Polynomial Size Research Articles

On the Fourier transform of a quantitative trait: Implications for compressive sensing

This paper explores the genotype-phenotype relationship. It outlines conditions under which the dependence of a quantitative trait on the genome might be predictable, based on measurement of a limited subset of genotypes. It uses the theory of real-valued Boolean functions in a systematic way to translate trait data into the Fourier domain. Important trait features, such as the roughness of the trait landscape or the modularity of a trait have a simple Fourier interpretation. Ruggedness at a gene location corresponds to high sensitivity to mutation, while a modular organization of gene activity reduces such sensitivity.Traits where rugged loci are rare will naturally compress gene data in the Fourier domain, leading to a sparse representation of trait data, concentrated in identifiable, low-level coefficients. This Fourier representation of a trait organizes epistasis in a form which is isometric to the trait data. As Fourier matrices are known to be maximally incoherent with the standard basis, this permits employing compressive sensing techniques to work from data sets that are relatively small—sometimes even of polynomial size—compared to the exponentially large sets of possible genomes.This theory provides a theoretical underpinning for systematic use of Boolean function machinery to dissect the dependency of a trait on the genome and environment.

Pseudo polynomial size LP formulation for calculating the least core value of weighted voting games

In this paper, we propose a pseudo polynomial size LP formulation for finding a payoff vector in the least core of a weighted voting game. The numbers of variables and constraints in our formulation are both bounded by O(nW+), where n is the number of players and W+ is the total sum of (integer) voting weights. When we employ our formulation, a commercial LP solver calculates a payoff vector in the least core of practical weighted voting games in a few seconds. We also extend our approach to vector weighted voting games.

Lower Bounds on OBDD Proofs with Several Orders

This article is motivated by seeking lower bounds on OBDD(∧, w, r) refutations, namely, OBDD refutations that allow weakening and arbitrary reorderings. We first work with 1 - NBP ∧ refutations based on read-once nondeterministic branching programs. These generalize OBDD(∧, r) refutations. There are polynomial size 1 - NBP(∧) refutations of the pigeonhole principle, hence 1-NBP(∧) is strictly stronger than OBDD}(∧, r). There are also formulas that have polynomial size tree-like resolution refutations but require exponential size 1-NBP(∧) refutations. As a corollary, OBDD}(∧, r) does not simulate tree-like resolution, answering a previously open question. The system 1-NBP(∧, ∃) uses projection inferences instead of weakening. 1-NBP(∧, ∃ k is the system restricted to projection on at most k distinct variables. We construct explicit constant degree graphs G n on n vertices and an ε > 0, such that 1-NBP(∧, ∃ ε n ) refutations of the Tseitin formula for G n require exponential size. Second, we study the proof system OBDD}(∧, w, r ℓ ), which allows ℓ different variable orders in a refutation. We prove an exponential lower bound on the complexity of tree-like OBDD(∧, w, r ℓ ) refutations for ℓ = ε log n , where n is the number of variables and ε > 0 is a constant. The lower bound is based on multiparty communication complexity.

Reliable Recurrence Algorithm for High-Order Krawtchouk Polynomials.

Krawtchouk polynomials (KPs) and their moments are promising techniques for applications of information theory, coding theory, and signal processing. This is due to the special capabilities of KPs in feature extraction and classification processes. The main challenge in existing KPs recurrence algorithms is that of numerical errors, which occur during the computation of the coefficients in large polynomial sizes, particularly when the KP parameter (p) values deviate away from 0.5 to 0 and 1. To this end, this paper proposes a new recurrence relation in order to compute the coefficients of KPs in high orders. In particular, this paper discusses the development of a new algorithm and presents a new mathematical model for computing the initial value of the KP parameter. In addition, a new diagonal recurrence relation is introduced and used in the proposed algorithm. The diagonal recurrence algorithm was derived from the existing n direction and x direction recurrence algorithms. The diagonal and existing recurrence algorithms were subsequently exploited to compute the KP coefficients. First, the KP coefficients were computed for one partition after dividing the KP plane into four. To compute the KP coefficients in the other partitions, the symmetry relations were exploited. The performance evaluation of the proposed recurrence algorithm was determined through different comparisons which were carried out in state-of-the-art works in terms of reconstruction error, polynomial size, and computation cost. The obtained results indicate that the proposed algorithm is reliable and computes lesser coefficients when compared to the existing algorithms across wide ranges of parameter values of p and polynomial sizes N. The results also show that the improvement ratio of the computed coefficients ranges from 18.64% to 81.55% in comparison to the existing algorithms. Besides this, the proposed algorithm can generate polynomials of an order ∼8.5 times larger than those generated using state-of-the-art algorithms.

The transit time constrained fixed charge multi-commodity network design problem

This paper introduces the transit time constrained fixed charge multi-commodity network design problem. Transit times are origin-to-destination time limits for the commodities, which appear for example in transport systems with perishable goods. We discuss how to model the problem and present three different formulations of it. The first formulation is an exponential size path formulation, which we solve with a branch-and-price algorithm. Several speed up techniques from the literature on fixed charge multi-commodity network design problems are implemented, such as lifted cover inequalities and the recently proposed deep dual-optimal inequalities. In an extensive set of computational experiments, we show that these inequalities significantly improve the performance of the algorithm. The other two formulations are of polynomial size: one uses path indices and the other uses time indices. While the branch-and-price algorithm outperforms solving the compact formulations with a general-purpose mixed-integer programming solver, the study of compact models helps better understand the problem, and we can use them as benchmarks. A detailed sensitivity analysis of the branch-and-price algorithm shows that longer transit times and an increased ratio of fixed charge to flow cost increase the difficulty of solving the problem whereas the arc capacity has less impact. We further discuss in-depth implementational details.

Approximate search for known gene clusters in new genomes using PQ-trees

Gene clusters are groups of genes that are co-locally conserved across various genomes, not necessarily in the same order. Their discovery and analysis is valuable in tasks such as gene annotation and prediction of gene interactions, and in the study of genome organization and evolution. The discovery of conserved gene clusters in a given set of genomes is a well studied problem, but with the rapid sequencing of prokaryotic genomes a new problem is inspired. Namely, given an already known gene cluster that was discovered and studied in one genomic dataset, to identify all the instances of the gene cluster in a given new genomic sequence. Thus, we define a new problem in comparative genomics, denoted PQ-Tree Search that takes as input a PQ-tree T representing the known gene orders of a gene cluster of interest, a gene-to-gene substitution scoring function h, integer arguments d_T and d_S, and a new sequence of genes S. The objective is to identify in S approximate new instances of the gene cluster; These instances could vary from the known gene orders by genome rearrangements that are constrained by T, by gene substitutions that are governed by h, and by gene deletions and insertions that are bounded from above by d_T and d_S, respectively. We prove that PQ-Tree Search is NP-hard and propose a parameterized algorithm that solves the optimization variant of PQ-Tree Search in O^*(2^{gamma }) time, where gamma is the maximum degree of a node in T and O^* is used to hide factors polynomial in the input size. The algorithm is implemented as a search tool, denoted PQFinder, and applied to search for instances of chromosomal gene clusters in plasmids, within a dataset of 1,487 prokaryotic genomes. We report on 29 chromosomal gene clusters that are rearranged in plasmids, where the rearrangements are guided by the corresponding PQ-trees. One of these results, coding for a heavy metal efflux pump, is further analysed to exemplify how PQFinder can be harnessed to reveal interesting new structural variants of known gene clusters.

Parameter concentrations in quantum approximate optimization

The quantum approximate optimization algorithm (QAOA) has become a cornerstone of contemporary quantum applications development. In QAOA, a quantum circuit is trained -- by repeatedly adjusting circuit parameters -- to solve a problem. Several recent findings have reported parameter concentration effects in QAOA and their presence has become one of folklore: while empirically observed, the concentrations have not been defined and analytical approaches remain scarce, focusing on limiting system and not considering parameter scaling as system size increases. We found that optimal QAOA circuit parameters concentrate as an inverse polynomial in the problem size, providing an optimistic result for improving circuit training. Our results are analytically demonstrated for variational state preparations at $p=1,2$ (corresponding to 2 and 4 tunable parameters respectively). The technique is also applicable for higher depths and the concentration effect is cross verified numerically. Parameter concentrations allow for training on a fraction $w < n$ of qubits to assert that these parameters are nearly optimal on $n$ qubits. Clearly this effect has significant practical importance.

Propositional proof systems based on maximum satisfiability

The paper describes the use of dual-rail MaxSAT systems to solve Boolean satisfiability (SAT), namely to determine if a set of clauses is satisfiable. The MaxSAT problem is the problem of satisfying the maximum number of clauses in an instance of SAT. The dual-rail encoding adds extra variables for the complements of variables, and allows encoding an instance of SAT as a Horn MaxSAT problem. We discuss three implementations of dual-rail MaxSAT: core-guided systems, minimal hitting set (MaxHS) systems, and MaxSAT resolution inference systems. All three of these can be more efficient than resolution and thus than conflict-driven clause learning (CDCL). All three systems can give polynomial size refutations for the pigeonhole principle, the doubled pigeonhole principle and the mutilated chessboard principles. The dual-rail MaxHS MaxSat system can give polynomial size proofs of the parity principle. However, dual-rail MaxSAT resolution requires exponential size proofs for the parity principle; this is proved by showing that constant depth Frege augmented with the pigeonhole principle can polynomially simulate dual-rail MaxSAT resolution. Consequently, dual-rail MaxSAT resolution does not simulate cutting planes. We further show that core-guided dual-rail MaxSAT and weighted dual-rail MaxSAT resolution polynomially simulate resolution. Finally, we report the results of experiments with core-guided dual-rail MaxSAT and MaxHS dual-rail MaxSAT showing strong performance by these systems.

Storing Set Families More Compactly with Top ZDDs

Zero-suppressed Binary Decision Diagrams (ZDDs) are data structures for representing set families in a compressed form. With ZDDs, many valuable operations on set families can be done in time polynomial in ZDD size. In some cases, however, the size of ZDDs for representing large set families becomes too huge to store them in the main memory. This paper proposes top ZDD, a novel representation of ZDDs which uses less space than existing ones. The top ZDD is an extension of the top tree, which compresses trees, to compress directed acyclic graphs by sharing identical subgraphs. We prove that navigational operations on ZDDs can be done in time poly-logarithmic in ZDD size, and show that there exist set families for which the size of the top ZDD is exponentially smaller than that of the ZDD. We also show experimentally that our top ZDDs have smaller sizes than ZDDs for real data.

Small Circuits and Dual Weak PHP in the Universal Theory of p-time Algorithms

We prove, under a computational complexity hypothesis, that it is consistent with the true universal theory of p-time algorithms that a specific p-time function extending bits to bits violates the dual weak pigeonhole principle: Every string equals the value of the function for some . The function is the truth-table function assigning to a circuit the table of the function it computes and the hypothesis is that every language in P has circuits of a fixed polynomial size .

The Running Intersection Relaxation of the Multilinear Polytope

The multilinear polytope of a hypergraph is the convex hull of a set of binary points satisfying a collection of multilinear equations. We introduce the running intersection inequalities, a new class of facet-defining inequalities for the multilinear polytope. Accordingly, we define a new polyhedral relaxation of the multilinear polytope, referred to as the running intersection relaxation, and identify conditions under which this relaxation is tight. Namely, we show that for kite-free beta-acyclic hypergraphs, a class that lies between gamma-acyclic and beta-acyclic hypergraphs, the running intersection relaxation coincides with the multilinear polytope and it admits a polynomial size extended formulation.

Lossless Dimension Expanders Via Linearized Polynomials and Subspace Designs

For a vector space \({\mathbb{F}^n}\) over a field \(\mathbb{F}\), an (η, β)-dimension expander of degree d is a collection of d linear maps \({\Gamma _j}:{\mathbb{F}^n} \rightarrow {\mathbb{F}^n}\) such that for every subspace U of \({\mathbb{F}^n}\) of dimension at most ηn, the image of U under all the maps, ∑ dj=1 Γj(U), has dimension at least α dim(U). Over a finite field, a random collection of d = O(1) maps Γj offers excellent “lossless” expansion whp: β≈d for η ≥ Ω(1/d). When it comes to a family of explicit constructions (for growing n), however, achieving even modest expansion factor β = 1+ ε with constant degree is a non-trivial goal.We present an explicit construction of dimension expanders over finite fields based on linearized polynomials and subspace designs, drawing inspiration from recent progress on list decoding in the rank metric. Our approach yields the following: Lossless expansion over large fields; more precisely β ≥ (1 − ε)d and \(\eta \ge {{1 - \varepsilon} \over d}\) with d = Oε(1), when \(\left| \mathbb{F} \right| \ge \Omega \left(n \right)\). Optimal up to constant factors expansion over fields of arbitrarily small polynomial size; more precisely β ≥ Ω(δd) and η ≥ Ω(1/(δd)) with d = Oδ(1), when \(\left| \mathbb{F} \right| \ge {n^\delta}\). Previously, an approach reducing to monotone expanders (a form of vertex expansion that is highly non-trivial to establish) gave (Ω(1), 1 + Ω(1))-dimension expanders of constant degree over all fields. An approach based on “rank condensing via subspace designs” led to dimension expanders with \(\beta \mathbin{\lower.3ex\hbox{$\buildrel>\over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} \sqrt d \) over large finite fields. Ours is the first construction to achieve lossless dimension expansion, or even expansion proportional to the degree.

An exact method for assortment optimization under the nested logit model

We study the problem of finding an optimal assortment of products maximizing the expected revenue, in which customer preferences are modeled using a nested logit choice model. This problem is known to be polynomially solvable in a specific case and NP-hard otherwise, with only approximation algorithms existing in the literature. We provide an exact general method that embeds a tailored Branch-and-Bound algorithm into a fractional programming framework. In contrast to the existing literature, in which assumptions are imposed on either the structure of nests or the combination and characteristics of products, no assumptions on the input data are imposed. Although our approach is not polynomial in input size, it can solve the most general problem setting for large-size instances. We show that the fractional programming scheme’s parameterized subproblem, a highly non-linear binary optimization problem, is decomposable by nests, which is the primary advantage of the approach. To solve the subproblem for each nest, we propose a two-stage approach. In the first stage, we fix a large set of variables based on the single-nest subproblem’s newly-derived structural properties. This can significantly reduce the problem size. In the second stage, we design a tailored Branch-and-Bound algorithm with problem-specific upper bounds. Numerical results show that the approach is able to solve assortment instances with five nests and with up to 5000 products per nest. The most challenging instances for our approach are those with a mix of nests’ dissimilarity parameters, where some of them are smaller than one and others are greater than one.

Sparktope: linear programs from algorithms

In a recent paper, Avis, Bremner, Tiwary and Watanabe gave a method for constructing linear programs (LPs) based on algorithms written in a simple programming language called Sparks. If an algorithm produces the solution x to a problem in polynomial time and space then the LP constructed is also of polynomial size and its optimum solution contains x as well as a complete execution trace of the algorithm. Their method led us to the construction of a compiler called sparktope which we describe in this paper. This compiler allows one to generate polynomial sized LPs for problems in that have exponential extension complexity, such as matching problems in non-bipartite graphs. In this paper, we describe sparktope, the language Sparks, and the assembler instructions and LP constraints it produces. This is followed by two concrete examples, the makespan problem and the problem of testing if a matching in a graph is maximum, both of which are known to have exponential extension complexity. Computational results are given. In discussing these examples, we make use of visualization techniques included in sparktope that may be of independent interest. The extremely large linear programs produced by the compiler appear to be quite challenging to solve using currently available software. Since the optimum LP solutions can be computed independently they may be useful as benchmarks. Further enhancements of the compiler and its application are also discussed.

An Exact Method for Assortment Optimization under the Nested Logit Model

We study the problem of finding an optimal assortment of products maximizing the expected revenue, in which customer preferences are modeled using a Nested Logit choice model. This problem is known to be polynomially solvable in a specific case and NP-hard otherwise, with only approximation algorithms existing in the literature. We provide an exact general method that embeds a tailored Branch-and-Bound algorithm into a fractional programming framework. Contrary to the existing literature, in which assumptions are imposed on either the structure of nests or the combination and characteristics of products, no assumptions on the input data are imposed. Although our approach is not polynomial in the input size, it can solve the most general problem setting for large-size instances. We show that the fractional programming scheme’s parameterized subproblem, a highly non-linear binary optimization problem, is decomposable by nests, which is the primary advantage of the approach. To solve the subproblem for each nest, we propose a two-stage approach. In the first stage, we fix a large set of variables based on the single-nest subproblem’s newly-derived structural properties. This can significantly reduce the problem size. In the second stage, we design a tailored Branch-and-Bound algorithm with problem-specific upper bounds. Numerical results show that the approach is able to solve assortment instances with five nests and with up to 5,000 products per nest. The most challenging instances for our approach are those with a mix of nests’ dissimilarity parameters, where some of them are smaller than one and others are greater than one. Keywords: combinatorial optimization, revenue management, assortment optimization, fractional programming, nested logit.

Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices

We study the Steiner Tree problem, in which a set of terminal vertices needs to be connected in the cheapest possible way in an edge-weighted graph. This problem has been extensively studied from the viewpoint of approximation and also parametrization. In particular, on one hand Steiner Tree is known to be APX-hard, and W[2]-hard on the other, if parameterized by the number of non-terminals (Steiner vertices) in the optimum solution. In contrast to this we give an efficient parameterized approximation scheme (EPAS), which circumvents both hardness results. Moreover, our methods imply the existence of a polynomial size approximate kernelization scheme (PSAKS) for the considered parameter. We further study the parameterized approximability of other variants of Steiner Tree, such as Directed Steiner Tree and Steiner Forest. For neither of these an EPAS is likely to exist for the studied parameter: for Steiner Forest an easy observation shows that the problem is APX-hard, even if the input graph contains no Steiner vertices. For Directed Steiner Tree we prove that approximating within any function of the studied parameter is W[1]-hard. Nevertheless, we show that an EPAS exists for Unweighted Directed Steiner Tree, but a PSAKS does not. We also prove that there is an EPAS and a PSAKS for Steiner Forest if in addition to the number of Steiner vertices, the number of connected components of an optimal solution is considered to be a parameter.

Hypersequent calculi for non-normal modal and deontic logics: countermodels and optimal complexity

AbstractWe present some hypersequent calculi for all systems of the classical cube and their extensions with axioms ${T}$, ${P}$ and ${D}$ and for every $n \geq 1$, rule ${RD}_n^+$. The calculi are internal as they only employ the language of the logic, plus additional structural connectives. We show that the calculi are complete with respect to the corresponding axiomatization by a syntactic proof of cut elimination. Then, we define a terminating proof search strategy in the hypersequent calculi and show that it is optimal for coNP-complete logics. Moreover, we show that from every failed proof of a formula or hypersequent it is possible to directly extract a countermodel of it in the bi-neighbourhood semantics of polynomial size for coNP logics, and for regular logics also in the relational semantics. We finish the paper by giving a translation between hypersequent rule applications and derivations in a labelled system for the classical cube.

Bounds on the size of PC and URC formulas

In this paper, we investigate CNF encodings, for which unit propagation is strong enough to derive a contradiction if the encoding is not consistent with a partial assignment of the variables (unit refutation complete or URC encoding) or additionally to derive all implied literals if the encoding is consistent with the partial assignment (propagation complete or PC encoding). We prove an exponential separation between the sizes of PC and URC encodings without auxiliary variables and strengthen the known results on their relationship to the PC and URC encodings that can use auxiliary variables. Besides of this, we prove that the sizes of any two irredundant PC formulas representing the same function differ at most by a factor polynomial in the number of the variables and present an example of a function demonstrating that a similar statement is not true for URC formulas. One of the separations above implies that a q-Horn formula may require an exponential number of additional clauses to become a URC formula. On the other hand, for every q-Horn formula, we present a polynomial size URC encoding of the same function using auxiliary variables. This encoding is not q-Horn in general.

Non-Self-Embedding Grammars, Constant-Height Pushdown Automata, and Limited Automata

Non-self-embedding grammars are a restriction of context-free grammars which does not allow to describe recursive structures and, hence, which characterizes only the class of regular languages. A double exponential gap in size from non-self-embedding grammars to deterministic finite automata is known. The same size gap is also known from constant-height pushdown automata and [Formula: see text]-limited automata to deterministic finite automata. Constant-height pushdown automata and [Formula: see text]-limited automata are compared with non-self-embedding grammars. It is proved that non-self-embedding grammars and constant-height pushdown automata are polynomially related in size. Furthermore, a polynomial size simulation by [Formula: see text]-limited automata is presented. However, the converse transformation is proved to cost exponential. Finally, a different simulation shows that also the conversion of deterministic constant-height pushdown automata into deterministic [Formula: see text]-limited automata costs polynomial.

On Lagrangian relaxation for constrained maximization and reoptimization problems

We prove a general result demonstrating the power of Lagrangian relaxation in solving constrained maximization problems with arbitrary objective functions. This yields a unified approach for solving a wide class of subset selection problems with linear constraints. Given a problem in this class and some small ε∈(0,1), we show that if there exists an r-approximation algorithm for the Lagrangian relaxation of the problem, for some r∈(0,1), then our technique achieves a ratio of rr+1−ε to the optimal, and this ratio is tight.Using the technique we obtain (re)approximation algorithms for natural (reoptimization) variants of classic subset selection problems, including real-time scheduling, the maximum generalized assignment problem (GAP) and maximum weight independent set. For all of the problems studied in this paper, the number of calls to the r-approximation algorithm, used by our algorithms, is linear in the input size and in log(1∕ε) for inputs with a cardinality constraint, and polynomial in the input size and in log(1∕ε) for inputs with an arbitrary linear constraint.