Deterministic Polynomial Time Algorithm Research Articles

Complexity of Strong Approximation on the Sphere

AbstractBy assuming some widely believed arithmetic conjectures, we show that the task of accepting a number that is representable as a sum of $d\geq 2$ squares subjected to given congruence conditions is NP-complete. On the other hand, we develop and implement a deterministic polynomial-time algorithm that represents a number as a sum of four squares with some restricted congruence conditions, by assuming a polynomial-time algorithm for factoring integers and Conjecture 1.1. As an application, we develop and implement a deterministic polynomial-time algorithm for navigating Lubotzky, Phillips, Sarnak (LPS) Ramanujan graphs, under the same assumptions.

Estimating the probability of meeting a deadline in schedules and plans

Given a plan (or schedule) with uncertain task times, we propose a deterministic polynomial (time and memory) algorithm for estimating the probability that it meets a deadline, or, equivalently, that its makespan is less than a given duration. Approximation is needed as it is known that this problem is NP-hard even for sequential plans (sum of random variables). In addition, we show two new complexity results: (1) Counting the number of events that do not cross the deadline is #P-hard; (2) Computing the expected makespan of a hierarchical plan is NP-hard. For the proposed approximation algorithm, we establish formal approximation bounds and show that the time and memory complexities grow polynomially with the required accuracy, the number of nodes in the plan, and with the size of the support of the random variables that represent the durations of the primitive tasks. We examine these approximation bounds empirically and demonstrate, using task networks taken from the literature, how our scheme outperforms sampling techniques and exact computation in terms of accuracy and run-time. As the empirical data shows much better error bounds than guaranteed, we also suggest a method for tightening the bounds in some cases.

Linear projections of the Vandermonde polynomial

An n-variate Vandermonde polynomial is the determinant of the n×n matrix where the ith column is the vector (1,xi,xi2,…,xin−1)T. Vandermonde polynomials play a crucial role in the theory of alternating polynomials and are useful in Lagrangian polynomial interpolation which arise in the theory of error correcting codes. In this work we study structural and computational aspects of linear projections of Vandermonde polynomials.First, we consider the problem of testing if a given polynomial is equivalent to the Vandermonde polynomial. We obtain a deterministic polynomial time algorithm to test if f is linearly equivalent to the Vandermonde polynomial when f is given as product of linear factors. In the case when f is given as a black-box our algorithm runs in randomized polynomial time.Exploring the structure of projections of Vandermonde polynomials further, we describe the group of symmetries of a Vandermonde polynomial and obtain a basis for the associated Lie algebra.Finally, we study arithmetic circuits built over projections of Vandermonde polynomials. We show universality property for some of the models and obtain a lower bounds against sum of projections of Vandermonde determinant.

Operator Scaling: Theory and Applications

In this paper, we present a deterministic polynomial time algorithm for testing whether a symbolic matrix in non-commuting variables over $${\mathbb {Q}}$$ is invertible or not. The analogous question for commuting variables is the celebrated polynomial identity testing (PIT) for symbolic determinants. In contrast to the commutative case, which has an efficient probabilistic algorithm, the best previous algorithm for the non-commutative setting required exponential time (Ivanyos et al. in Comput Complex 26(3):717–763, 2017) (whether or not randomization is allowed). The algorithm efficiently solves the “word problem” for the free skew field, and the identity testing problem for arithmetic formulae with division over non-commuting variables, two problems which had only exponential time algorithms prior to this work. The main contribution of this paper is a complexity analysis of an existing algorithm due to Gurvits (J Comput Syst Sci 69(3):448–484, 2004), who proved it was polynomial time for certain classes of inputs. We prove it always runs in polynomial time. The main component of our analysis is a simple (given the necessary known tools) lower bound on central notion of capacity of operators (introduced by Gurvits 2004). We extend the algorithm to actually approximate capacity to any accuracy in polynomial time, and use this analysis to give quantitative bounds on the continuity of capacity (the latter is used in a subsequent paper on Brascamp–Lieb inequalities). We also extend the algorithm to compute not only singularity, but actually the (non-commutative) rank of a symbolic matrix, yielding a factor 2 approximation of the commutative rank. This naturally raises a relaxation of the commutative PIT problem to achieving better deterministic approximation of the commutative rank. Symbolic matrices in non-commuting variables, and the related structural and algorithmic questions, have a remarkable number of diverse origins and motivations. They arise independently in (commutative) invariant theory and representation theory, linear algebra, optimization, linear system theory, quantum information theory, approximation of the permanent and naturally in non-commutative algebra. We provide a detailed account of some of these sources and their interconnections. In particular, we explain how some of these sources played an important role in the development of Gurvits’ algorithm and in our analysis of it here.

On expansion and contraction of DL-Lite knowledge bases

Knowledge bases (KBs) are not static entities: new information constantly appears and some of the previous knowledge becomes obsolete. In order to reflect this evolution of knowledge, KBs should be expanded with the new knowledge and contracted from the obsolete one. This problem is well-studied for propositional but much less for first-order KBs. In this work we investigate knowledge expansion and contraction for KBs expressed in DL-Lite, a family of description logics (DLs) that underlie the tractable fragment OWL2QL of the Web Ontology Language OWL2. We start with a novel knowledge evolution framework and natural postulates that evolution should respect, and compare our postulates to the well-established AGM postulates. We then review well-known model and formula-based approaches for expansion and contraction for propositional theories and show how they can be adapted to the case of DL-Lite. In particular, we show intrinsic limitations of model-based approaches: besides the fact that some of them do not respect the postulates we have established, they ignore the structural properties of KBs. This leads to undesired properties of evolution results: evolution of DL-Lite KBs cannot be captured in DL-Lite. Moreover, we show that well-known formula-based approaches are also not appropriate for DL-Lite expansion and contraction: they either have a high complexity of computation, or they produce logical theories that cannot be expressed in DL-Lite. Thus, we propose a novel formula-based approach that respects our principles and for which evolution is expressible in DL-Lite. For this approach we also propose polynomial time deterministic algorithms to compute evolution of DL-Lite KBs when evolution affects only factual data.

On Expansion and Contraction of DL-Lite Knowledge Bases

Knowledge bases (KBs) are not static entities: new information constantly appears and some of the previous knowledge becomes obsolete. In order to reflect this evolution of knowledge, KBs should be expanded with the new knowledge and contracted from the obsolete one. This problem is well-studied for propositional but much less for first-order KBs. In this work we investigate knowledge expansion and contraction for KBs expressed in DL-Lite, a family of description logics (DLs) that underlie the tractable fragment OWL2 QL of the Web Ontology Language OWL2. We start with a novel knowledge evolution framework and natural postulates that evolution should respect, and compare our postulates to the well-established AGM postulates. We then review well-known model and formula-based approaches for expansion and contraction for propositional theories and show how they can be adapted to the case of DL-Lite. In particular, we show intrinsic limitations of model-based approaches: besides the fact that some of them do not respect the postulates we have established, they ignore the structural properties of KBs. This leads to undesired properties of evolution results: evolution of DL-Lite KBs cannot be captured in DL-Lite. Moreover, we show that well-known formula-based approaches are also not appropriate for DL-Lite expansion and contraction: they either have a high complexity of computation, or they produce logical theories that cannot be expressed in DL-Lite. Thus, we propose a novel formula-based approach that respects our principles and for which evolution is expressible in DL-Lite. For this approach we also propose polynomial time deterministic algorithms to compute evolution of DL-Lite KBs when evolution affects only factual data.

Efficient Approximations for the Online Dispersion Problem

The dispersion problem has been widely studied in computational geometry and facility location and is closely related to the packing problem. The goal is to locate $n$ points (e.g., facilities or persons) in a $k$-dimensional polytope, so that they are far away from each other and from the boundary of the polytope. In many real-world scenarios, however, the points arrive and depart at different times, and decisions must be made without knowing future events. Therefore, we study, for the first time in the literature, the online dispersion problem in Euclidean space. There are two natural objectives when time is involved: the all-time worst-case (ATWC) problem tries to maximize the minimum distance that ever appears at any time; and the cumulative distance (CD) problem tries to maximize the integral of the minimum distance throughout the whole time interval. Interestingly, the online problems are highly nontrivial even on a segment. For cumulative distance, this remains the case even when the problem is time-dependent but offline, with all the arriving and departure times given in advance. For the online ATWC problem on a segment, we construct a deterministic polynomial-time algorithm which is $(2\ln 2+\epsilon)$-competitive, where $\epsilon>0$ can be arbitrarily small and the algorithm's running time is polynomial in $\frac{1}{\epsilon}$. We show that this algorithm is actually optimal. For the same problem in a square, we provide a $1.591$-competitive algorithm and a $1.183$ lower bound. Furthermore, for arbitrary $k$-dimensional polytopes with $k\geq 2$, we provide a $\frac{2}{1-\epsilon}$-competitive algorithm and a $\frac{7}{6}$ lower bound. All our lower bounds come from the structure of the online problems and hold even when computational complexity is not a concern. Interestingly, for the offline CD problem in arbitrary $k$-dimensional polytopes, we provide a polynomial-time black-box reduction to the online ATWC problem, and the resulting competitive ratio increases by a factor of at most 2. Our techniques also apply to online dispersion problems with different boundary conditions.

Qsymm: algorithmic symmetry finding and symmetric Hamiltonian generation

Symmetry is a guiding principle in physics that allows us to generalize conclusions between many physical systems. In the ongoing search for new topological phases of matter, symmetry plays a crucial role by protecting topological phases. We address two converse questions relevant to the symmetry classification of systems: is it possible to generate all possible single-body Hamiltonians compatible with a given symmetry group? Is it possible to find all the symmetries of a given family of Hamiltonians? We present numerically stable, deterministic polynomial time algorithms to solve both of these problems. Our treatment extends to all continuous or discrete symmetries of non-interacting lattice or continuum Hamiltonians. We implement the algorithms in the Qsymm Python package, and demonstrate their usefulness through applications in active research areas of condensed matter physics, including Majorana wires and Kekule graphene.

An Improved Bound for Minimizing the Total Weighted Completion Time of Coflows in Datacenters

In data-parallel computing frameworks, intermediate parallel data is often produced at various stages which needs to be transferred among servers in the datacenter network (e.g., the shuffle phase in MapReduce). A stage often cannot start or be completed unless all the required data pieces from the preceding stage are received. Coflow is a recently proposed networking abstraction to capture such communication patterns. We consider the problem of efficiently scheduling coflows with release dates in a shared datacenter network so as to minimize the total weighted completion time of coflows. Several heuristics have been proposed recently to address this problem, as well as a few polynomial-time approximation algorithms with provable performance guarantees. Our main result in this paper is a polynomial-time deterministic algorithm that improves the prior known results. Specifically, we propose a deterministic algorithm with approximation ratio of 5, which improves the prior best known ratio of 12. For the special case when all coflows are released at time zero, our deterministic algorithm obtains approximation ratio of 4 which improves the prior best known ratio of 8. The key ingredient of our approach is an improved linear program formulation for sorting the coflows followed by a simple list scheduling policy. Extensive simulation results, using both synthetic and real traffic traces, are presented that verify the performance of our algorithm and show improvement over the prior approaches.

Complete Derandomization of Identity Testing and Reconstruction of Read-Once Formulas

In this article, we study the identity testing problem of arithmetic read-once formulas (ROFs) and some related models. An ROF is a formula (a circuit whose underlying graph is a tree) in which the operations are { +, × } and such that every input variable labels at most one leaf. We obtain the first polynomial-time deterministic identity testing algorithm that operates in the black-box setting for ROFs, as well as some other related models. As an application, we obtain the first polynomial-time deterministic reconstruction algorithm for such formulas. Our results are obtained by improving and extending the analysis of the algorithm of Shpilka and Yolkovich [51].

Constructive non-commutative rank computation is in deterministic polynomial time

We extend the techniques developed in Ivanyos et al. (Comput Complex 26(3):717–763, 2017) to obtain a deterministic polynomial-time algorithm for computing the non-commutative rank of linear spaces of matrices over any field. The key new idea that causes a reduction in the time complexity of the algorithm in Ivanyos et al. (2017) from exponential time to polynomial time is a reduction procedure that keeps the blow-up parameter small, and there are two methods to implement this idea: the first one is a greedy argument that removes certain rows and columns, and the second one is an efficient algorithmic version of a result of Derksen & Makam (Adv Math 310:44–63, 2017b), who were the first to observe that the blow-up parameter can be controlled. Both methods rely crucially on the regularity lemma from Ivanyos et al. (2017). In this note, we improve that lemma by removing a coprime condition there.

A Polynomial-Time Algorithm for Pliable Index Coding

In pliable index coding, we consider a server with $m$ messages and $n$ clients, where each client has as side information a subset of the messages. We seek to minimize the number of broadcast transmissions, so that each client can recover any one unknown message she does not already have. Previous work has shown that the pliable index coding problem is NP-hard and requires at most $\mathcal {O}(\log ^{2}(n))$ broadcast transmissions, which indicates exponential savings over the conventional index coding that requires in the worst case $\mathcal {O}(n)$ transmissions. In this paper, building on a decoding criterion that we propose, we first design a deterministic polynomial-time algorithm that can realize the exponential benefits, by achieving, in the worst case, a performance upper bounded by $\mathcal {O}(\log ^{2}(n))$ broadcast transmissions. We extend our algorithm to the $t$ -requests case, where each client requires $t$ unknown messages that she does not have, and show that our algorithm requires at most $\mathcal {O}(t\log (n)+\log ^{2}(n))$ broadcast transmissions. We construct lower bound instances that require at least $\Omega (\log (n))$ transmissions for linear pliable index coding and at least $\Omega (t+\log (n))$ transmissions for the $t$ -requests case, indicating that both our upper and lower bounds are polynomials of $\log (n)$ and differ within a factor of $\mathcal {O}(\log (n))$ . We provide a probabilistic analysis over random instances and show that the required number of transmissions is almost surely $\Theta (\log (n))$ , as compared with the $\Theta (n/\log (n))$ for index coding. In addition, we show that these upper and lower bounds also hold for vector pliable index coding in the worst case instances and the random graph instances, implying that vector coding does not provide benefits in terms of these bounds. Our numerical experiments show that our algorithm outperforms existing algorithms for pliable index coding by up to 50% less transmissions.

A Load-Balanced Parallelization of AKS Algorithm

The best known deterministic polynomial-time algorithm for primality testing right now is due to Agrawal, Kayal, and Saxena. This algorithm has a time complexity O (log^{ 15 / 2} ( n )) . Although this algorithm is polynomial, its reliance on the congruence of large polynomials results in enormous computational requirement. In this paper, we propose a parallelization technique for this algorithm based on message-passing parallelism together with four workload-distribution strategies. We perform a series of experiments on an implementation of this algorithm in a high-performance computing system consisting of 15 nodes, each with 4 CPU cores. The experiments indicate that our proposed parallelization technique introduce a significant speedup on existing implementations. Furthermore, the dynamic workload-distribution strategy performs better than the others. Overall, the experiments show that the parallelization obtains up to 36 times speedup.

Finding fixed point free elements and small bases in permutation groups

In this paper we study the parameterized complexity of two well-known permutation group problems which are NP-complete.1.Given a permutation group G=〈S〉≤Sn and a parameter k, find a permutation π∈G such that |{i∈[n]|π(i)≠i}|≥k. This generalizes the NP-complete problem of finding a fixed-point free permutation in G[8,17] (this is the case when k=n). We show that this problem with parameter k is fixed-parameter tractable. In the process, we give a simple deterministic polynomial-time algorithm for finding a fixed point free element in a transitive permutation group, answering an open question of Cameron [9,8] (also see [10]).2.A base for G is a subset B⊆[n] such that the subgroup of G that fixes B pointwise is trivial. We consider the parameterized complexity of checking if a given permutation group G=〈S〉≤Sn has a base of size k, where k is the parameter for the problem. This problem is known to be NP-complete [5]. We show that it is fixed-parameter tractable for cyclic permutation groups and for permutation groups of constant orbit size. For more general classes of permutation groups we do not know whether the problem is in FPT or is W[1]-hard.

The Word Problem for Omega-Terms over the Trotter-Weil Hierarchy

For two given $\omega$-terms $\alpha$ and $\beta$, the word problem for $\omega$-terms over a variety $\boldsymbol{\mathrm{V}}$ asks whether $\alpha=\beta$ in all monoids in $\boldsymbol{\mathrm{V}}$. We show that the word problem for $\omega$-terms over each level of the Trotter-Weil Hierarchy is decidable. More precisely, for every fixed variety in the Trotter-Weil Hierarchy, our approach yields an algorithm in nondeterministic logarithmic space (NL). In addition, we provide deterministic polynomial time algorithms which are more efficient than straightforward translations of the NL-algorithms. As an application of our results, we show that separability by the so-called corners of the Trotter-Weil Hierarchy is witnessed by $\omega$-terms (this property is also known as $\omega$-reducibility). In particular, the separation problem for the corners of the Trotter-Weil Hierarchy is decidable.

Polynomial degree bounds for matrix semi-invariants

We study the left–right action of SLn×SLn on m-tuples of n×n matrices with entries in an infinite field K. We show that invariants of degree n2−n define the null cone. Consequently, invariants of degree ≤n6 generate the ring of invariants if char(K)=0. We also prove that for m≫0, invariants of degree at least n⌊n+1⌋ are required to define the null cone. We generalize our results to matrix invariants of m-tuples of p×q matrices, and to rings of semi-invariants for quivers. For the proofs, we use new techniques such as the regularity lemma by Ivanyos, Qiao and Subrahmanyam, and the concavity property of the tensor blow-ups of matrix spaces. We will discuss several applications to algebraic complexity theory, such as a deterministic polynomial time algorithm for non-commutative rational identity testing, and the existence of small division-free formulas for non-commutative polynomials.

Powers of Hamilton cycles in pseudorandom graphs

We study the appearance of powers of Hamilton cycles in pseudorandom graphs, using the following comparatively weak pseudorandomness notion. A graph G is (e,p,k,l)-pseudorandom if for all disjoint X, Y ⊂ V(G) with |X| ≥ ep k n and |Y| ≥ ep l n we have e(X,Y) = (1±e)p|X||Y|. We prove that for all β > 0 there is an e > 0 such that an (e,p,1,2)-pseudorandom graph on n vertices with minimum degree at least βpn contains the square of a Hamilton cycle. In particular, this implies that (n,d,λ)-graphs with λ ≪ d 5/2 n − 3/2 contain the square of a Hamilton cycle, and thus a triangle factor if n is a multiple of 3. This improves on a result of Krivelevich, Sudakov and Szabo [Triangle factors in sparse pseudo-random graphs, Combinatorica 24 (2004), no. 3, 403–426]. We also obtain results for higher powers of Hamilton cycles and establish corresponding counting versions. Our proofs are constructive, and yield deterministic polynomial time algorithms.

Algorithms for Commutative Algebras Over the Rational Numbers

The algebras considered in this paper are commutative rings of which the additive group is a finite-dimensional vector space over the field of rational numbers. We present deterministic polynomial-time algorithms that, given such an algebra, determine its nilradical, all of its prime ideals, as well as the corresponding localizations and residue class fields, its largest separable subalgebra, and its primitive idempotents. We also solve the discrete logarithm problem in the multiplicative group of the algebra. While deterministic polynomial-time algorithms were known earlier, our approach is different from previous ones. One of our tools is a primitive element algorithm; it decides whether the algebra has a primitive element and, if so, finds one, all in polynomial time. A methodological novelty is the use of derivations to replace a Hensel–Newton iteration. It leads to an explicit formula for lifting idempotents against nilpotents that is valid in any commutative ring.

On the Limitations of Greedy Mechanism Design for Truthful Combinatorial Auctions

We study mechanisms for the combinatorial auction (CA) problem, in which m objects are sold to rational agents and the goal is to maximize social welfare. Of particular interest is the special case of s -CAs, where agents are interested in sets of size at most s , for which a simple greedy algorithm obtains an s + 1 approximation, but no polynomial time deterministic truthful mechanism is known to attain an approximation ratio better than O ( m /√log m ). We view this not only as an extreme gap between the power of greedy auctions and truthful greedy auctions, but also as exemplifying the gap between the known power of truthful and non-truthful polynomial time deterministic algorithms. We associate the notion of greediness with a broad class of algorithms, known as priority algorithms, which encapsulate many natural auction methods. This motivates us to ask: how well can a truthful greedy algorithm approximate the optimal social welfare for CA problems? We show that no truthful greedy priority algorithm can obtain an approximation to the CA problem that is sublinear in m , even for s -CAs with s ≥ 2. Our inapproximations are independent of any time constraints on the mechanism and are purely a consequence of the greedy-type restriction. We conclude that any truthful combinatorial auction mechanism with a non-trivial approximation factor must fall outside the scope of many natural auction methods.

Lattices with Symmetry

For large ranks, there is no good algorithm that decides whether a given lattice has an orthonormal basis. But when the lattice is given with enough symmetry, we can construct a provably deterministic polynomial-time algorithm to accomplish this, based on the work of Gentry and Szydlo. The techniques involve algorithmic algebraic number theory, analytic number theory, commutative algebra, and lattice basis reduction.