The Power of Human–Algorithm Collaboration in Solving Combinatorial Optimization Problems

Many combinatorial optimization problems are often considered intractable to solve exactly or by approximation. An example of such a problem is maximum clique, which—under standard assumptions in complexity theory—cannot be solved in sub-exponential time or be approximated within the polynomial factor efficiently. However, we show that if a polynomial time algorithm can query informative Gaussian priors from an expert poly(n) times, then a class of combinatorial optimization problems can be solved efficiently up to a multiplicative factor ϵ, where ϵ is arbitrary constant. In this paper, we present proof of our claims and show numerical results to support them. Our methods can cast new light on how to approach optimization problems in domains where even the approximation of the problem is not feasible. Furthermore, the results can help researchers to understand the structures of these problems (or whether these problems have any structure at all!). While the proposed methods can be used to approximate combinatorial problems in NPO, we note that the scope of the problems solvable might well include problems that are provable intractable (problems in EXPTIME).

Exact Algorithms for the Minimum Load Spanning Tree Problem

In a minimum load spanning tree (MLST) problem, we are given an undirected graph and nondecreasing load functions for nodes defined on nodes’ degrees in a spanning tree, and the objective is to find a spanning tree that minimizes the maximum load among all nodes. We propose the first [Formula: see text] time exact algorithm for the MLST problem, where [Formula: see text] is the number of nodes and [Formula: see text] ignores polynomial factor. The algorithm is obtained by repeatedly querying whether a candidate objective value is feasible, where each query can be formulated as a bounded degree spanning tree problem (BDST). We propose a novel solution to BDST by extending an inclusion-exclusion based algorithm. To further enhance the time efficiency of the previous algorithm, we then propose a faster algorithm by generalizing the concept of branching walks. In addition, for the purpose of comparison, we give the first mixed integer linear programming formulation for MLST. In numerical analysis, we consider various load functions on a randomly generated network. The results verify the effectiveness of the proposed algorithms. Summary of Contribution: Minimum load spanning tree (MLST) plays an important role in various applications such as wireless sensor networks (WSNs). In many applications of WSNs, we often need to collect data from all sensors to some specified sink. In this paper, we propose the first exact algorithms for the MLST problem. Besides having theoretical guarantees, our algorithms have extraordinarily good performance in practice. We believe that our results make significant contributions to the field of graph theory, internet of things, and WSNs.

STRICTLY REAL FUNDAMENTAL THEOREM OF ALGEBRA USING POLYNOMIAL INTERLACING

Without resorting to complex numbers or any advanced topological arguments, we show that any real polynomial of degree greater than two always has a real quadratic polynomial factor, which is equivalent to the fundamental theorem of algebra. The proof uses interlacing of bivariate polynomials similar to Gauss's first proof of the fundamental theorem of algebra using complex numbers, but in a different context of division residues of strictly real polynomials. This shows the sufficiency of basic real analysis as the minimal platform to prove the fundamental theorem of algebra.

Algorithms for CRT-variant of Approximate Greatest Common Divisor Problem

The approximate greatest common divisor problem (ACD) and its variants have been used to construct many cryptographic primitives. In particular, the variants of the ACD problem based on Chinese remainder theorem (CRT) are being used in the constructions of a batch fully homomorphic encryption to encrypt multiple messages in one ciphertext. Despite the utility of the CRT-variant scheme, the algorithms that secures its security foundation have not been probed well enough.In this paper, we propose two algorithms and the results of experiments in which the proposed algorithms were used to solve the variant problem. Both algorithms take the same time complexity $\begin{array}{} \displaystyle 2^{\tilde{O}(\frac{\gamma}{(\eta-\rho)^2})} \end{array}$ up to a polynomial factor to solve the variant problem for the bit size of samples γ, secret primes η, and error bound ρ. Our algorithm gives the first parameter condition related to η and γ size. From the results of the experiments, it has been proved that the proposed algorithms work well both in theoretical and experimental terms.

Orthogonality of quasi-orthogonal polynomials

A result of P?lya states that every sequence of quadrature formulas Qn(f) with n nodes and positive Cotes numbers converges to the integral I(f) of a continuous function f provided Qn(f) = I(f) for a space of algebraic polynomials of certain degree that depends on n. The classical case when the algebraic degree of precision is the highest possible is well-known and the quadrature formulas are the Gaussian ones whose nodes coincide with the zeros of the corresponding orthogonal polynomials and the Cotes (Christoffel) numbers are expressed in terms of the so-called kernel polynomials. In many cases it is reasonable to relax the requirement for the highest possible degree of precision in order to gain the possibility to either approximate integrals of more specific continuous functions that contain a polynomial factor or to include additional fixed nodes. The construction of such quadrature processes is related to quasi-orthogonal polynomials. Given a sequence {Pn}n?0 of monic orthogonal polynomials and a fixed integer k, we establish necessary and sufficient conditions so that the quasi-orthogonal polynomials {Qn}n?0 defined by Qn(x) = Pn(x) + ?k-1,i=1 bi,nPn-i(x), n ? 0, with bi,n ? R, and bk-1,n ? 0 for n ? k-1, also constitute a sequence of orthogonal polynomials. Therefore we solve the inverse problem for linearly related orthogonal polynomials. The characterization turns out to be equivalent to some nice recurrence formulas for the coefficients bi,n. We employ these results to establish explicit relations between various types of quadrature rules from the above relations. A number of illustrative examples are provided.

COMPLEXITY OF SHORT GENERATING FUNCTIONS

We give complexity analysis for the class of short generating functions. Assuming #P$\not \subseteq$FP/poly, we show that this class is not closed under taking many intersections, unions or projections of generating functions, in the sense that these operations can increase the bit length of coefficients of generating functions by a super-polynomial factor. We also prove that truncated theta functions are hard for this class.

On Saturated $k$-Sperner Systems

Given a set $X$, a collection $\mathcal{F}\subseteq\mathcal{P}(X)$ is said to be $k$-Sperner if it does not contain a chain of length $k+1$ under set inclusion and it is saturated if it is maximal with respect to this property. Gerbner et al. conjectured that, if $|X|$ is sufficiently large with respect to $k$, then the minimum size of a saturated $k$-Sperner system $\mathcal{F}\subseteq\mathcal{P}(X)$ is $2^{k-1}$. We disprove this conjecture by showing that there exists $\varepsilon>0$ such that for every $k$ and $|X| \geq n_0(k)$ there exists a saturated $k$-Sperner system $\mathcal{F}\subseteq\mathcal{P}(X)$ with cardinality at most $2^{(1-\varepsilon)k}$.A collection $\mathcal{F}\subseteq \mathcal{P}(X)$ is said to be an oversaturated $k$-Sperner system if, for every $S\in\mathcal{P}(X)\setminus\mathcal{F}$, $\mathcal{F}\cup\{S\}$ contains more chains of length $k+1$ than $\mathcal{F}$. Gerbner et al. proved that, if $|X|\geq k$, then the smallest such collection contains between $2^{k/2-1}$ and $O\left(\frac{\log{k}}{k}2^k\right)$ elements. We show that if $|X|\geq k^2+k$, then the lower bound is best possible, up to a polynomial factor.

APPROXIMATING ASYMMETRIC TSP IN EXPONENTIAL TIME

Let G be a complete directed graph with n vertices and integer edge weights in range [0,M]. It is well known that an optimal Traveling Salesman Problem (TSP) in G can be solved in 2n time and space (all bounds are given within a polynomial factor of the input length, i.e., poly(n, log M)) and this is still the fastest known algorithm. If we allow a polynomial space only, then the best known algorithm has running time 4nnlog n. For TSP with bounded weights there is an algorithm with 1.657n · M running time. It is a big challenge to develop an algorithm with 2n time and polynomial space. Also, it is well-known that TSP cannot be approximated within any polynomial time computable function unless P=NP. In this short note we propose a very simple algorithm that, for any 0 < ε < 1, finds (1+ε)-approximation to asymmetric TSP in 2nε−1 time and ε−1 · poly(n, log M) space. Thereby, for any fixed ε, the algorithm needs 2n steps and polynomial space to compute (1 + ε)-approximation.