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Register a new userHard Problems That Quickly Become Very Easy
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Added by
Daniel Paulusma
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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A graph class is hereditary if it is closed under vertex deletion. We give examples of NP-hard, PSPACE-complete and NEXPTIME-complete problems that become constant-time solvable for every hereditary graph class that is not equal to the class of all graphs.
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A heuristic model procedure for determining satisfiability of CNF-formulae is set up and described by nonlinear recursion relations for m (number of clauses), n (number of variables) and clause filling k. The system mimicked by the recursion undergoes a sharp transition from bounded running times (easy) to uncontrolled runaway behaviour (hard). Thus the parameter space turns out to be separated into regions with qualitatively different efficiency of the model procedure. The transition results from a competition of exponential blow up by branching versus growing number of orthogonal clauses.
We prove essentially tight lower bounds, conditionally to the Exponential Time Hypothesis, for two fundamental but seemingly very different cutting problems on surface-embedded graphs: the Shortest Cut Graph problem and the Multiway Cut problem. A cut graph of a graph $G$ embedded on a surface $S$ is a subgraph of $G$ whose removal from $S$ leaves a disk. We consider the problem of deciding whether an unweighted graph embedded on a surface of genus $g$ has a cut graph of length at most a given value. We prove a time lower bound for this problem of $n^{Omega(g/log g)}$ conditionally to ETH. In other words, the first $n^{O(g)}$-time algorithm by Erickson and Har-Peled [SoCG 2002, Discr. Comput. Geom. 2004] is essentially optimal. We also prove that the problem is W[1]-hard when parameterized by the genus, answering a 17-year old question of these authors. A multiway cut of an undirected graph $G$ with $t$ distinguished vertices, called terminals, is a set of edges whose removal disconnects all pairs of terminals. We consider the problem of deciding whether an unweighted graph $G$ has a multiway cut of weight at most a given value. We prove a time lower bound for this problem of $n^{Omega(sqrt{gt + g^2+t}/log(g+t))}$, conditionally to ETH, for any choice of the genus $gge0$ of the graph and the number of terminals $tge4$. In other words, the algorithm by the second author [Algorithmica 2017] (for the more general multicut problem) is essentially optimal; this extends the lower bound by the third author [ICALP 2012] (for the planar case). Reductions to planar problems usually involve a grid-like structure. The main novel idea for our results is to understand what structures instead of grids are needed if we want to exploit optimally a certain value $g$ of the genus.
We prove PSPACE-completeness of two classic types of Chess problems when generalized to n-by-n boards. A retrograde problem asks whether it is possible for a position to be reached from a natural starting position, i.e., whether the position is valid or legal or reachable. Most real-world retrograde Chess problems ask for the last few moves of such a sequence; we analyze the decision question which gets at the existence of an exponentially long move sequence. A helpmate problem asks whether it is possible for a player to become checkmated by any sequence of moves from a given position. A helpmate problem is essentially a cooperative form of Chess, where both players work together to cause a particular player to win; it also arises in regular Chess games, where a player who runs out of time (flags) loses only if they could ever possibly be checkmated from the current position (i.e., the helpmate problem has a solution). Our PSPACE-hardness reductions are from a variant of a puzzle game called Subway Shuffle.
Differential equations parameterized by neural networks become expensive to solve numerically as training progresses. We propose a remedy that encourages learned dynamics to be easier to solve. Specifically, we introduce a differentiable surrogate for the time cost of standard numerical solvers, using higher-order derivatives of solution trajectories. These derivatives are efficient to compute with Taylor-mode automatic differentiation. Optimizing this additional objective trades model performance against the time cost of solving the learned dynamics. We demonstrate our approach by training substantially faster, while nearly as accurate, models in supervised classification, density estimation, and time-series modelling tasks.
For relational structures A, B of the same signature, the Promise Constraint Satisfaction Problem PCSP(A,B) asks whether a given input structure maps homomorphically to A or does not even map to B. We are promised that the input satisfies exactly one of these two cases. If there exists a structure C with homomorphisms $Ato Cto B$, then PCSP(A,B) reduces naturally to CSP(C). To the best of our knowledge all known tractable PCSPs reduce to tractable CSPs in this way. However Barto showed that some PCSPs over finite structures A, B require solving CSPs over infinite C. We show that even when such a reduction to finite C is possible, this structure may become arbitrarily large. For every integer $n>1$ and every prime p we give A, B of size n with a single relation of arity $n^p$ such that PCSP(A, B) reduces via a chain of homomorphisms $ Ato Cto B$ to a tractable CSP over some C of size p but not over any smaller structure. In a second family of examples, for every prime $pgeq 7$ we construct A, B of size $p-1$ with a single ternary relation such that PCSP(A, B) reduces via $Ato Cto B$ to a tractable CSP over some C of size p but not over any smaller structure. In contrast we show that if A, B are graphs and PCSP(A,B) reduces to tractable CSP(C) for some finite C, then already A or B has tractable CSP. This extends results and answers a question of Deng et al.
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