We study the Ordered Covering (OC) problem. The input is a finite set of n elements X, a color function and a collection of subsets of X. A solution consists of an ordered tuple of sets from which covers X, and a colo...
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We study the Ordered Covering (OC) problem. The input is a finite set of n elements X, a color function and a collection of subsets of X. A solution consists of an ordered tuple of sets from which covers X, and a coloring such that , the first set covering x in the tuple, namely with , has color . The minimization version is to find a solution using the minimum number of sets. Variants of OC include OC in which each element of color appears in at most sets of , and k-OC in which the first set of the solution is required to have color 0, and there are at most alternations of colors in the solution. Among other results we showThere is a polynomialtime approximation algorithm for Min-OC(2, 2) with approximation ratio 2. (This is best possible unless Vertex Cover can be approximated within a ratio better than 2.) Moreover, Min-OC(2, 2) can be solved optimally in polynomialtime if the underlying instance is bipartite. For every , there is a polynomialtime approximation algorithm for Min-3-OC with approximation . Unless the unique games conjecture is false, this is best possible.
Boolean function f is k-interval if - input vector viewed as n-bit number - f is true for and only for inputs from given (at most) k intervals. Recognition of k-interval fuction given its DNF representation is coNP-ha...
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Boolean function f is k-interval if - input vector viewed as n-bit number - f is true for and only for inputs from given (at most) k intervals. Recognition of k-interval fuction given its DNF representation is coNP-hard problem. This thesis shows that for DNFs from a given solvable class (class C of DNFs is solvable if we can for any DNF F ∈ C decide F ≡ 1 in polynomialtime and C is closed under partial assignment) and fixed k we can decide whether F represents k-interval function in polynomialtime. 1
We study the circuit evaluation problem (also known as the compressed word problem) for finitely generated linear groups. The best upper bound for this problem is coRP (the complements of problems in randomized polyno...
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We study the circuit evaluation problem (also known as the compressed word problem) for finitely generated linear groups. The best upper bound for this problem is coRP (the complements of problems in randomized polynomialtime), which is shown by a reduction to polynomial identity testing for arithmetic circuits. Conversely, the compressed word problem for the linear group is equivalent to polynomial identity testing. In the paper, we show that the compressed word problem for every finitely generated nilpotent group is in . Within the larger class of polycyclic groups we find examples where the compressed word problem is at least as hard as polynomial identity testing for skew arithmetic circuits. It is a major open problem, whether polynomial identity testing for skew arithmetic circuits can be solved in polynomialtime.
This work addresses four single-machine scheduling problems (SMSPs) with learning effects and variable maintenance activity. The processing times of the jobs are simultaneously determined by a decreasing function of t...
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This work addresses four single-machine scheduling problems (SMSPs) with learning effects and variable maintenance activity. The processing times of the jobs are simultaneously determined by a decreasing function of their corresponding scheduled positions and the sum of the processing times of the already processed jobs. Maintenance activity must start before a deadline and its duration increases with the starting time of the maintenance activity. This work proposes a polynomial-time algorithm for optimally solving two SMSPs to minimize the total completion time and the total tardiness with a common due date.
We show that one can approximate the least fixed point solution for a multivariate system of monotone probabilistic polynomial equations in timepolynomial in both the encoding size of the system of equations and in l...
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We show that one can approximate the least fixed point solution for a multivariate system of monotone probabilistic polynomial equations in timepolynomial in both the encoding size of the system of equations and in log(1/c), where c > 0 is the desired additive error bound of the solution. (The model of computation is the standard Turing machine model.) We use this result to resolve several open problems regarding the computational complexity of computing key quantities associated with some classic and well studied stochastic processes, including multitype branching processes and stochastic context-free grammars.
The celebrated Brascamp-Lieb (BL) inequalities [BL76,Lie90], and their reverse form of Barthe [Bar98], are an important mathematical tool, unifying and generalizing numerous inequalities in analysis, convex geometry a...
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The celebrated Brascamp-Lieb (BL) inequalities [BL76,Lie90], and their reverse form of Barthe [Bar98], are an important mathematical tool, unifying and generalizing numerous inequalities in analysis, convex geometry and information theory, with many used in computer science. While their structural theory is very well understood, far less is known about computing their main parameters (which we later define below). Prior to this work, the best known algorithms for any of these optimization tasks required at least exponential time. In this work, we give polynomial time algorithms to compute: (1) Feasibility of BL-datum, (2) Optimal BL-constant, (3) Weak separation oracle for BL-polytopes. What is particularly exciting about this progress, beyond the better understanding of BL-inequalities, is that the objects above naturally encode rich families of optimization problems which had no prior efficient algorithms. In particular, the BL-constants (which we efficiently compute) are solutions to non-convex optimization problems, and the BL-polytopes (for which we provide efficient membership and separation oracles) are linear programs with exponentially many facets. Thus we hope that new combinatorial optimization problems can be solved via reductions to the ones above, and make modest initial steps in exploring this possibility. Our algorithms are obtained by a simple efficient reduction of a given BL-datum to an instance of the Operator Scaling problem defined by [Gur04]. To obtain the results above, we utilize the two (very recent and different) algorithms for the operator scaling problem [GGOW16,IQS15]. Our reduction implies algorithmic versions of many of the known structural results on BL-inequalities, and in some cases provide proofs that are different or simpler than existing ones. Further, the analytic properties of the [GGOW16] algorithm provide new, effective bounds on the magnitude and continuity of BL-constants;prior work relied on compactness, and thus provided
We consider the bounded parallel-batch scheduling with proportional-linear deterioration and outsourcing, in which the actual processing time is p(j) = alpha(j)(A+Dt) or p(j) = alpha(j)t. A job is either accepted and ...
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We consider the bounded parallel-batch scheduling with proportional-linear deterioration and outsourcing, in which the actual processing time is p(j) = alpha(j)(A+Dt) or p(j) = alpha(j)t. A job is either accepted and processed in batches on a singlemachine bymanufactures themselves or outsourced to the third party with a certain penalty having to be paid. The objective is to minimize the maximum completion time of the accepted jobs and the total penalty of the outsourced jobs. For the p(j) = alpha(j)(A+Dt) model, when all the jobs are released at time zero, we show that the problem is NP-hard and present a pseudo-polynomialtime algorithm, respectively. For the p(j) - alpha(j)t model, when the jobs have distinct m(
#SMT, or model counting for logical theories, is a well-known hard problem that generalizes such tasks as counting the number of satisfying assignments to a Boolean formula and computing the volume of a polytope. In t...
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#SMT, or model counting for logical theories, is a well-known hard problem that generalizes such tasks as counting the number of satisfying assignments to a Boolean formula and computing the volume of a polytope. In the realm of satisfiability modulo theories (SMT) there is a growing need for model counting solvers, coming from several application domains (quantitative information flow, static analysis of probabilistic programs). In this paper, we show a reduction from an approximate version of #SMT to SMT. We focus on the theories of integer arithmetic and linear real arithmetic. We propose model counting algorithms that provide approximate solutions with formal bounds on the approximation error. They run in polynomialtime and make a polynomial number of queries to the SMT solver for the underlying theory, exploiting "for free" the sophisticated heuristics implemented within modern SMT solvers. We have implemented the algorithms and used them to solve the value problem for a model of loop-free probabilistic programs with nondeterminism.
One of the central challenges in the study of quantum many-body systems is the complexity of simulating them on a classical computer. A recent advance (Landau et al. in Nat Phys, 2015) gave a polynomialtime algorithm...
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One of the central challenges in the study of quantum many-body systems is the complexity of simulating them on a classical computer. A recent advance (Landau et al. in Nat Phys, 2015) gave a polynomialtime algorithm to compute a succinct classical description for unique ground states of gapped 1D quantum systems. Despite this progress many questions remained unsolved, including whether there exist efficient algorithms when the ground space is degenerate (and of polynomial dimension in the system size), or for the polynomially many lowest energy states, or even whether such states admit succinct classical descriptions or area laws. In this paper we give a new algorithm, based on a rigorously justified RG type transformation, for finding low energy states for 1D Hamiltonians acting on a chain of n particles. In the process we resolve some of the aforementioned open questions, including giving a polynomialtime algorithm for poly(n) degenerate ground spaces and an n (O(log) n) algorithm for the poly(n) lowest energy states (under a mild density condition). For these classes of systems the existence of a succinct classical description and area laws were not rigorously proved before this work. The algorithms are natural and efficient, and for the case of finding unique ground states for frustration-free Hamiltonians the running time is , where M(n) is the time required to multiply two n x n matrices.
We give a quasi-polynomialtime classical algorithm for estimating the ground state energy and for computing low energy states of quantum impurity models. Such models describe a bath of free fermions coupled to a smal...
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We give a quasi-polynomialtime classical algorithm for estimating the ground state energy and for computing low energy states of quantum impurity models. Such models describe a bath of free fermions coupled to a small interacting subsystem called an impurity. The full system consists of n fermionic modes and has a Hamiltonian H = H-0 + H-imp, where H-0 is quadratic in creation-annihilation operators and H-imp is an arbitrary Hamiltonian acting on a subset of O(1) modes. We show that the ground energy of H can be approximated with an additive error 2(-b) in time n(3) exp [O(b(3))]. Our algorithm also finds a low energy state that achieves this approximation. The low energy state is represented as a superposition of exp [O(b(3))] fermionic Gaussian states. To arrive at this result we prove several theorems concerning exact ground states of impurity models. In particular, we show that eigenvalues of the ground state covariance matrix decay exponentially with the exponent depending very mildly on the spectral gap of H-0. A key ingredient of our proof is Zolotarev's rational approximation to the root x function. We anticipate that our algorithms may be used in hybrid quantum-classical simulations of strongly correlated materials based on dynamical mean field theory. We implemented a simplified practical version of our algorithm and benchmarked it using the single impurity Anderson model.
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