Computer Science

Assuming the Unique Games Conjecture, we show that existing approximation algorithms for some Boolean Max-2-CSPs with cardinality constraints are optimal. In particular, we prove that Max-Cut with cardinality constraints is UG-hard to approximate within \approx 0.858, and that Max-2-Sat with cardinality constraints is UG-hard to approximate within \approx 0.929. In both cases, the previous best hardness results were the same as the hardness of the corresponding unconstrained Max-2-CSP (\approx 0.878 for Max-Cut, and \approx 0.940 for Max-2-Sat). The hardness obtained for Max-2-Sat applies to monotone Max-2-Sat instances, meaning that we also obtain tight inapproximability for the Max-k-Vertex-Cover problem.

A previous article shows that any linear height bounded normal proof of a tautology in the Natural Deduction for Minimal implicational logic M ⊃ is as huge as it is redundant. More precisely, any proof in a family of super-polynomially sized and linearly height bounded proofs have a sub-derivation that occurs super-polynomially many times in it. In this article, we show that by collapsing all the repeated sub-derivations we obtain a smaller structure, a rooted Directed Acyclic Graph (r-DAG), that is polynomially upper-bounded on the size of α and it is a certificate that α is a tautology that can be verified in polynomial time. In other words, for every huge proof of a tautology in M ⊃ , we obtain a succinct certificate for its validity. Moreover, we show an algorithm able to check this validity in polynomial time on the certificate's size. Comments on how the results in this article are related to a proof of the conjecture NP=CoNP appears in conclusion.

We study homomorphism polynomials, which are polynomials that enumerate all homomorphisms from a pattern graph H to n -vertex graphs. These polynomials have received a lot of attention recently for their crucial role in several new algorithms for counting and detecting graph patterns, and also for obtaining natural polynomial families which are complete for algebraic complexity classes VBP , VP , and VNP . We discover that, in the monotone setting, the formula complexity, the ABP complexity, and the circuit complexity of such polynomial families are exactly characterized by the treedepth, the pathwidth, and the treewidth of the pattern graph respectively. Furthermore, we establish a single, unified framework, using our characterization, to collect several known results that were obtained independently via different methods. For instance, we attain superpolynomial separations between circuits, ABPs, and formulas in the monotone setting, where the polynomial families separating the classes all correspond to well-studied combinatorial problems. Moreover, our proofs rediscover fine-grained separations between these models for constant-degree polynomials. The characterization additionally yields new space-time efficient algorithms for several pattern detection and counting problems.

Let G=(V,E) be a vertex-colored graph, where C is the set of colors used to color V . The Graph Motif (or GM) problem takes as input G , a multiset M of colors built from C , and asks whether there is a subset S⊆V such that (i) G[S] is connected and (ii) the multiset of colors obtained from S equals M . The Colorful Graph Motif (or CGM) problem is the special case of GM in which M is a set, and the List-Colored Graph Motif (or LGM) problem is the extension of GM in which each vertex v of V may choose its color from a list L(v)⊆C of colors. We study the three problems GM, CGM, and LGM, parameterized by the dual parameter ℓ:=|V|−|M| . For general graphs, we show that, assuming the strong exponential time hypothesis, CGM has no (2−ϵ ) ℓ ⋅|V | O(1) -time algorithm, which implies that a previous algorithm, running in O( 2 ℓ ⋅|E|) time is optimal [Betzler et al., IEEE/ACM TCBB 2011]. We also prove that LGM is W[1]-hard with respect to ℓ even if we restrict ourselves to lists of at most two colors. If we constrain the input graph to be a tree, then we show that GM can be solved in O( 3 ℓ ⋅|V|) time but admits no polynomial-size problem kernel, while CGM can be solved in O( 2 – √ ℓ +|V|) time and admits a polynomial-size problem kernel.

We consider bounded width CNF-formulas where the width is measured by popular graph width measures on graphs associated to CNF-formulas. Such restricted graph classes, in particular those of bounded treewidth, have been extensively studied for their uses in the design of algorithms for various computational problems on CNF-formulas. Here we consider the expressivity of these formulas in the model of clausal encodings with auxiliary variables. We first show that bounding the width for many of the measures from the literature leads to a dramatic loss of expressivity, restricting the formulas to such of low communication complexity. We then show that the width of optimal encodings with respect to different measures is strongly linked: there are two classes of width measures, one containing primal treewidth and the other incidence cliquewidth, such that in each class the width of optimal encodings only differs by constant factors. Moreover, between the two classes the width differs at most by a factor logarithmic in the number of variables. Both these results are in stark contrast to the setting without auxiliary variables where all width measures we consider here differ by more than constant factors and in many cases even by linear factors.

We consider the following string matching problem on a node-labeled graph G=(V,E) : given a pattern string P , decide whether there exists a path in G whose concatenation of node labels equals P . This is a basic primitive in various problems in bioinformatics, graph databases, or networks. The hardness results of Backurs and Indyk (FOCS 2016) imply that this problem cannot be solved in better than O(|E||P|) time, under the Orthogonal Vectors Hypothesis (OVH), and this holds even under various restrictions on the graph (Equi et al., ICALP 2019). In this paper we consider its offline version, namely the one in which we are allowed to index the graph in order to support time-efficient string matching queries. Indeed, it was tantalizing in the string matching community to believe that sub-quadratic time queries can be achieved, e.g. at the cost of a high-degree polynomial-time indexing. We disprove this belief, showing that, under OVH, no polynomial-time index can support querying P in time O(|E | δ |P | β ) , with either δ<1 or β<1 . We prove this tight bound employing a known self-reducibility technique, e.g. from the field of dynamic algorithms, which translates conditional lower bounds for an online problem to its offline version. As a side-contribution, we formalize this technique with the notion of linear independent-components reduction, allowing for a simple proof of our result. As another illustration of our technique, we also translate the quadratic conditional lower bound of Backurs and Indyk (STOC 2015) for the problem of matching a query string inside a text, under edit distance. We obtain an analogous tight quadratic lower bound for its offline version, improving the recent result of Cohen-Addad, Feuilloley and Starikovskaya (SODA 2019), but with a slightly different boundary condition.

We show that there is a dense set $\ourset\subseteq \mathbb{N}$ of group orders and a constant c such that for every $n\in \ourset$ we can decide in time O( n 2 (logn ) c ) whether two n×n multiplication tables describe isomorphic groups of order n . This improves significantly over the general n O(logn) -time complexity and shows that group isomorphism can be tested efficiently for almost all group orders n . We also show that in time O( n 2 (logn ) c ) it can be decided whether an n×n multiplication table describes a group; this improves over the known O( n 3 ) complexity. Our complexities are calculated for a deterministic multi-tape Turing machine model. We give the implications to a RAM model in the promise hierarchy as well.

The first-fit coloring is a heuristic that assigns to each vertex, arriving in a specified order σ , the smallest available color. The problem Grundy Coloring asks how many colors are needed for the most adversarial vertex ordering σ , i.e., the maximum number of colors that the first-fit coloring requires over all possible vertex orderings. Since its inception by Grundy in 1939, Grundy Coloring has been examined for its structural and algorithmic aspects. A brute-force f(k) n 2 k−1 -time algorithm for Grundy Coloring on general graphs is not difficult to obtain, where k is the number of colors required by the most adversarial vertex ordering. It was asked several times whether the dependency on k in the exponent of n can be avoided or reduced, and its answer seemed elusive until now. We prove that Grundy Coloring is W[1]-hard and the brute-force algorithm is essentially optimal under the Exponential Time Hypothesis, thus settling this question by the negative. The key ingredient in our W[1]-hardness proof is to use so-called half-graphs as a building block to transmit a color from one vertex to another. Leveraging the half-graphs, we also prove that b-Chromatic Core is W[1]-hard, whose parameterized complexity was posed as an open question by Panolan et al. [JCSS '17]. A natural follow-up question is, how the parameterized complexity changes in the absence of (large) half-graphs. We establish fixed-parameter tractability on K t,t -free graphs for b-Chromatic Core and Partial Grundy Coloring, making a step toward answering this question. The key combinatorial lemma underlying the tractability result might be of independent interest.

This paper shows NP-completeness for finding Hamiltonian cycles in induced subgraphs of the dual graphs of semi-regular tessilations. It also shows NP-hardness for a new, wide class of graphs called augmented square grids. This work follows up on prior studies of the complexity of finding Hamiltonian cycles in regular and semi-regular grid graphs.

The relationship between the complexity classes P and NP is an unsolved question in the field of theoretical computer science. In the first part of this paper, a lattice framework is proposed to handle the 3-CNF-SAT problems, known to be in NP . In the second section, we define a multi-linear descriptor function H φ for any 3-CNF-SAT problem φ of size n , in the sense that H φ :{0,1 } n →{0,1 } n is such that Im H φ is the set of all the solutions of φ . A new merge operation H φ ⋀ H ψ is defined, where ψ is a single 3-CNF clause. Given H φ [but this can be of exponential complexity], the complexity needed for the computation of Im H φ , the set of all solutions, is shown to be polynomial for hard 3-CNF-SAT problems, i.e. the one with few ( ≤ 2 k ) or no solutions. The third part uses the relation between H φ and the indicator function 1 S φ for the set of solutions, to develop a greedy polynomial algorithm to solve hard 3-CNF-SAT problems.