arXiv subjects

Artem Kaznatcheev

Publications and source records attributed to Artem Kaznatcheev.

15 recordsLinked to original sources

arXiv2026

Vertex cover number of valued constraints is a structural parameter for efficient local search

Many local search methods for problems in artificial intelligence can be viewed as an uphill climb on a corresponding discrete fitness landscapes. Finding even local peaks in these fitness landscapes is computationally intractable in theory, but often works in practice. So what features of fitness landscapes allow for efficient local search? Since any fitness landscapes can be represented by a (hyper)graph of valued constraints, I re-frame this as a question of parameterized complexity: what structural parameter of a valued constraint graph guarantees that a strict local search will find a local peak in the corresponding fitness landscape efficiently? Given a valued constrain graph of vertex cover number k, I prove that greedy local search will find the a local fitness peak in at most $2^{2k}\cdot(n - k + 1)$ steps and random uphill local search will find a local fitness peak in an expected number of at most $2^k\cdot n(n - k)$ steps. I also show that these results are asymptotically good for strict local search because there are valued constraint graphs of vertex cover number $k$ where every ascent from some initial assignment has a length of $\frac{9}{128} \cdot 2^k \cdot (n - k)$ or greater. This suggests vertex cover number as a good structural parameter for the complexity of local search.

cs.DM↗

arXiv2026

Binary constraints on one additional variable can create exponential ascents for local search

Local search in combinatorial optimisation can be viewed as an uphill climb on a corresponding fitness landscape, where the assignments visited by a strict local search follow an ascent. This hill-climbing is sometimes surprisingly efficient, but not always. Since fitness landscapes can be succinctly represented by Boolean valued constraint satisfaction problems (VCSPs), we ask: what properties of VCSPs ensure that all ascents are polynomial? Or alternatively, what are the "simplest" VCSPs with exponential ascents? Prior examples of VCSPs with exponential ascents were built up as a chain of gadgets of constraints. Here we investigate what happens for simpler starlike graphs. To establish lower bounds on the \emph{longest} ascent, we construct: (1) a star of binary constraints with a quadratic longest ascent, (2) a binary VCSP of treedepth 3 on 4n + 1 Boolean variables (made by gluing 2n triangles of constraints at a common centre variable) with an exponential longest ascent of length $10\cdot2^n - 9$, and (3) starlike VCSP of logarithmic vertex cover number with an exponential longest ascent. For steepest ascent, we prove (4) a tight bound of 2(n - 1) for VCSPs with star constraint graphs, and construct (5) a starlike tree VCSP with treedepth 3 with a quadratic steepest ascent. Finally, we prove upper bounds on the shortest ascent from any initial assignment: (6) linear for VCSPs of treedepth 3; and (7) $O(2^k (n - k))$ for VCSPs of vertex cover number k. Together (2) and (6) establish treedepth 3, and (3) and (7) establish logarithmic vertex cover number, as the first structural graph parameters for which the longest and shortest ascents can be exponentially separated. We discuss the consequences of our results for the parameterized complexity of local search.

cs.DM↗

arXiv2026

Local search for valued constraint satisfaction parameterized by treedepth

Sometimes local search algorithms cannot efficiently find even local peaks. To understand why, I look at the structure of ascents in fitness landscapes from valued constraint satisfaction problems (VCSPs) parameterized by the treedepth of their constraint graphs. There are existing constructions of VCSPs with logarithm treedepth that represent fitness landscapes where all ascents are exponential from some initial assignment. I improve these bounds by showing that with loglog treedepth, superpolynomial ascents exist; and for polylog treedepth, there are initial assignments from which all ascents are superpolynomial. My hope is that these examples of sparse VCSPs can help us better understand the barriers to efficient local search.

cs.DM↗

arXiv2026

All ascents exponential from valued constraint graphs of pathwidth three

Many combinatorial optimization problems can be formulated as finding an assignment that maximizes some pseudo-Boolean function (that we call the fitness function). Strict local search starts with some assignment and follows some update rule to proceed to an adjacent assignment of strictly higher fitness. This means that strict local search algorithms follow ascents in the fitness landscape of the pseudo-Boolean function. The complexity of the pseudo-Boolean function (and the fitness landscapes that it represents) can be parameterized by properties of the valued constraint satisfaction problem (VCSP) that encodes the pseudo-Boolean function. We focus on properties of the constraint graphs of the VCSP, with the intuition that spare graphs are less complex than dense ones. Specifically, we argue that pathwidth is the natural sparsity parameter for understanding limits on the power of strict local search. We show that prior constructions of sparse VCSPs where all ascents are exponentially long had pathwidth greater than or equal to four. We improve this this with our controlled doubling construction: a valued constraint satisfaction problem of pathwidth three where all ascents are exponentially long from a designated initial assignment. We conclude that all strict local search algorithms can be forced to take an exponential number of steps even on simple valued constraint graphs of pathwidth three.

cs.DM↗

arXiv2026

When is local search both effective and efficient?

Combinatorial optimization problems implicitly define fitness landscapes that combine the numeric structure of the 'fitness' function to be maximized with the combinatorial structure of which assignments are 'adjacent'. Local search starts at an assignment in this landscape and successively moves assignments until no further improvement is possible among the adjacent assignments. Classic analyses of local search algorithms have focused more on the question of effectiveness ("did we find a good solution?") and often implicitly assumed that there are no doubts about their efficiency ("did we find it quickly?"). But there are many reasons to doubt the efficiency of local search. Even if we focus on fitness landscapes on the hypercube that are single peaked on every subcube (i.e., semismooth fitness landscapes) where effectiveness is obvious, many local search algorithms are known to be inefficient. Since fitness landscapes are unwieldy exponentially large objects, we focus on their polynomial-sized representations by instances of valued constraint satisfaction problems (VCSP). We define a "direction" for valued constraints such that directed VCSPs generate semismooth fitness landscapes. We call VCSPs oriented if they do not have any pair of variables with arcs in both directions. Since recognizing if a VCSP-instance is directed or oriented is coNP-complete, we generalized oriented VCSPs as conditionally-smooth fitness landscapes that are recognizable in polynomial time for a VCSP-instance. We prove that many popular local search algorithms like random ascent, simulated annealing, history-based rules, jumping rules, and the Kernighan-Lin heuristic are very efficient on conditionally-smooth landscapes. But conditionally-smooth landscapes are still expressive enough so that algorithms like steepest ascent and random facet require a super-polynomial number of steps to find the fitness peak.

cs.DS↗

arXiv2025

Greed is slow on sparse graphs of oriented valued constraints

Greedy local search is especially popular for solving valued constraint satisfaction problems (VCSPs). Since any method will be slow for some VCSPs, we ask: what is the simplest VCSP on which greedy local search is slow? We construct a VCSP on 6n Boolean variables for which greedy local search takes 7(2^n - 1) steps to find the unique peak. Our VCSP is simple in two ways. First, it is very sparse: its constraint graph has pathwidth 2 and maximum degree 3. This is the simplest VCSP on which some local search could be slow. Second, it is "oriented" - there is an ordering on the variables such that later variables are conditionally-independent of earlier ones. Being oriented allows many non-greedy local search methods to find the unique peak in a quadratic number of steps. Thus, we conclude that - among local search methods - greed is particularly slow.

cs.DM↗

arXiv2024

Exponential Steepest Ascent from Valued Constraint Graphs of Pathwidth Four

We examine the complexity of maximising fitness via local search on valued constraint satisfaction problems (VCSPs). We consider two kinds of local ascents: (1) steepest ascents, where each step changes the domain that produces a maximal increase in fitness; and (2) $\prec$-ordered ascents, where -- of the domains with available fitness increasing changes -- each step changes the $\prec$-minimal domain. We provide a general padding argument to simulate any ordered ascent by a steepest ascent. We construct a VCSP that is a path of binary constraints between alternating 2-state and 3-state domains with exponentially long ordered ascents. We apply our padding argument to this VCSP to obtain a Boolean VCSP that has a constraint (hyper)graph of arity 5 and pathwidth 4 with exponential steepest ascents. This is an improvement on the previous best known construction for long steepest ascents, which had arity 8 and pathwidth 7.

cs.DM↗

arXiv2022

Nothing makes sense in deep learning, except in the light of evolution

Deep Learning (DL) is a surprisingly successful branch of machine learning. The success of DL is usually explained by focusing analysis on a particular recent algorithm and its traits. Instead, we propose that an explanation of the success of DL must look at the population of all algorithms in the field and how they have evolved over time. We argue that cultural evolution is a useful framework to explain the success of DL. In analogy to biology, we use `development' to mean the process converting the pseudocode or text description of an algorithm into a fully trained model. This includes writing the programming code, compiling and running the program, and training the model. If all parts of the process don't align well then the resultant model will be useless (if the code runs at all!). This is a constraint. A core component of evolutionary developmental biology is the concept of deconstraints -- these are modification to the developmental process that avoid complete failure by automatically accommodating changes in other components. We suggest that many important innovations in DL, from neural networks themselves to hyperparameter optimization and AutoGrad, can be seen as developmental deconstraints. These deconstraints can be very helpful to both the particular algorithm in how it handles challenges in implementation and the overall field of DL in how easy it is for new ideas to be generated. We highlight how our perspective can both advance DL and lead to new insights for evolutionary biology.

cs.LG↗

arXiv2021

Weighted automata are compact and actively learnable

We show that weighted automata over the field of two elements can be exponentially more compact than non-deterministic finite state automata. To show this, we combine ideas from automata theory and communication complexity. However, weighted automata are also efficiently learnable in Angluin's minimal adequate teacher model in a number of queries that is polynomial in the size of the minimal weighted automaton.. We include an algorithm for learning WAs over any field based on a linear algebraic generalization of the Angluin-Schapire algorithm. Together, this produces a surprising result: weighted automata over fields are structured enough that even though they can be very compact, they are still efficiently learnable.

cs.FL↗

arXiv2020

Representing fitness landscapes by valued constraints to understand the complexity of local search

Local search is widely used to solve combinatorial optimisation problems and to model biological evolution, but the performance of local search algorithms on different kinds of fitness landscapes is poorly understood. Here we consider how fitness landscapes can be represented using valued constraints, and investigate what the structure of such representations reveals about the complexity of local search. First, we show that for fitness landscapes representable by binary Boolean valued constraints there is a minimal necessary constraint graph that can be easily computed. Second, we consider landscapes as equivalent if they allow the same (improving) local search moves; we show that a minimal constraint graph still exists, but is NP-hard to compute. We then develop several techniques to bound the length of any sequence of local search moves. We show that such a bound can be obtained from the numerical values of the constraints in the representation, and show how this bound may be tightened by considering equivalent representations. In the binary Boolean case, we prove that a degree 2 or tree-structured constraint graph gives a quadratic bound on the number of improving moves made by any local search; hence, any landscape that can be represented by such a model will be tractable for any form of local search. Finally, we build two families of examples to show that the conditions in our tractability results are essential. With domain size three, even just a path of binary constraints can model a landscape with an exponentially long sequence of improving moves. With a treewidth-two constraint graph, even with a maximum degree of three, binary Boolean constraints can model a landscape with an exponentially long sequence of improving moves.

cs.DS↗

arXiv2019

Steepest ascent can be exponential in bounded treewidth problems

We investigate the complexity of local search based on steepest ascent. We show that even when all variables have domains of size two and the underlying constraint graph of variable interactions has bounded treewidth (in our construction, treewidth 7), there are fitness landscapes for which an exponential number of steps may be required to reach a local optimum. This is an improvement on prior recursive constructions of long steepest ascents, which we prove to need constraint graphs of unbounded treewidth.

cs.DM↗

arXiv2016

Cancer treatment scheduling and dynamic heterogeneity in social dilemmas of tumour acidity and vasculature

Background: Tumours are diverse ecosystems with persistent heterogeneity in various cancer hallmarks like self-sufficiency of growth factor production for angiogenesis and reprogramming of energy-metabolism for aerobic glycolysis. This heterogeneity has consequences for diagnosis, treatment, and disease progression. Methods: We introduce the double goods game to study the dynamics of these traits using evolutionary game theory. We model glycolytic acid production as a public good for all tumour cells and oxygen from vascularization via VEGF production as a club good benefiting non-glycolytic tumour cells. This results in three viable phenotypic strategies: glycolytic, angiogenic, and aerobic non-angiogenic. Results: We classify the dynamics into three qualitatively distinct regimes: (1) fully glycolytic, (2) fully angiogenic, or (3) polyclonal in all three cell types. The third regime allows for dynamic heterogeneity even with linear goods, something that was not possible in prior public good models that considered glycolysis or growth-factor production in isolation. Conclusion: The cyclic dynamics of the polyclonal regime stress the importance of timing for anti-glycolysis treatments like lonidamine. The existence of qualitatively different dynamic regimes highlights the order effects of treatments. In particular, we consider the potential of vascular renormalization as a neoadjuvant therapy before follow up with interventions like buffer therapy.

q-bio.PE↗

arXiv2015

Edge effects in game theoretic dynamics of spatially structured tumours

Background: Analysing tumour architecture for metastatic potential usually focuses on phenotypic differences due to cellular morphology or specific genetic mutations, but often ignore the cell's position within the heterogeneous substructure. Similar disregard for local neighborhood structure is common in mathematical models. Methods: We view the dynamics of disease progression as an evolutionary game between cellular phenotypes. A typical assumption in this modeling paradigm is that the probability of a given phenotypic strategy interacting with another depends exclusively on the abundance of those strategies without regard local heterogeneities. We address this limitation by using the Ohtsuki-Nowak transform to introduce spatial structure to the go vs. grow game. Results: We show that spatial structure can promote the invasive (go) strategy. By considering the change in neighbourhood size at a static boundary -- such as a blood-vessel, organ capsule, or basement membrane -- we show an edge effect that allows a tumour without invasive phenotypes in the bulk to have a polyclonal boundary with invasive cells. We present an example of this promotion of invasive (EMT positive) cells in a metastatic colony of prostate adenocarcinoma in bone marrow. Interpretation: Pathologic analyses that do not distinguish between cells in the bulk and cells at a static edge of a tumour can underestimate the number of invasive cells. We expect our approach to extend to other evolutionary game models where interaction neighborhoods change at fixed system boundaries.

q-bio.PE↗

arXiv2014

Evolving useful delusions: Subjectively rational selfishness leads to objectively irrational cooperation

We introduce a framework within evolutionary game theory for studying the distinction between objective and subjective rationality and apply it to the evolution of cooperation on 3-regular random graphs. In our simulations, agents evolve misrepresentations of objective reality that help them cooperate and maintain higher social welfare in the Prisoner's dilemma. These agents act rationally on their subjective representations of the world, but irrationally from the perspective of an external observer. We model misrepresentations as subjective perceptions of payoffs and quasi-magical thinking as an inferential bias, finding that the former is more conducive to cooperation. This highlights the importance of internal representations, not just observed behavior, in evolutionary thought. Our results provide support for the interface theory of perception and suggest that the individual's interface can serve not only the individual's aims, but also society as a whole, offering insight into social phenomena such as religion.

q-bio.PE↗

arXiv2013

Complexity of evolutionary equilibria in static fitness landscapes

A fitness landscape is a genetic space -- with two genotypes adjacent if they differ in a single locus -- and a fitness function. Evolutionary dynamics produce a flow on this landscape from lower fitness to higher; reaching equilibrium only if a local fitness peak is found. I use computational complexity to question the common assumption that evolution on static fitness landscapes can quickly reach a local fitness peak. I do this by showing that the popular NK model of rugged fitness landscapes is PLS-complete for K >= 2; the reduction from Weighted 2SAT is a bijection on adaptive walks, so there are NK fitness landscapes where every adaptive path from some vertices is of exponential length. Alternatively -- under the standard complexity theoretic assumption that there are problems in PLS not solvable in polynomial time -- this means that there are no evolutionary dynamics (known, or to be discovered, and not necessarily following adaptive paths) that can converge to a local fitness peak on all NK landscapes with K = 2. Applying results from the analysis of simplex algorithms, I show that there exist single-peaked landscapes with no reciprocal sign epistasis where the expected length of an adaptive path following strong selection weak mutation dynamics is $e^{O(n^{1/3})}$ even though an adaptive path to the optimum of length less than n is available from every vertex. The technical results are written to be accessible to mathematical biologists without a computer science background, and the biological literature is summarized for the convenience of non-biologists with the aim to open a constructive dialogue between the two disciplines.

q-bio.PE↗