Computer Science

Brouwer's fixed point theorem states that any continuous function from a compact convex space to itself has a fixed point. Roughgarden and Weinstein (FOCS 2016) initiated the study of fixed point computation in the two-player communication model, where each player gets a function from [0,1 ] n to [0,1 ] n , and their goal is to find an approximate fixed point of the composition of the two functions. They left it as an open question to show a lower bound of 2 Ω(n) for the (randomized) communication complexity of this problem, in the range of parameters which make it a total search problem. We answer this question affirmatively. Additionally, we introduce two natural fixed point problems in the two-player communication model. ∙ Each player is given a function from [0,1 ] n to [0,1 ] n/2 , and their goal is to find an approximate fixed point of the concatenation of the functions. ∙ Each player is given a function from [0,1 ] n to [0,1 ] n , and their goal is to find an approximate fixed point of the interpolation of the functions. We show a randomized communication complexity lower bound of 2 Ω(n) for these problems (for some constant approximation factor). Finally, we initiate the study of finding a panchromatic simplex in a Sperner-coloring of a triangulation (guaranteed by Sperner's lemma) in the two-player communication model: A triangulation T of the d -simplex is publicly known and one player is given a set S A ⊂T and a coloring function from S A to {0,…,d/2} , and the other player is given a set S B ⊂T and a coloring function from S B to {d/2+1,…,d} , such that S A ∪ ˙ S B =T , and their goal is to find a panchromatic simplex. We show a randomized communication complexity lower bound of |T | Ω(1) for the aforementioned problem as well (when d is large).

We point out that the computation of true \emph{proof} and \emph{disproof} numbers for proof number search in arbitrary directed acyclic graphs is NP-hard, an important theoretical result for proof number search. The proof requires a reduction from SAT, which demonstrates that finding true proof/disproof number for arbitrary DAG is at least as hard as deciding if arbitrary SAT instance is satisfiable, thus NP-hard.

We study the problem of counting the number of homomorphisms from an input graph G to a fixed (quantum) graph H ¯ in any finite field of prime order Z p . The subproblem with graph H was introduced by Faben and Jerrum~[ToC'15] and its complexity is still uncharacterised despite active research, e.g. the very recent work of Focke, Goldberg, Roth, and Zivný~[SODA'21]. Our contribution is threefold. First, we introduce the study of quantum graphs to the study of modular counting homomorphisms. We show that the complexity for a quantum graph H ¯ collapses to the complexity criteria found at dimension 1: graphs. Second, in order to prove cases of intractability we establish a further reduction to the study of bipartite graphs. Lastly, we establish a dichotomy for all bipartite ( K 3,3 ∖{e},domino) -free graphs by a thorough structural study incorporating both local and global arguments. This result subsumes all results on bipartite graphs known for all prime moduli and extends them significantly. Even for the subproblem with p=2 this establishes new results.

We study the arithmetic circuit complexity of some well-known family of polynomials through the lens of parameterized complexity. Our main focus is on the construction of explicit algebraic branching programs (ABP) for determinant and permanent polynomials of the \emph{rectangular} symbolic matrix in both commutative and noncommutative settings. The main results are: 1. We show an explicit O ∗ (( n ↓k/2 )) -size ABP construction for noncommutative permanent polynomial of k×n symbolic matrix. We obtain this via an explicit ABP construction of size O ∗ (( n ↓k/2 )) for S ∗ n,k , noncommutative symmetrized version of the elementary symmetric polynomial S n,k . 2. We obtain an explicit O ∗ ( 2 k ) -size ABP construction for the commutative rectangular determinant polynomial of the k×n symbolic matrix. 3. In contrast, we show that evaluating the rectangular noncommutative determinant over rational matrices is W[1] -hard.

We show that there is no subexponential time algorithm for computing the exact solution of the maximum independent set problem in d-regular graphs unless ETH fails. We expand our method to show that it helps to provide lower bounds for other covering problems such as vertex cover and clique. We utilize the construction to show the NP-hardness of MIS on 5-regular planar graphs, closing the exact complexity status of the problem on regular planar graphs.

The main result is that: function descriptions are not made equal, and they can be categorised in at least two categories using various computational methods for function evaluation. The result affects Kolmogorov complexity and Random Oracle Model notions. More precisely, the idea that the size of an object and the size of the smallest computer program defining that object is a ratio that represents the object complexity needs additional definitions to hold its original assertions.

In the (k,h) -SetCover problem, we are given a collection S of sets over a universe U , and the goal is to distinguish between the case that S contains k sets which cover U , from the case that at least h sets in S are needed to cover U . Lin (ICALP'19) recently showed a gap creating reduction from the (k,k+1) -SetCover problem on universe of size O k (log|S|) to the (k, log|S| loglog|S| − − − − − − √ k ⋅k) -SetCover problem on universe of size |S| . In this paper, we prove a more scalable version of his result: given any error correcting code C over alphabet [q] , rate ρ , and relative distance δ , we use C to create a reduction from the (k,k+1) -SetCover problem on universe U to the (k, 2 1−δ − − − √ 2k ) -SetCover problem on universe of size log|S| ρ ⋅|U | q k . Lin established his result by composing the input SetCover instance (that has no gap) with a special threshold graph constructed from extremal combinatorial object called universal sets, resulting in a final SetCover instance with gap. Our reduction follows along the exact same lines, except that we generate the threshold graphs specified by Lin simply using the basic properties of the error correcting code C . We use the same threshold graphs mentioned above to prove inapproximability results, under W[1] ≠ FPT and ETH, for the k -MaxCover problem introduced by Chalermsook et al. (SICOMP'20). Our inapproximaiblity results match the bounds obtained by Karthik et al. (JACM'19), although their proof framework is very different, and involves generalization of the distributed PCP framework. Prior to this work, it was not clear how to adopt the proof strategy of Lin to prove inapproximability results for k -MaxCover.

List k-Coloring (Li k-Col) is the decision problem asking if a given graph admits a proper coloring compatible with a given list assignment to its vertices with colors in {1,2,..,k}. The problem is known to be NP-hard even for k=3 within the class of 3-regular planar bipartite graphs and for k=4 within the class of chordal bipartite graphs. In 2015, Huang, Johnson and Paulusma asked for the complexity of Li 3-Col in the class of chordal bipartite graphs. In this paper we give a partial answer to this question by showing that Li k-Col is polynomial in the class of convex bipartite graphs. We show first that biconvex bipartite graphs admit a multichain ordering, extending the classes of graphs where a polynomial algorithm of Enright, Stewart and Tardos (2014) can be applied to the problem. We provide a dynamic programming algorithm to solve the Li k-Col in the calss of convex bipartite graphs. Finally we show how our algorithm can be modified to solve the more general Li H-Col problem on convex bipartite graphs.

We prove that the equivalence of two fundamental problems in the theory of computing. For every polynomial t(n)≥(1+ε)n,ε>0 , the following are equivalent: - One-way functions exists (which in turn is equivalent to the existence of secure private-key encryption schemes, digital signatures, pseudorandom generators, pseudorandom functions, commitment schemes, and more); - t -time bounded Kolmogorov Complexity, K t , is mildly hard-on-average (i.e., there exists a polynomial p(n)>0 such that no PPT algorithm can compute K t , for more than a 1− 1 p(n) fraction of n -bit strings). In doing so, we present the first natural, and well-studied, computational problem characterizing the feasibility of the central private-key primitives and protocols in Cryptography.

I generalize a well-known result that P = NP fails for monotone polynomial circuits - more precisely, that the clique problem CLIQUE(k^4,k) is not solvable by Boolean (AND,OR)-circuits of the size polynomial in k. In the other words, there is no Boolean (AND,OR)-formula F expressing that a given graph with k^4 vertices contains a clique of k elements, provided that the circuit length of F, cl(F), is polynomial in k. In fact, for any solution F in question, cl(F) must be exponential in k. Moreover this holds also for DeMorgan normal (abbr.: DMN) (AND,OR)-formulas F that allow negated variables. Based on the latter observation I consider an arbitrary (AND,OR,NOT)-formula F and recall that standard NOT-conversions to DMN at most double its circuit length. Hence for any Boolean solution F of CLIQUE(k^4,k), cl(F) is exponential in k. I conclude that CLIQUE(k^4,k) is not solvable by polynomial-size Boolean circuits, and hence P is not NP. The entire proof is formalizable by standard methods in the exponential function arithmetic EFA.