Abstract. The no low-energy trivial state (NLTS) conjecture of Freedman and Hastings [ Quantum Inf. Comput., 14 (2014), pp. 144–180] posits that there exist families of Hamiltonians with all low energy states of nontrivial complexity (with complexity measured by the quantum circuit depth preparing the state). We prove this conjecture by showing that families of constant-rate and linear-distance qLDPC codes correspond to NLTS local Hamiltonians.
Abstract. We present an algorithm for approximating the edit distance between two strings of length [Formula: see text] in time [Formula: see text] up to a constant factor for any [Formula: see text]. Our result completes a research direction set forth in the recent breakthrough paper [ CDG[Formula: see text]18 ], which showed the first constant-factor approximation algorithm with a (strongly) subquadratic running time. Recent results [ KS20b , BR20 ] have shown near-linear time algorithms that obtain an additive approximation, near-linear in [Formula: see text] (equivalently, constant-factor approximation when the edit distance value is close to [Formula: see text]). In contrast, our algorithm obtains a constant-factor approximation in near-linear time for any input string. In contrast to prior algorithms, which are mostly recursing over smaller substrings, our algorithm gradually smoothes out the local contribution to the edit distance over progressively larger substrings. To accomplish this, we iteratively construct a distance oracle data structure for the metric of edit distance on all substrings of input strings of length [Formula: see text] for [Formula: see text]. The distance oracle approximates the edit distance over these substrings in a certain average sense, just enough to estimate the overall edit distance.
Abstract. The paper presents fault-tolerant (FT) labeling schemes for general graphs, as well as improved FT routing schemes. For a given [Formula: see text]-vertex graph [Formula: see text] and a bound [Formula: see text] on the number of faults, an [Formula: see text]-FT connectivity labeling scheme is a distributed data structure that assigns to each of the graph edges and vertices a short label, such that given the labels of a vertex pair [Formula: see text] and [Formula: see text], and the labels of at most [Formula: see text] failing edges [Formula: see text], one can determine if [Formula: see text] and [Formula: see text] are connected in [Formula: see text]. The primary complexity measure is the length of the individual labels. Since their introduction by [Courcelle, Twigg, STACS ’07], compact FT labeling schemes have been devised only for a limited collection of graph families. In this work, we fill in this gap by proposing two (independent) FT connectivity labeling schemes for general graphs, with a nearly optimal label length. This serves the basis for providing also FT approximate distance labeling schemes, and ultimately also routing schemes. Our main results for an [Formula: see text]-vertex graph and a fault bound [Formula: see text] are (1) There is a randomized FT connectivity labeling scheme with a label length of [Formula: see text] bits, hence optimal for [Formula: see text]. This scheme is based on the notion of cycle space sampling [Pritchard, Thurimella, TALG ’11]. (2) There is a randomized FT connectivity labeling scheme with a label length of [Formula: see text] bits (independent of the number of faults [Formula: see text]). This scheme is based on the notion of linear sketches of [Ahn et al., SODA ’12]. (3) For a given stretch parameter [Formula: see text], there is a randomized routing scheme that routes a message from [Formula: see text] to [Formula: see text] in the presence of a set [Formula: see text] of faulty edges (unknown to [Formula: see text]) over a path of length [Formula: see text]. The routing labels have [Formula: see text] bits, the header size is [Formula: see text] bits, and each routing table has only [Formula: see text] bits. (Throughout the paper, we use the notation [Formula: see text] to hide poly-logarithmic in [Formula: see text] terms.) The results also hold for weighted graphs with positive polynomial weights. This significantly improves over the state-of-the-art bounds by [Chechik, ICALP ’11], providing the first scheme with sublinear FT labeling and routing schemes for general graphs.
Abstract. The minimum circuit size problem ([Formula: see text]) asks, given the truth table of a Boolean function [Formula: see text] and an integer [Formula: see text], if there is a circuit computing [Formula: see text] of size at most [Formula: see text]. It is a long-standing open question whether [Formula: see text] is [Formula: see text]-complete. We give, in our view, the strongest evidence yet that [Formula: see text] is in fact [Formula: see text]-complete. Specifically, we show that, with probability one, there is a [Formula: see text] reduction from the [Formula: see text]-hard problem of approximating vertex cover on hypergraphs to [Formula: see text] on circuits that have access to a uniformly random oracle [Formula: see text] (the reduction can be made uniform if it is given access to [Formula: see text]). Our reduction yields near-optimal additive hardness of approximation and extends to computing time-bounded Kolmogorov complexity ([Formula: see text]). Heuristically “instantiating” [Formula: see text] with real-world cryptographic hash functions, we get a plethora of candidate uniform deterministic polynomial-time many-one reductions from [Formula: see text] to [Formula: see text] and [Formula: see text] in the standard unrelativized world. To our knowledge, no candidate reduction from [Formula: see text] to [Formula: see text] or [Formula: see text] was known previously. Moreover, our results hold in the regime where [Formula: see text] has a non–black-box worst-case to average-case reduction [Hirahara, Non-black-box worst-case to average-case reductions within NP, 2018]. Thus, intriguingly, the existence of sufficiently “unstructured” functions implies that a problem with a known (non–black-box) worst-case to average-case reduction is [Formula: see text]-complete.
Abstract. Assuming the unique games conjecture (UGC), the best approximation ratio that can be obtained in polynomial time for the max cut problem is [Formula: see text], obtained by the celebrated SDP-based approximation algorithm of Goemans and Williamson. The current best approximation algorithm for max di-cut, i.e., the max cut problem in directed graphs, achieves a ratio of about 0.87401, leaving open the question of whether max di-cut can be approximated as well as max cut. We obtain a slightly improved algorithm for max di-cut and a new UGC-hardness for it, showing that [Formula: see text], where [Formula: see text] is the best approximation ratio that can be obtained in polynomial time for max di-cut under UGC. Our new upper bound shows that max di-cut cannot be approximated as well as max cut, which separates max di-cut from max cut and resolves a question raised by Feige and Goemans. A natural generalization of max di-cut is the max [Formula: see text]-and problem in which each constraint is of the form [Formula: see text], where [Formula: see text] and [Formula: see text] are literals, i.e., variables or their negations (in max di-cut each constraint is of the form [Formula: see text] where [Formula: see text] and [Formula: see text] are variables). Austrin separated max [Formula: see text]-and from max cut by showing that [Formula: see text] and conjectured that max [Formula: see text]-and and max di-cut have the same approximation ratio. Our new lower bound on max di-cut refutes this conjecture, completing the separation of the three problems max [Formula: see text]-and, max di-cut, and max cut. We also obtain a new lower bound for max [Formula: see text]-and, showing that [Formula: see text]. Our upper bound on max di-cut is achieved via a simple, analytical proof. The new lower bounds on max di-cut and max [Formula: see text]-and, i.e., the new approximation algorithms, use experimentally discovered distributions of rounding functions which are then verified via computer-assisted proofs. Code for the project is available at https://github.com/jbrakensiek/max-dicut .
We introduce a new measure for the performance of online algorithms in Bayesian settings, where the input is drawn from a known prior, but the realizations are revealed one-by-one in an online fashion. Our new measure is called an order-competitive ratio. It is defined as the worst case (over all distribution sequences) ratio between the performance of the best order-unaware and order-aware algorithms, and quantifies the loss that is incurred due to lack of knowledge of the arrival order. Despite the growing interest in the role of the arrival order on the performance of online algorithms, this loss has been overlooked thus far. We study the order-competitive ratio in the paradigmatic prophet inequality problem, for the two common objective functions of (i) maximizing the expected value, and (ii) maximizing the probability of obtaining the largest value; and with respect to two families of algorithms, namely, (i) adaptive algorithms, and (ii) single-threshold algorithms. We provide tight bounds for all four combinations, with respect to deterministic algorithms, and preliminary results for randomized algorithms. Our analysis requires new ideas and departs from standard techniques. In particular, our adaptive algorithms inevitably go beyond single-threshold algorithms. In contrast to the classic competitive ratio measure, where the optimal performance is obtained by deterministic single-threshold algorithms, our results for order-competitive ratio capture the intuition that adaptive algorithms may be more powerful than single-threshold ones, and randomized algorithms outperform deterministic ones.
A k-vertex connectivity oracle for an undirected graph G is a data structure that, given u, v \in V (G), reports min{\kappa(u, v), k + 1}, where \kappa (u, v) is the pairwise vertex connectivity between u, v. There are three main measures of efficiency: construction time, query time, and space. Prior work of Izsak and Nutov [Inform. Process. Lett., 112 (2012), pp. 39--43] produced a data structure of O(kn log n) words, which can even be encoded as a O(k log3 n)-bit labeling scheme, that can answer vertex connectivity queries in O(k log n) time. The construction time is polynomial but unspecified. In this paper, we address the top three complexity measures. (1) Space: We prove that any k-vertex connectivity oracle requires \Omega(kn) bits of space for any k \leqn. This proves that the Iszak-- Nutov data structure is optimal up to polylogarithmic factors for every k and that the sparsifiers of Nagamochi and Ibaraki [Algorithmica, 7 (1992), pp. 583--596] are optimal compression schemes for k-vertex connectivity up to a logarithmic factor. In particular, whereas all edge connectivities can be efficiently compressed (as a weighted (n-1)-edge Gomory--Hu tree), vertex connectivity admits no asymptotic compression: \Omega(n2) bits are necessary. We design a variation on Izsak and Nutov's data structure that uses O(min{kn log n,m log n log k}) words of space. (2) Query time: We answer queries in O (log n) time, improving on the \Omega(k log n) time bound of Izsak and Nutov [Inform. Process. Lett., 112 (2012), pp. 39--43]. The main idea is to build instances of SetIntersection data structures, with additional structure based on affine planes. This structure allows for query time that is linear in the output size, which evades some conditional lower bounds that are polynomial in the query set sizes [42, 51]. (3) Construction time: Our data structure can be constructed in the time of O(k3 polylog(n)) max-flow computations, namely, k3m1+o(1) time, using the recent near-linear time flow algorithm of [13]. The main technical contribution here is a fast algorithm to compute a (1+e)-approximate Gomory--Hu tree for element connectivity in the time of O(e-1 polylog(n)) max-flow computations. Element connectivity is a notion that generalizes edge and vertex connectivity.
The polytope scheduling problem (PSP) was introduced by Im, Kulkarni, and Munagala [J. ACM, 65 (2018), pp. 3:1--3:33] as a very general abstraction of resource allocation over time, where jobs can receive processing rates subject to arbitrary packing constraints. It captures many well-studied problems, including classical unrelated machine scheduling, multidimensional scheduling, and broadcast scheduling. An elegant and well-known algorithm for instantaneous rate allocation with good fairness and efficiency properties is the proportional fairness (PF) algorithm, which was analyzed for PSP by Im, Kulkarni, and Munagala. We drastically improve the analysis of PF for both the general PSP and several of its important special cases subject to the objective of minimizing the sum of weighted completion times. We reduce the upper bound on the competitive ratio from 128 to 27 for general PSP and to 4 for the prominent class of monotone PSP. For certain heterogeneous machine environments, we even close the substantial gap to the lower bound of 2 for nonclairvoyant scheduling. Our analysis also gives the first polynomial-time improvement over the nearly 30-year-old bounds on the competitive ratio of the doubling framework, which was introduced by Hall, Shmoys, and Wein (SODA 1996) for clairvoyant online preemptive scheduling on unrelated machines. Somewhat surprisingly, we achieve this improvement by a nonclairvoyant algorithm, thereby demonstrating that nonclairvoyance is not a (significant) hurdle. Our improvements are based on exploiting monotonicity properties of PSP, providing tight dual fitting arguments on structured instances, and showing new algebraic properties of the optimal objective value for scheduling on unrelated machines. Finally, we establish new connections between PF and matching markets and thereby provide new insights on equilibria and their computational complexity.
Eigenvalue transformations, which include solving time-dependent differential equations as a special case, have a wide range of applications in scientific and engineering computation. While quantum algorithms for singular value transformations are well studied, eigenvalue transformations are distinct, especially for non-normal matrices. We propose an efficient quantum algorithm for performing a class of eigenvalue transformations that can be expressed as a certain type of matrix Laplace transformation. This allows us to significantly extend the recently developed linear combination of Hamiltonian simulation (LCHS) method [An, Liu, Lin, Phys. Rev. Lett. 131, 150603, 2023; An, Childs, Lin, arXiv:2312.03916] to represent a wider class of eigenvalue transformations, such as powers of the matrix inverse, A^-k, and the exponential of the matrix inverse, e^-A^-1. The latter can be interpreted as the solution of a mass-matrix differential equation of the form A u'(t)=-u(t). We demonstrate that our eigenvalue transformation approach can solve this problem without explicitly inverting A, reducing the computational complexity.
For every n, we construct a sum-of-squares identity (\sumni=1 x2i)(\sumjn=1 yj2 ) = \sumsk =1 fk2, where fk are bilinear forms with complex coefficients and s = O(n1.62). Previously, such a construction was known with s = O(n2/ log n). The same bound holds over any field of positive characteristic. As an application to complexity of noncommutative computation, we show that the polynomial IDn = \sumi,j\in[n] xiyjxiyj in 2n noncommuting variables can be computed by a noncommutative arithmetic circuit of size O(n1.96). This holds over any field of characteristic different from two. The same bound applies to noncommutative versions of the elementary symmetric polynomial of degree four and the rectangular permanent of a 4 imes n matrix.
A dot-product proof (DPP) is a simple probabilistic proof system in which the input statement x and the proof pi are vectors over a finite field F, and the proof is verified by making a single dot-product query < q, (x parallel to pi)> jointly to x and pi. A DPP can be viewed as a 1-query fully linear PCP. We study the feasibility and efficiency of DPPs, obtaining the following results: Small-field DPP. For any finite field F and Boolean circuit C of size S, there is a DPP for proving that there exists w such that C(x, w) = 1 with a proof pi of length S . poly(vertical bar F vertical bar) and soundness error epsilon = O(1/root vertical bar F vertical bar). We show this error to be asymptotically optimal. In particular, and in contrast to the best known PCPs, there exist strictly linear-length DPPs over constant-size fields. Large-field DPP. If vertical bar F vertical bar >= poly(S/epsilon), there is a similar DPP with soundness error epsilon and proof length O(S) (in field elements). The above results do not rely on the PCP theorem and their proofs are considerably simpler. We apply our DPP constructions toward two kinds of applications. Hardness of approximation. We obtain a simple proof for the NP-hardness of approximating MAXLIN (with dense instances) over any finite field F up to some constant factor c > 1, independent of F. Unlike previous PCP-based proofs, our proof yields exponential-time hardness under the exponential time hypothesis (ETH). Succinct arguments. We improve the concrete efficiency of succinct interactive arguments in the generic group model using input-independent preprocessing. In particular, the communication is comparable to sending two group elements and the verifier's computation is dominated by a single group exponentiation. We also show how to use DPPs together with linear-only encryption to construct succinct commit-and-prove arguments.
The fundamental theorem of Goldreich, Micali, and Wigderson (J. ACM 1991) shows that the existence of a one-way function is sufficient for constructing computational zero knowledge (CZK) proofs for all languages in NP. We prove its converse, thereby establishing characterizations of one-way functions based on the worst-case complexities of zero knowledge. Specifically, we prove that the following are equivalent: center dot A one-way function exists. center dot NP. CZK and NP is hard in the worst case. center dot CZK is hard in the worst case and the problem GapMCSP of approximating circuit complexity is in CZK. The characterization above also holds for statistical and computational zero-knowledge argument systems. We further extend this characterization to a proof system with knowledge complexity O(log n). In particular, we show that the existence of a one-way function is characterized by the worst-case hardness of CZK if GapMCSP has a proof system with knowledge complexity O( log n). We complement this result by showing that NP admits an interactive proof system with knowledge complexity omega( log n) under the existence of an exponentially hard auxiliary-input one-way function (which is a weaker primitive than an exponentially hard one-way function).We also characterize the existence of a robustly-often nonuniformly computable one-way function by the non-deterministic hardness of CZK under the weak assumption that PSPACE not subset of AM. We present two applications of our results. First, we simplify the proof of the recent characterization of a one-way function by NP-hardness of a meta-computational problem and the worst-case hardness of NP given by Hirahara (STOC'23). Second, we show that if NP has a laconic zero-knowledge argument system, then there exists a public-key encryption scheme whose security can be based on the worst-case hardness of NP. This improves previous results which assume the existence of an indistinguishable obfuscation.
We prove that a binary linear code of block length $n$ that is locally correctable with $3$ queries against a fraction $\delta > 0$ of adversarial errors must have dimension at most $O_{\delta}(\log^2 n \cdot \log \log n)$. This is almost tight in view of quadratic Reed-Muller codes being a $3$-query locally correctable code (LCC) with dimension $\Theta(\log^2 n)$. Our result improves, for the binary field case, the $O_{\delta}(\log^8 n)$ bound obtained in the recent breakthrough of (Kothari and Manohar, 2023) (arXiv:2311.00558) (and the more recent improvement to $O_{\delta}(\log^4 n)$ for binary linear codes announced in (Yankovitz, 2024)). Previous bounds for $3$-query linear LCCs proceed by constructing a $2$-query locally decodable code (LDC) from the $3$-query linear LCC/LDC and applying the strong bounds known for the former. Our approach is more direct and proceeds by bounding the covering radius of the dual code, borrowing inspiration from (Iceland and Samorodnitsky, 2018) (arXiv:1802.01184). That is, we show that if $x \mapsto (v_1 \cdot x, v_2 \cdot x, \ldots, v_n \cdot x)$ is an arbitrary encoding map $\mathbb{F}_2^k \to \mathbb{F}_2^n$ for the $3$-query LCC, then all vectors in $\mathbb{F}_2^k$ can be written as a $\widetilde{O}_{\delta}(\log n)$-sparse linear combination of the $v_i$'s, which immediately implies $k \le \widetilde{O}_{\delta}((\log n)^2)$. The proof of this fact proceeds by iteratively reducing the size of any arbitrary linear combination of at least $\widetilde{\Omega}_{\delta}(\log n)$ of the $v_i$'s. We achieve this using the recent breakthrough result of (Alon, Buci\'c, Sauermann, Zakharov, and Zamir, 2023) (arXiv:2309.04460) on the existence of rainbow cycles in properly edge-colored graphs, applied to graphs capturing the linear dependencies underlying the local correction property.
Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are necessary to prove a given theorem. In this work, we systematically explore the reverse mathematics of complexity lower bounds. We explore reversals in the setting of bounded arithmetic, with Cook's theory PV1 as the base theory, and show that several natural lower bound statements about communication complexity, error correcting codes, and Turing machines are equivalent to widely investigated combinatorial principles such as the weak pigeonhole principle for polynomial-time functions and its variants. As a consequence, complexity lower bounds can be formally seen as fundamental mathematical axioms with far-reaching implications. The proof-theoretic equivalence between complexity lower bound statements and combinatorial principles yields several new implications for the (un)provability of lower bounds. Among other results, we derive the following consequences: Under a plausible cryptographic assumption, the classical single-tape Turing machine Omega(n(2))-time lower bound for Palindrome is unprovable in Jerabek's theory APC(1). The conditional unprovability of this simple lower bound goes against the intuition shared by some researchers that most complexity lower bounds could be established in APC(1). While APC(1) proves one-way communication lower bounds for Set Disjointness, it does not prove one-way communication lower bounds for Equality, under a plausible cryptographic assumption. An amplification phenomenon connected to the (un)provability of some lower bounds, under which a quantitatively weak n(1+epsilon) lower bound is provable if and only if a stronger (and often tight) n(c) lower bound is provable. Feasibly definable randomized algorithms can be feasibly defined deterministically (APC(1) is for all Sigma(b)(1)-conservative over PV1) if and only if one-way communication complexity lower bound for Set Disjointness are provable in PV1.
We design polynomial size, constant depth (namely, $\mathsf{AC}^0$) arithmetic formulae for the greatest common divisor (GCD) of two polynomials, as well as the related problems of the discriminant, resultant, B\'ezout coefficients, squarefree decomposition, and the inversion of structured matrices like Sylvester and B\'ezout matrices. Our GCD algorithm extends to any number of polynomials. Previously, the best known arithmetic formulae for these problems required super-polynomial size, regardless of depth. These results are based on new algorithmic techniques to compute various symmetric functions in the roots of polynomials, as well as manipulate the multiplicities of these roots, without having access to them. These techniques allow $\mathsf{AC}^0$ computation of a large class of linear and polynomial algebra problems, which include the above as special cases. We extend these techniques to problems whose inputs are multivariate polynomials, which are represented by $\mathsf{AC}^0$ arithmetic circuits. Here too we solve problems such as computing the GCD and squarefree decomposition in $\mathsf{AC}^0$.
We consider the task of certifying that a random d-dimensional subspace X in R-n is well-spread-every vector x is an element of X satisfies c root n & Vert;x & Vert;2 <=& Vert;x & Vert;1 <=root n & Vert;x & Vert;2. In a seminal work, Barak et al. [Proceedings of the Forty-Fourth Annual ACM Symposium on Theory of Computing, ACM, New York, 2012, pp. 307-326] showed a polynomial-time certification algorithm when d <= O(root n). On the other hand, when d >>root n, the certification task is information-theoretically possible but there is evidence that it is computationally hard [C. Mao and A. S. Wein, Optimal Spectral Recovery of a Planted Vector in a Subspace, preprint, arXiv:2105.15081, 2021; H. Chen and T. d'Orsi, Proc. Mach. Learn, Res, (PMLR), 178 (2022), pp. 1-31], a phenomenon known as the information-computation gap. In this paper, we give subexponential-time certification algorithms in the d >>root n regime. Our algorithm runs in time exp((O)(n(epsilon))) when d <=(O)(n1+epsilon/2), establishing a smooth tradeoff between runtime and the dimension. Our techniques naturally extend to the related planted problem, where the task is to recover a sparse vector planted in a random subspace. Our algorithm achieves the same runtime and dimension tradeoff for this task.
We consider the time and space required for quantum computers to solve a wide variety of problems involving matrices, many of which have only been analyzed classically in prior work. Our main results show that for a range of linear algebra problems -- including matrix-vector product, matrix inversion, matrix multiplication and powering -- existing classical time-space tradeoffs, several of which are tight for every space bound, also apply to quantum algorithms. For example, for almost all matrices $A$, including the discrete Fourier transform (DFT) matrix, we prove that quantum circuits with at most $T$ input queries and $S$ qubits of memory require $T=\Omega(n^2/S)$ to compute matrix-vector product $Ax$ for $x \in \{0,1\}^n$. We similarly prove that matrix multiplication for $n\times n$ binary matrices requires $T=\Omega(n^3 / \sqrt{S})$. Because many of our lower bounds match deterministic algorithms with the same time and space complexity, we show that quantum computers cannot provide any asymptotic advantage for these problems with any space bound. We obtain matching lower bounds for the stronger notion of quantum cumulative memory complexity -- the sum of the space per layer of a circuit. We also consider Boolean (i.e. AND-OR) matrix multiplication and matrix-vector products, improving the previous quantum time-space tradeoff lower bounds for $n\times n$ Boolean matrix multiplication to $T=\Omega(n^{2.5}/S^{1/4})$ from $T=\Omega(n^{2.5}/S^{1/2})$. Our improved lower bound for Boolean matrix multiplication is based on a new coloring argument that extracts more from the strong direct product theorem used in prior work. Our tight lower bounds for linear algebra problems require adding a new bucketing method to the recording-query technique of Zhandry that lets us apply classical arguments to upper bound the success probability of quantum circuits.
Rational Identity Testing (RIT) is the decision problem of determining whether or not a noncommutative rational formula computes zero in the free skew field. It admits a deterministic polynomial-time white-box algorithm [Garg, Gurvits, Oliveira, and Wigderson (2016); Ivanyos, Qiao, Subrahmanyam (2018); Hamada and Hirai (2021)], and a randomized polynomial-time algorithm [Derksen and Makam (2017)] in the black-box setting, via singularity testing of linear matrices over the free skew field. Indeed, a randomized NC algorithm for RIT in the white-box setting follows from the result of Derksen and Makam (2017). Designing an efficient deterministic black-box algorithm for RIT and understanding the parallel complexity of RIT are major open problems in this area. Despite being open since the work of Garg, Gurvits, Oliveira, and Wigderson (2016), these questions have seen limited progress. In fact, the only known result in this direction is the construction of a quasipolynomial-size hitting set for rational formulas of only inversion height two [Arvind, Chatterjee, and Mukhopadhyay (2022)]. In this paper, we significantly improve the black-box complexity of this problem and obtain the first quasipolynomial-size hitting set for all rational formulas of polynomial size. Our construction also yields the first deterministic quasi-NC upper bound for RIT in the white-box setting.
The minimal faithful permutation degree \mu(G) of a finite group G is the smallest integer m for which there is an injective homomorphism \phi from G to Sm. The main result of this paper is a randomized polynomial-time algorithm for computing the minimal faithful permutation degree groups without abelian normal subgroups. Additionally, we show that: 1. For any primitive permutation group G, \mu(G) can be computed in quasi-polynomial time. 2. For a group G given by its Cayley table, \mu(G) can be computed in DSPACE(log3 | G| ).
