Given an n -vertex undirected graph G = ( V , E ) and positive edge weights { w e } e∈E , a linear arrangement is a permutation π : V → {1, 2, …, n }. The value of the arrangement is val ( G , π) := 1/n∑ e ={ u, v } ∈ E w e |π( u ) − π ( v )|. In the minimum linear arrangement problem, the goal is to find a linear arrangement π * that achieves val ( G , π * ) = MLA( G ) := min π val ( G , π). In this article, we show that for any ϵ > 0 and positive integer r , there is an n O ( r /ϵ) -time randomized algorithm that, given a graph G , returns a linear arrangement π, such that val ( G , π) ≤ (1 + 2/(1 − ε)λ r ( L )) MLA( G ) + O (√log n / n ∑ e ∈ E w e ) with high probability, where L is the normalized Laplacian of G and λ r ( L ) is the r th smallest eigenvalue of L . Our algorithm gives a constant factor approximation for regular graphs that are weak expanders.

We consider the problem of solving systems of multivariate polynomial equations of degree k over a finite field. For every integer k ≤ 2 and finite field q where q = pd for a prime p, we give, to the best of our knowledge, the first algorithms that achieve an exponential speedup over the brute force O(qn) time algorithm in the worst case. We present two algorithms, a randomized algorithm with running time qn+o(n) · q−n/O(k) time if q < 24ekd, and otherwise, where e = 2.718… is Napier's constant, and a deterministic algorithm for counting solutions with running time qn+o(n) · q−n/O(kq6/7d). For the important special case of quadratic equations in F2, our randomized algorithm has running time O(20.8765n). For systems over 2 we also consider the case where the input polynomials do not have bounded degree, but instead can be efficiently represented as a ΣΠΣ circuit, i.e., a sum of products of sums of variables. For this case we present a deterministic algorithm running in time 2n-dn for δ = 1/O(log(s/n)) for instances with s product gates in total and n variables. Our algorithms adapt several techniques recently developed via the polynomial method from circuit complexity. The algorithm for systems of ΣΠΣ polynomials also introduces a new degree reduction method that takes an instance of the problem and outputs a subexponential-sized set of instances, in such a way that feasibility is preserved and every polynomial among the output instances has degree O(log(s/n)).

Most of the known lower bounds for binary Boolean circuits with unrestricted depth are proved by the gate elimination method. The most efficient known algorithms for the #SAT problem on binary Boolean circuits use similar case analyses to the ones in gate elimination. Chen and Kabanets recently showed that the known case analyses can also be used to prove average case circuit lower bounds, that is, lower bounds on the size of approximations of an explicit function. In this paper, we provide a general framework for proving worst/average case lower bounds for circuits and upper bounds for #SAT that is built on ideas of Chen and Kabanets. A proof in such a framework goes as follows. One starts by fixing three parameters: a class of circuits, a circuit complexity measure, and a set of allowed substitutions. The main ingredient of a proof goes as follows: by going through a number of cases, one shows that for any circuit from the given class, one can find an allowed substitution such that the given measure of the circuit reduces by a sufficient amount. This case analysis immediately implies an upper bound for #SAT. To~obtain worst/average case circuit complexity lower bounds one needs to present an explicit construction of a function that is a disperser/extractor for the class of sources defined by the set of substitutions under consideration. We show that many known proofs (of circuit size lower bounds and upper bounds for #SAT) fall into this framework. Using this framework, we prove the following new bounds: average case lower bounds of 3.24n and 2.59n for circuits over U_2 and B_2, respectively (though the lower bound for the basis B_2 is given for a quadratic disperser whose explicit construction is not currently known), and faster than 2^n #SAT-algorithms for circuits over U_2 and B_2 of size at most 3.24n and 2.99n, respectively. Here by B_2 we mean the set of all bivariate Boolean functions, and by U_2 the set of all bivariate Boolean functions except for parity and its complement.

A Boolean function f:{0,1}^n -> {0,1} is weighted symmetric if there exist a function g: Z -> {0,1} and integers w_0, w_1, ..., w_n such that f(x_1, ...,x_n) = g(w_0+sum_{i=1}^n w_i x_i) holds. In this paper, we present algorithms for the circuit satisfiability problem of bounded depth circuits with AND, OR, NOT gates and a limited number of weighted symmetric gates. Our algorithms run in time super-polynomially faster than 2^n even when the number of gates is super-polynomial and the maximum weight of symmetric gates is nearly exponential. With an additional trick, we give an algorithm for the maximum satisfiability problem that runs in time poly(n^t)*2^{n-n^{1/O(t)}} for instances with n variables, O(n^t) clauses and arbitrary weights. To the best of our knowledge, this is the first moderately exponential time algorithm even for Max 2SAT instances with arbitrary weights. Through the analysis of our algorithms, we obtain average-case lower bounds and compression algorithms for such circuits and worst-case lower bounds for majority votes of such circuits, where all the lower bounds are against the generalized Andreev function. Our average-case lower bounds might be of independent interest in the sense that previous ones for similar circuits with arbitrary symmetric gates rely on communication complexity lower bounds while ours are based on the restriction method.

We present improved exponential time exact algorithms for Max SAT. Our algorithms run in time of the form O(2^{(1-mu(c))n}) for instances with n variables and m=cn clauses. In this setting, there are three incomparable currently best algorithms: a deterministic exponential space algorithm with mu(c)=1/O(c * log(c)) due to Dantsin and Wolpert [SAT 2006], a randomized polynomial space algorithm with mu(c)=1/O(c * log^3(c)) and a deterministic polynomial space algorithm with mu(c)=1/O(c^2 * log^2(c)) due to Sakai, Seto and Tamaki [Theory Comput. Syst., 2015]. Our first result is a deterministic polynomial space algorithm with mu(c)=1/O(c * log(c)) that achieves the previous best time complexity without exponential space or randomization. Furthermore, this algorithm can handle instances with exponentially large weights and hard constraints. The previous algorithms and our deterministic polynomial space algorithm run super-polynomially faster than 2^n only if m=O(n^2). Our second results are deterministic exponential space algorithms for Max SAT with mu(c)=1/O((c * log(c))^{2/3}) and for Max 3-SAT with mu(c)=1/O(c^{1/2}) that run super-polynomially faster than 2^n when m=o(n^{5/2}/log^{5/2}(n)) and m=o(n^3/log^2(n)) respectively.

A temporal constraint language G is a set of relations with first-order definitions in (Q; <). Let CSP(G) denote the set of constraint satisfaction problem instances with relations from G. CSP(G) admits robust approximation if, for any e >= 0, given a (1-e)-satisfiable instance of CSP(G), we can compute an assignment that satisfies at least a (1-f(e))-fraction of constraints in polynomial time. Here, f(e) is some function satisfying f(0)=0 and f(e) goes 0 as e goes 0. Firstly, we give a qualitative characterization of robust approximability: Assuming the Unique Games Conjecture, we give a necessary and sufficient condition on G under which CSP(G) admits robust approximation. Secondly, we give a quantitative characterization of robust approximability: Assuming the Unique Games Conjecture, we precisely characterize how f(e) depends on e for each G. We show that our robust approximation algorithms can be run in almost linear time.

We present a moderately exponential time algorithm for the satisfiability of Boolean formulas over the full binary basis. For formulas of size at most cn, our algorithm runs in time 2(1-μc)nfor some constant μc>; 0. As a byproduct of the running time analysis of our algorithm, we get strong average-case hardness of affine extractors for linear-sized formulas over the full binary basis.

We say that an algorithm robustly decides a constraint satisfaction problem Π if it distinguishes at-least-(1 -ε)-satisfiable instances from less-than-(1 - r(ε))-satisfiable instances for some function r(ε) with r(ε) → 0 as ε → 0. In this paper we show that the canonical linear programming relaxation robustly decides Π if and only if Π has "width 1" (in the sense of Feder and Vardi).