440 research outputs found
On Minimum Maximal Distance-k Matchings
We study the computational complexity of several problems connected with
finding a maximal distance- matching of minimum cardinality or minimum
weight in a given graph. We introduce the class of -equimatchable graphs
which is an edge analogue of -equipackable graphs. We prove that the
recognition of -equimatchable graphs is co-NP-complete for any fixed . We provide a simple characterization for the class of strongly chordal
graphs with equal -packing and -domination numbers. We also prove that
for any fixed integer the problem of finding a minimum weight
maximal distance- matching and the problem of finding a minimum weight
-independent dominating set cannot be approximated in polynomial
time in chordal graphs within a factor of unless
, where is a fixed constant (thereby
improving the NP-hardness result of Chang for the independent domination case).
Finally, we show the NP-hardness of the minimum maximal induced matching and
independent dominating set problems in large-girth planar graphs.Comment: 15 pages, 4 figure
The Degree of a Finite Set of Words
We generalize the notions of the degree and composition from uniquely decipherable codes to arbitrary finite sets of words. We prove that if X = Y?Z is a composition of finite sets of words with Y complete, then d(X) = d(Y) ? d(Z), where d(T) is the degree of T. We also show that a finite set is synchronizing if and only if its degree equals one.
This is done by considering, for an arbitrary finite set X of words, the transition monoid of an automaton recognizing X^* with multiplicities. We prove a number of results for such monoids, which generalize corresponding results for unambiguous monoids of relations
Universality and Forall-Exactness of Cost Register Automata with Few Registers
The universality problem asks whether a given finite state automaton accepts all the input words. For quantitative models of automata, where input words are mapped to real values, this is naturally extended to ask whether all the words are mapped to values above (or below) a given threshold. This is known to be undecidable for commonly studied examples such as weighted automata over the positive rational (plus-times) or the integer tropical (min-plus) semirings, or equivalently cost register automata (CRAs) over these semirings. In this paper, we prove that when restricted to CRAs with only three registers, the universality problem is still undecidable, even with additional restrictions for the CRAs to be copyless linear with resets.
In contrast, we show that, assuming the unary encoding of updates, the ?-exact problem (does the CRA output zero on all the words?) for integer min-plus linear CRAs can be decided in polynomial time if the number of registers is constant. Without the restriction on the number of registers this problem is known to be PSPACE-complete
Reachability in Fixed VASS: Expressiveness and Lower Bounds
The recent years have seen remarkable progress in establishing the complexity
of the reachability problem for vector addition systems with states (VASS),
equivalently known as Petri nets. Existing work primarily considers the case in
which both the VASS as well as the initial and target configurations are part
of the input. In this paper, we investigate the reachability problem in the
setting where the VASS is fixed and only the initial configuration is variable.
We show that fixed VASS fully express arithmetic on initial segments of the
natural numbers. It follows that there is a very weak reduction from any fixed
such number-theoretic predicate (e.g. primality or square-freeness) to
reachability in fixed VASS where configurations are presented in unary. If
configurations are given in binary, we show that there is a fixed VASS with
five counters whose reachability problem is PSPACE-hard
Universality and Forall-Exactness of Cost Register Automata with Few Registers
The universality problem asks whether a given finite state automaton accepts all the input words. For quantitative models of automata, where input words are mapped to real values, this is naturally extended to ask whether all the words are mapped to values above (or below) a given threshold. This is known to be undecidable for commonly studied examples such as weighted automata over the positive rational (plus-times) or the integer tropical (min-plus) semirings, or equivalently cost register automata (CRAs) over these semirings. In this paper, we prove that when restricted to CRAs with only three registers, the universality problem is still undecidable, even with additional restrictions for the CRAs to be copyless linear with resets. In contrast, we show that, assuming the unary encoding of updates, the ForAll-exact problem (does the CRA output zero on all the words?) for integer min-plus linear CRAs can be decided in polynomial time if the number of registers is constant. Without the restriction on the number of registers this problem is known to be PSPACE-complete
Synchronizing Strongly Connected Partial DFAs
We study synchronizing partial DFAs, which extend the classical concept of
synchronizing complete DFAs and are a special case of synchronizing unambiguous
NFAs. A partial DFA is called synchronizing if it has a word (called a reset
word) whose action brings a non-empty subset of states to a unique state and is
undefined for all other states. While in the general case the problem of
checking whether a partial DFA is synchronizing is PSPACE-complete, we show
that in the strongly connected case this problem can be efficiently reduced to
the same problem for a complete DFA. Using combinatorial, algebraic, and formal
languages methods, we develop techniques that relate main synchronization
problems for strongly connected partial DFAs with the same problems for
complete DFAs. In particular, this includes the \v{C}ern\'{y} and the rank
conjectures, the problem of finding a reset word, and upper bounds on the
length of the shortest reset words of literal automata of finite prefix codes.
We conclude that solving fundamental synchronization problems is equally hard
in both models, as an essential improvement of the results for one model
implies an improvement for the other.Comment: Full version of the paper at STACS 202
Formal and Empirical Studies of Counting Behaviour in ReLU RNNs
In recent years, the discussion about systematicity of neural network learning has gained renewed interest, in particular the formal analysis of neural network behaviour. In this paper, we investigate the capability of single-cell ReLU RNN models to demonstrate precise counting behaviour. Formally, we start by characterising the semi-Dyck-1 language and semi-Dyck-1 counter machine that can be implemented by a single Rectified Linear Unit (ReLU) cell. We define three Counter Indicator Conditions (CICs) on the weights of a ReLU cell and show that fulfilling these conditions is equivalent to accepting the semi-Dyck-1 language, i.e. to perform exact counting. Empirically, we study the ability of single-cell ReLU RNNs to learn to count by training and testing them on different datasets of Dyck-1 and semi-Dyck-1 strings. While networks that satisfy the CICs count exactly and thus correctly even on very long strings, the trained networks exhibit a wide range of results and never satisfy the CICs exactly. We investigate the effect of deviating from the CICs and find that configurations that fulfil the CICs are not at a minimum of the loss function in the most common setups. This is consistent with observations in previous research indicating that training ReLU networks for counting tasks often leads to poor results. We finally discuss implications of these results and possible avenues for improving network behaviour
Finding Short Synchronizing Words for Prefix Codes
We study the problems of finding a shortest synchronizing word and its length for a given prefix code. This is done in two different settings: when the code is defined by an arbitrary decoder recognizing its star and when the code is defined by its literal decoder (whose size is polynomially equivalent to the total length of all words in the code). For the first case for every epsilon > 0 we prove n^(1 - epsilon)-inapproximability for recognizable binary maximal prefix codes, Theta(log n)-inapproximability for finite binary maximal prefix codes and n^(1/2 - epsilon)-inapproximability for finite binary prefix codes. By c-inapproximability here we mean the non-existence of a c-approximation polynomial time algorithm under the assumption P != NP, and by n the number of states of the decoder in the input. For the second case, we propose approximation and exact algorithms and conjecture that for finite maximal prefix codes the problem can be solved in polynomial time. We also study the related problems of finding a shortest mortal and a shortest avoiding word
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