Computer Science

The present work argues that strong arithmetic circuit lower bounds yield Boolean circuit lower bounds. In particular we show that the De Morgan Boolean formula complexity upper-bounds algebraic variants of the Kolomogorov complexity measure of partial differential incarnations of Turing machines. We devise from this connection new non-trivial upper and lower bounds for the De Morgan Boolean formula complexity of some familiar Boolean functions.

We study approximability of regular constraint satisfaction problems, i.e., CSPs where each variable in an instance has the same number of occurrences. In particular, we show that for any CSP Λ , existence of an α approximation algorithm for unweighted regular Max-CSP Λ implies existence of an α−o(1) approximation algorithm for weighted Max-CSP Λ in which regularity of the instances is not imposed. We also give an analogous result for Min-CSPs, and therefore show that up to arbitrarily small error it is sufficient to conduct the study of approximability of CSPs only on regular unweighted instances.

In this paper we study the separation between two complexity measures: the degree of a Boolean function as a polynomial over the reals and its block sensitivity. We show that separation between these two measures can be improved from d 2 (f)?�bs(f) , established by Tal, to d 2 (f)?? 10 ????????)bs(f) . As a corollary, we show that separations between some other complexity measures are not tight as well, for instance, we can improve recent sensitivity conjecture result by Huang to s 4 (f)?? 10 ????????)bs(f) . Our techniques are based on paper by Nisan and Szegedy and include more detailed analysis of a symmetrization polynomial. In our next result we show the same type of improvement in the separation between the approximate degree of a Boolean function and its block sensitivity: we show that de g 2 1/3 (f)??6/101 ????????????bs(f) and improve the previous result by Nisan and Szegedy de g 1/3 (f)??bs(f)/6 ??????????????. In addition, we construct an example which shows that gap between constants in the lower bound and in the known upper bound is less than 0.2 . In our last result we study the properties of conjectured fully sensitive function on 10 variables of degree 4, existence of which would lead to improvement of the biggest known gap between these two measures. We prove that there is the only univariate polynomial that can be achieved by symmetrization of this function by using the combination of interpolation and linear programming techniques.

The Ulam's metric is the minimal number of moves consisting in removal of one element from a permutation and its subsequent reinsertion in different place, to go between two given permutations. Thet elements that are not moved create longest common subsequence of permutations. Aldous and Diaconis, in their paper, pointed that Ulam's metric had been introduced in the context of questions concerning sorting and tossing cards. In this paper we define and study Ulam's metric in highier dimensions: for dimension one the considered object is a pair of permutations, for dimension k it is a pair of k-tuples of permutations. Over encodings by k-tuples of permutations we define two dually related hierarchies. Our very first motivation come from Murata at al. paper, in which pairs of permutations were used as representation of topological relation between rectangles packed into minimal area with application to VLSI physical design. Our results concern hardness, approximability, and parametrized complexity inside the hierarchies.

We provide a computational complexity analysis for the Sinkhorn algorithm that solves the entropic regularized Unbalanced Optimal Transport (UOT) problem between two measures of possibly different masses with at most n components. We show that the complexity of the Sinkhorn algorithm for finding an ε -approximate solution to the UOT problem is of order O ˜ ( n 2 /ε) , which is near-linear time. To the best of our knowledge, this complexity is better than the complexity of the Sinkhorn algorithm for solving the Optimal Transport (OT) problem, which is of order O ˜ ( n 2 / ε 2 ) . Our proof technique is based on the geometric convergence of the Sinkhorn updates to the optimal dual solution of the entropic regularized UOT problem and some properties of the primal solution. It is also different from the proof for the complexity of the Sinkhorn algorithm for approximating the OT problem since the UOT solution does not have to meet the marginal constraints.

Romashchenko and Zimand~\cite{rom-zim:c:mutualinfo} have shown that if we partition the set of pairs (x,y) of n -bit strings into combinatorial rectangles, then I(x:y)≥I(x:y∣t(x,y))−O(logn) , where I denotes mutual information in the Kolmogorov complexity sense, and t(x,y) is the rectangle containing (x,y) . We observe that this inequality can be extended to coverings with rectangles which may overlap. The new inequality essentially states that in case of a covering with combinatorial rectangles, I(x:y)≥I(x:y∣t(x,y))−logρ−O(logn) , where t(x,y) is any rectangle containing (x,y) and ρ is the thickness of the covering, which is the maximum number of rectangles that overlap. We discuss applications to communication complexity of protocols that are nondeterministic, or randomized, or Arthur-Merlin, and also to the information complexity of interactive protocols.

In the Token Jumping problem we are given a graph G=(V,E) and two independent sets S and T of G , each of size k≥1 . The goal is to determine whether there exists a sequence of k -sized independent sets in G , ⟨ S 0 , S 1 ,…, S ℓ ⟩ , such that for every i , | S i |=k , S i is an independent set, S= S 0 , S ℓ =T , and | S i Δ S i+1 |=2 . In other words, if we view each independent set as a collection of tokens placed on a subset of the vertices of G , then the problem asks for a sequence of independent sets which transforms S to T by individual token jumps which maintain the independence of the sets. This problem is known to be PSPACE-complete on very restricted graph classes, e.g., planar bounded degree graphs and graphs of bounded bandwidth. A closely related problem is the Token Sliding problem, where instead of allowing a token to jump to any vertex of the graph we instead require that a token slides along an edge of the graph. Token Sliding is also known to be PSPACE-complete on the aforementioned graph classes. We investigate the parameterized complexity of both problems on several graph classes, focusing on the effect of excluding certain cycles from the input graph. In particular, we show that both Token Sliding and Token Jumping are fixed-parameter tractable on C 4 -free bipartite graphs when parameterized by k . For Token Jumping, we in fact show that the problem admits a polynomial kernel on { C 3 , C 4 } -free graphs. In the case of Token Sliding, we also show that the problem admits a polynomial kernel on bipartite graphs of bounded degree. We believe both of these results to be of independent interest. We complement these positive results by showing that, for any constant p≥4 , both problems are W[1]-hard on { C 4 ,…, C p } -free graphs and Token Sliding remains W[1]-hard even on bipartite graphs.

Let a polyhedron P be defined by one of the following ways: (i) P={x∈ R n :Ax≤b} , where A∈ Z (n+k)×n , b∈ Z (n+k) and rankA=n ; (ii) P={x∈ R n + :Ax=b} , where A∈ Z k×n , b∈ Z k and rankA=k . And let all rank order minors of A be bounded by Δ in absolute values. We show that the short rational generating function for the power series ∑ m∈P∩ Z n x m can be computed with the arithmetic complexity O( T SNF (d)⋅ d k ⋅ d log 2 Δ ), where k and Δ are fixed, d=dimP , and T SNF (m) is the complexity to compute the Smith Normal Form for m×m integer matrix. In particular, d=n for the case (i) and d=n−k for the case (ii). The simplest examples of polyhedra that meet conditions (i) or (ii) are the simplicies, the subset sum polytope and the knapsack or multidimensional knapsack polytopes. We apply these results to parametric polytopes, and show that the step polynomial representation of the function c P (y)=| P y ∩ Z n | , where P y is parametric polytope, can be computed by a polynomial time even in varying dimension if P y has a close structure to the cases (i) or (ii). As another consequence, we show that the coefficients e i (P,m) of the Ehrhart quasi-polynomial |mP∩ Z n |= ∑ j=0 n e i (P,m) m j can be computed by a polynomial time algorithm for fixed k and Δ .

We study parity decision trees for Boolean functions. The motivation of our study is the log-rank conjecture for XOR functions and its connection to Fourier analysis and parity decision tree complexity. Let f be a Boolean function with Fourier support S and Fourier sparsity k. 1) We prove via the probabilistic method that there exists a parity decision tree of depth O(sqrt k) that computes f. This matches the best known upper bound on the parity decision tree complexity of Boolean functions (Tsang, Wong, Xie, and Zhang, FOCS 2013). Moreover, while previous constructions (Tsang et al., FOCS 2013, Shpilka, Tal, and Volk, Comput. Complex. 2017) build the trees by carefully choosing the parities to be queried in each step, our proof shows that a naive sampling of the parities suffices. 2) We generalize the above result by showing that if the Fourier spectra of Boolean functions satisfy a natural "folding property", then the above proof can be adapted to establish existence of a tree of complexity polynomially smaller than O(sqrt k). We make a conjecture in this regard which, if true, implies that the communication complexity of an XOR function is bounded above by the fourth root of the rank of its communication matrix, improving upon the previously known upper bound of square root of rank (Tsang et al., FOCS 2013, Lovett, J. ACM. 2016). 3) It can be shown by elementary techniques that for any Boolean function f and all pairs (alpha, beta) of parities in S, there exists another pair (gamma, delta) of parities in S such that alpha + beta = gamma + delta. We show, among other results, that there must exist several gamma in F_2^n such that there are at least three pairs (alpha_1, alpha_2) of parities in S with alpha_1 + alpha_2 = gamma.

The subject logic in computer science should entail proof theoretic applications. So the question arises whether open problems in computational complexity can be solved by advanced proof theoretic techniques. In particular, consider the complexity classes NP, coNP and PSPACE. It is well-known that NP and coNP are contained in PSPACE, but till recently precise characterization of these relationships remained open. Now [2], [3] (see also [4]) presented proofs of corresponding equalities NP = coNP = PSPACE. These results were obtained by appropriate proof theoretic tree-to-dag compressing techniques to be briefy explained below. [2] L. Gordeev, E. H. Haeusler, Proof Compression and NP Versus PSPACE, Studia Logica (107) (1): 55{83 (2019) [3] L. Gordeev, E. H. Haeusler, Proof Compression and NP Versus PSPACE II, Bulletin of the Section of Logic (49) (3): 213{230 (2020) this http URL [4] L. Gordeev, E. H. Haeusler, Proof Compression and NP Versus PSPACE II: Addendum, arXiv:2011.09262 (2020)