Computer Science

We study a new reconfiguration problem inspired by classic mechanical puzzles: a colored token is placed on each vertex of a given graph; we are also given a set of distinguished cycles on the graph. We are tasked with rearranging the tokens from a given initial configuration to a final one by using cyclic shift operations along the distinguished cycles. We first investigate a large class of graphs, which generalizes several classic puzzles, and we give a characterization of which final configurations can be reached from a given initial configuration. Our proofs are constructive, and yield efficient methods for shifting tokens to reach the desired configurations. On the other hand, when the goal is to find a shortest sequence of shifting operations, we show that the problem is NP-hard, even for puzzles with tokens of only two different colors.

We consider the cyclotomic identity testing (CIT) problem: given a polynomial f( x 1 ,…, x k ) , decide whether f( ζ e 1 n ,…, ζ e k n ) is zero, where ζ n = e 2πi/n is a primitive complex n -th root of unity and e 1 ,…, e k are integers, represented in binary. When f is given by an algebraic circuit, we give a randomized polynomial-time algorithm for CIT assuming the generalised Riemann hypothesis (GRH), and show that the problem is in coNP unconditionally. When f is given by a circuit of polynomially bounded degree, we give a randomized NC algorithm. In case f is a linear form we show that the problem lies in NC. Towards understanding when CIT can be solved in deterministic polynomial-time, we consider so-called diagonal depth-3 circuits, i.e., polynomials f= ∑ m i=1 g d i i , where g i is a linear form and d i a positive integer given in unary. We observe that a polynomial-time algorithm for CIT on this class would yield a sub-exponential-time algorithm for polynomial identity testing. However, assuming GRH, we show that if the linear forms~ g i are all identical then CIT can be solved in polynomial time. Finally, we use our results to give a new proof that equality of compressed strings, i.e., strings presented using context-free grammars, can be decided in randomized NC.

In this paper, we investigate the relative power of several conjectures that attracted recently lot of interest. We establish a connection between the Network Coding Conjecture (NCC) of Li and Li and several data structure like problems such as non-adaptive function inversion of Hellman and the well-studied problem of polynomial evaluation and interpolation. In turn these data structure problems imply super-linear circuit lower bounds for explicit functions such as integer sorting and multi-point polynomial evaluation.

A decision list is an ordered list of rules. Each rule is specified by a term, which is a conjunction of literals, and a value. Given an input, the output of a decision list is the value corresponding to the first rule whose term is satisfied by the input. Decision lists generalize both CNFs and DNFs, and have been studied both in complexity theory and in learning theory. The size of a decision list is the number of rules, and its width is the maximal number of variables in a term. We prove that decision lists of small width can always be approximated by decision lists of small size, where we obtain sharp bounds. This in particular resolves a conjecture of Gopalan, Meka and Reingold (Computational Complexity, 2013) on DNF sparsification. An ingredient in our proof is a new random restriction lemma, which allows to analyze how DNFs (and more generally, decision lists) simplify if a small fraction of the variables are fixed. This is in contrast to the more commonly used switching lemma, which requires most of the variables to be fixed.

In a recent paper, Kim and Kopparty (Theory of Computing, 2017) gave a deterministic algorithm for the unique decoding problem for polynomials of bounded total degree over a general grid. We show that their algorithm can be adapted to solve the unique decoding problem for the general family of Downset codes. Here, a downset code is specified by a family D of monomials closed under taking factors: the corresponding code is the space of evaluations of all polynomials that can be written as linear combinations of monomials from D.

We prove logarithmic depth lower bounds in Stabbing Planes for the classes of combinatorial principles known as the Pigeonhole principle and the Tseitin contradictions. The depth lower bounds are new, obtained by giving almost linear length lower bounds which do not depend on the bit-size of the inequalities and in the case of the Pigeonhole principle are tight. The technique known so far to prove depth lower bounds for Stabbing Planes is a generalization of that used for the Cutting Planes proof system. In this work we introduce two new approaches to prove length/depth lower bounds in Stabbing Planes: one relying on Sperner's Theorem which works for the Pigeonhole principle and Tseitin contradictions over the complete graph; a second proving the lower bound for Tseitin contradictions over a grid graph, which uses a result on essential coverings of the boolean cube by linear polynomials, which in turn relies on Alon's combinatorial Nullenstellensatz.

Given a graph property Φ , we consider the problem EdgeSub(Φ) , where the input is a pair of a graph G and a positive integer k , and the task is to decide whether G contains a k -edge subgraph that satisfies Φ . Specifically, we study the parameterized complexity of EdgeSub(Φ) and of its counting problem #EdgeSub(Φ) with respect to both approximate and exact counting. We obtain a complete picture for minor-closed properties Φ : the decision problem EdgeSub(Φ) always admits an FPT algorithm and the counting problem #EdgeSub(Φ) always admits an FPTRAS. For exact counting, we present an exhaustive and explicit criterion on the property Φ which, if satisfied, yields fixed-parameter tractability and otherwise #W[1] -hardness. Additionally, most of our hardness results come with an almost tight conditional lower bound under the so-called Exponential Time Hypothesis, ruling out algorithms for #EdgeSub(Φ) that run in time f(k)⋅|G | o(k/logk) for any computable function f . As a main technical result, we gain a complete understanding of the coefficients of toroidal grids and selected Cayley graph expanders in the homomorphism basis of #EdgeSub(Φ) . This allows us to establish hardness of exact counting using the Complexity Monotonicity framework due to Curticapean, Dell and Marx (STOC'17). Our methods can also be applied to a parameterized variant of the Tutte Polynomial T k G of a graph G , to which many known combinatorial interpretations of values of the (classical) Tutte Polynomial can be extended. As an example, T k G (2,1) corresponds to the number of k -forests in the graph G . Our techniques allow us to completely understand the parametrized complexity of computing the evaluation of T k G at every pair of rational coordinates (x,y) .

We prove a complexity dichotomy theorem for a class of Holant problems on 3-regular bipartite graphs. Given an arbitrary nonnegative weighted symmetric constraint function f=[ x 0 , x 1 , x 2 , x 3 ] , we prove that the bipartite Holant problem Holant(f∣( = 3 )) is \emph{either} computable in polynomial time \emph{or} # P-hard. The dichotomy criterion on f is explicit.

The complexity of graph homomorphisms has been a subject of intense study [11, 12, 4, 42, 21, 17, 6, 20]. The partition function Z A (⋅) of graph homomorphism is defined by a symmetric matrix A over C . We prove that the complexity dichotomy of [6] extends to bounded degree graphs. More precisely, we prove that either G↦ Z A (G) is computable in polynomial-time for every G , or for some Δ>0 it is #P-hard over (simple) graphs G with maximum degree Δ(G)≤Δ . The tractability criterion on A for this dichotomy is explicit, and can be decided in polynomial-time in the size of A . We also show that the dichotomy is effective in that either a P-time algorithm for, or a reduction from #SAT to, Z A (⋅) can be constructed from A , in the respective cases.

In k -Digraph Coloring we are given a digraph and are asked to partition its vertices into at most k sets, so that each set induces a DAG. This well-known problem is NP-hard, as it generalizes (undirected) k -Coloring, but becomes trivial if the input digraph is acyclic. This poses the natural parameterized complexity question what happens when the input is "almost" acyclic. In this paper we study this question using parameters that measure the input's distance to acyclicity in either the directed or the undirected sense. It is already known that, for all k≥2 , k -Digraph Coloring is NP-hard on digraphs of DFVS at most k+4 . We strengthen this result to show that, for all k≥2 , k -Digraph Coloring is NP-hard for DFVS k . Refining our reduction we obtain two further consequences: (i) for all k≥2 , k -Digraph Coloring is NP-hard for graphs of feedback arc set (FAS) at most k 2 ; interestingly, this leads to a dichotomy, as we show that the problem is FPT by k if FAS is at most k 2 −1 ; (ii) k -Digraph Coloring is NP-hard for graphs of DFVS k , even if the maximum degree Δ is at most 4k−1 ; we show that this is also almost tight, as the problem becomes FPT for DFVS k and Δ≤4k−3 . We then consider parameters that measure the distance from acyclicity of the underlying graph. We show that k -Digraph Coloring admits an FPT algorithm parameterized by treewidth, whose parameter dependence is (tw!) k tw . Then, we pose the question of whether the tw! factor can be eliminated. Our main contribution in this part is to settle this question in the negative and show that our algorithm is essentially optimal, even for the much more restricted parameter treedepth and for k=2 . Specifically, we show that an FPT algorithm solving 2 -Digraph Coloring with dependence t d o(td) would contradict the ETH.