {+1, -1} cases and problems in further generalizations of the lower bounds.

  • Courtesy of Zuse Institute Berlin:
    Friday Feb 12 at 14:00
    Introduction to IP presolving techniques in PaPILO
    (Alexander Hoen, Zuse Institute Berlin)

    Presolving is an essential part contributing to the performance of modern MIP solvers. PaPILO, a new C++ library, provides presolving routines for MIP and LP problems. This talk will give an introduction to basic IP-presolving techniques and provide insights into important design choices enabling PaPILO's capabilities, in particular regarding its use of parallel hardware. While presolving itself is designed to be fast, this can come at the cost of failing to find important reductions due to working limits and heuristic filtering. Yet, even most commercial solvers do not use multi-threading for the preprocessing step as of today. PaPILO's design facilitates use of parallel hardware to allow for more aggressive presolving and presolving of huge problems. The architecture of PaPILO allows presolvers generally to run in parallel without requiring expensive copies of the problem and without special synchronization in the presolvers themselves. Additionally, the use of Intel's TBB library aids PaPILO to efficiently exploit recursive parallelism within expensive presolving routines, such as probing, dominated columns, or constraint sparsification. Despite PaPILO's use of parallelization, its results are guaranteed to be deterministic independently of the number of threads available.

MIAO seminars autumn 2020

  • Friday Oct 9 at 13:15
    Model counting with probabilistic component caching
    (Shubham Sharma and Kuldeep S. Meel, National University of Singapore)

    Given a Boolean formula F, the problem of model counting, also referred to as #SAT, seeks to compute the number of solutions of F. Model counting is a fundamental problem with a wide variety of applications ranging from planning, quantified information flow to probabilistic reasoning and the like. The modern #SAT solvers tend to be either based on static decomposition, dynamic decomposition, or a hybrid of the two. Despite dynamic decomposition based #SAT solvers sharing much of their architecture with SAT solvers, the core design and heuristics of dynamic decomposition-based #SAT solvers has remained constant for over a decade. In this paper, we revisit the architecture of the state-of-the-art dynamic decomposition-based #SAT tool, sharpSAT, and demonstrate that by introducing a new notion of probabilistic component caching and the usage of universal hashing for exact model counting along with the development of several new heuristics can lead to significant performance improvement over state-of-the-art model-counters. In particular, we develop GANAK, a new scalable probabilistic exact model counter that outperforms state-of-the-art exact and approximate model counters for a wide variety of instances.

  • Friday Sep 25 at 13:15
    Manthan: A data-driven approach for Boolean function synthesis
    (Priyanka Golia and Kuldeep S. Meel, National University of Singapore)

    Boolean functional synthesis is a fundamental problem in computer science with wide-ranging applications and has witnessed a surge of interest resulting in progressively improved techniques over the past decade. Despite intense algorithmic development, a large number of problems remain beyond the reach of the current state-of-the-art techniques. Motivated by the progress in machine learning, we propose Manthan, a novel data-driven approach to Boolean functional synthesis. Manthan views functional synthesis as a classification problem, relying on advances in constrained sampling for data generation, and advances in automated reasoning for a novel proof-guided refinement and provable verification. On an extensive and rigorous evaluation over 609 benchmarks, we demonstrate that Manthan significantly improves upon the current state of the art, solving 356 benchmarks in comparison to 280, which is the most solved by a state-of-the-art technique; thereby, we demonstrate an increase of 76 benchmarks over the current state of the art. The significant performance improvements, along with our detailed analysis, highlights several interesting avenues of future work at the intersection of machine learning, constrained sampling, and automated reasoning.

  • Friday Aug 28 at 13:15
    Tuning Sat4j PB solvers for decision problems
    (Romain Wallon, Université d'Artois)

    During the last decades, many improvements in CDCL SAT solvers have made possible to solve efficiently large problems containing millions of variables and clauses. Despite this practical efficiency, some instances remain out of reach for such solvers. This is particularly true when the input formula requires an exponential size refutation proof in the resolution proof system (as for pigeonhole formulae). This observation motivated the development of another kind of solvers, known as pseudo-Boolean (PB) solvers. These solvers take as inputs a conjunction of PB constraints (integral linear inequations over Boolean variables) and benefit from the cutting planes proof system, which is (in theory) stronger than the resolution proof system. To implement this proof system, current PB solvers follow the direction of modern SAT solvers, by implementing a conflict analysis procedure relying on the application of cutting planes rules. However, adapting the CDCL architecture to take into account PB constraints is not as straightforward as it may look. In particular, many CDCL invariants and properties do not hold anymore as long as PB constraints are considered. While some of them may be safely ignored (e.g., when they affect the decision heuristic, the deletion strategy or the restart policy), some others must be fixed to ensure the soundness of the solver (e.g., by ensuring to preserve the conflict during its analysis).

    In this talk, we will give an overview of the main differences between PB solving and classical SAT solving, and present different approaches that have been designed to take these differences into account to extend the CDCL architecture to PB problems. We will in particular discuss the pros and cons of these approaches, and we will explain how they can be enhanced to improve the practical performance of PB solvers.

  • Friday Aug 21 at 10:15
    Using proofs to analyze SAT solvers
    (Janne Kokkala, Lund University and University of Copenhagen)

    The main idea of this seminar is to give an overview of what one can ask and what one can learn about SAT solvers when analyzing the proof to estimate what part of the work was useful. I will start by giving a 17-ish minute alpha test run for my presentation for our CP paper [1]. After that, I will discuss earlier works that use the same or a similar idea for general insights [2], analyzing VSIDS usefulness [3], and clause exchange for parallel solvers [4,5]. To conclude, we can discuss how this approach could be used for pseudo-Boolean solvers.

    [1] J. I. Kokkala, J. Nordström. Using Resolution Proofs to Analyse CDCL SAT solvers.
    [2] L. Simon. Post Mortem Analysis of SAT Solver Proofs.
    [3] S. Malik, V. Ying. On the efficiency of the VSIDS decision heuristic. (Workshop presentation.)
    [4] G. Audemard, L. Simon. Lazy clause exchange policy for parallel SAT solvers.
    [5] G. Katsirelos, A. Sabharwal, H. Samulowitz, L. Simon. Resolution and parallelizability: Barriers to the efficient parallelization of SAT solvers.

  • Wednesday Aug 19 at 13:15
    On computational aspects of the antibandwidth problem
    (Markus Sinnl, Johannes Kepler University Linz)

    In this talk, we consider the antibandwidth problem, also known as dual bandwidth problem, separation problem and maximum differential coloring problem. Given a labeled graph (i.e., a numbering of the vertices of a graph), the antibandwidth of a node is defined as the minimum absolute difference of its labeling to the labeling of all its adjacent vertices. The goal in the antibandwidth problem is to find a labeling maximizing the antibandwidth. The problem is NP-hard in general graphs and has applications in diverse areas like scheduling, radio frequency assignment, obnoxious facility location and map-coloring.

    There has been much work on deriving theoretical bounds for the problem and also in the design of metaheuristics in recent years. However, the optimality gaps between the best known solution values and reported upper bounds for the HarwellBoeing Matrix-instances, which are the commonly used benchmark instances for this problem, were often very large (e.g., up to 577%).

    We present new mixed-integer programming approaches for the problem, including one approach, which does not directly formulate the problem as optimization problem, but as a series of feasibility problems. We also discuss how these feasibility problems can be encoded with various SAT-encodings, including a new and specialised encoding which exploits a certain staircase-structure occuring in the problem formulation. We present computational results for all the algorithms, including a comparison of the MIP and SAT-approaches. Our developed approaches allow to find the proven optimal solution for eight instances from literature, where the optimal solution was unknown and also provide reduced gaps for eleven additional instances, including improved solution values for seven instances, the largest optimality gap is now 46%. Instances based on the problem were submitted to the SAT Competition 2020.

MIAO seminars spring 2020

  • Wednesday Jul 15 at 15:00
    Naïve algorithm selection for SAT solving
    (Stephan Gocht, Lund University and University of Copenhagen)

    Although solving propositional formulas is an NP-complete problem, state-of-the-art SAT solvers are able to solve formulas with millions of variables. To obtain good performance it is necessary to configure parameters for heuristic decisions. However, there is no single parameter configuration that is perfect for all formulas, and choosing the parameters is a difficult task. The standard approach is to evaluate different configurations on some formulas and to choose the single configuration, that performs best overall. This configuration, which is called single best solver, is then used to solve new unseen formulas. In this paper we demonstrate how random forests can be used to choose a configuration dynamically based on simple features of the formula. The evaluation shows that our approach is able to outperform the single best solver on formulas that are similar to the training set, but not on formulas from completely new domains.

  • Friday Jun 26 at 10:00
    Behind the scenes of chronological CDCL
    (Sibylle Möhle and Armin Biere, Johannes Kepler University Linz)

    Combining conflict-driven clause learning (CDCL) with chronological backtracking is challenging: Multiple invariants considered crucial in modern SAT solvers are violated, if after conflict analysis the solver does not jump to the assertion level but to a higher decision level instead. In their SAT'18 paper "Chronological Backtracking", Alexander Nadel and Vadim Ryvchim provide a fix to this issue. Moreover, their SAT solver implementing chronological backtracking won the main track of the SAT Competition 2018. In our SAT'19 paper, "Backing Backtracking", we present a formalization and generalization of this method. We prove its correctness and provide an independent implementation.

    In this seminar, we demonstrate the working of chronological CDCL by means of an example. In this example, after a conflict the conflicting clause contains only one literal at conflict level. It is therefore used as a reason for backtracking, thus saving the effort of conflict analysis. We further show which invariants are violated and present new ones followed by a discussion of the rules of our formal framework. We also shed light onto implementation details, including those which are not mentioned in our SAT'19 paper.

  • Monday Jun 22 at 13:30
    McSplit: A partitioning algorithm for maximum common subgraph problems
    (Ciaran McCreesh and James Trimble, University of Glasgow)

    We will give a short introduction to McSplit, an algorithm for the maximum common (connected) subgraph problem coauthored with Patrick Prosser and presented at IJCAI 2017. McSplit resembles a forward-checking constraint programming algorithm, but uses a partitioning data structure to store domains which greatly reduces memory use and time per search node. We will also present our recent work with Stephan Gocht and Jakob Nordström on adding proof logging to the algorithm, turning McSplit into a certifying algorithm whose outputs can be independently verified.

  • Thursday Jun 18 at 20:30
    A pseudo-Boolean approach to nonlinear verification
    (Vincent Liew, University of Washington)

    We discuss some new experimental results showing the promise of using pseudo-Boolean solvers, rather than SAT solvers, to verify bit-vector problems containing multiplication. We use this approach to efficiently verify the commutativity of a multiplier output bit by output bit. We also give some examples of simple bit-vector inequalities where the pseudo-Boolean approach significantly outperformed SAT solvers and even bit-vector solvers. Finally, we give some of our observations on the strengths and weaknesses of different methods of conflict analysis used by pseudo-Boolean solvers.

  • Friday May 8 at 13:15
    Pseudo-Boolean solvers for answer set programming
    (Bart Bogaerts, Vrije Universiteit Brussel)

    Answer set programming (ASP) is a well-established knowledge representation formalism. Most ASP solvers are based on (extensions of) technology from Boolean satisfiability solving. While these solvers have shown to be very successful in many practical applications, their strength is limited by their underlying proof system, resolution. In this research, we present a new tool LP2PB that translates ASP programs into pseudo-Boolean theories, for which solvers based on the (stronger) cutting plane proof system exist. We evaluate our tool, and the potential of cutting-plane-based solving for ASP on traditional ASP benchmarks as well as benchmarks from pseudo-Boolean solving. Our results are mixed: overall, traditional ASP solvers still outperform our translational approach, but several benchmark families are identified where the balance shifts the other way, thereby suggesting that further investigation into a stronger proof system for ASP is valuable.

Published by: Jakob Nordström <jn~at-sign~di~dot~ku~dot~dk>
Updated 2022-09-28