Complexity of finite-variable fragments of products with K

2020 ◽  
Author(s):  
Mikhail Rybakov ◽  
Dmitry Shkatov
Keyword(s):  
Lower Bound ◽  
Polynomial Time ◽  
Modal Logics ◽  
Single Variable ◽  
Dynamic Logics ◽  
Polymodal Logic

Abstract We show that products and expanding relativized products of propositional modal logics where one component is the minimal monomodal logic K are polynomial-time reducible to their single-variable fragments. Therefore, the known lower-bound complexity and undecidability results for such logics are extended to their single-variable fragments. Similar results are obtained for products where one component is a polymodal logic with a K-style modality; these include products with propositional dynamic logics.

SIMULATIONS BY TIME-BOUNDED COUNTER MACHINES

2011 ◽  
Vol 22 (02) ◽  
pp. 395-409 ◽  
Author(s):  
HOLGER PETERSEN
Keyword(s):  
Lower Bound ◽  
Real Time ◽  
Polynomial Time ◽  
Time Complexity ◽  
Linear Time ◽  
Fixed Number ◽  
Exponential Time ◽  
New Family ◽  
Counter Machines ◽  
Time Bounds

We investigate the efficiency of simulations of storages by several counters. A simulation of a pushdown store is described which is optimal in the sense that reducing the number of counters of a simulator leads to an increase in time complexity. The lower bound also establishes a tight counter hierarchy in exponential time. Then we turn to simulations of a set of counters by a different number of counters. We improve and generalize a known simulation in polynomial time. Greibach has shown that adding s + 1 counters increases the power of machines working in time ns. Using a new family of languages we show here a tight hierarchy result for machines with the same polynomial time-bound. We also prove hierarchies for machines with a fixed number of counters and with growing polynomial time-bounds. For machines with one counter and an additional "store zero" instruction we establish the equivalence of real-time and linear time. If at least two counters are available, the classes of languages accepted in real-time and linear time can be separated.

scholarly journals Augmenting the Power of (Partial) MaxSat Resolution with Extension

2020 ◽  
Vol 34 (02) ◽  
pp. 1561-1568 ◽  
Author(s):  
Javier Larrosa ◽  
Emma Rollon
Keyword(s):  
Lower Bound ◽  
Objective Function ◽  
Lower Bounds ◽  
Polynomial Time ◽  
Proof System ◽  
Extension Rule ◽  
Resolution Proof

The refutation power of SAT and MaxSAT resolution is challenged by problems like the soft and hard Pigeon Hole Problem PHP for which short refutations do not exist. In this paper we augment the MaxSAT resolution proof system with an extension rule. The new proof system MaxResE is sound and complete, and more powerful than plain MaxSAT resolution, since it can refute the soft and hard PHP in polynomial time. We show that MaxResE refutations actually subtract lower bounds from the objective function encoded by the formulas. The resulting formula is the residual after the lower bound extraction. We experimentally show that the residual of the soft PHP (once its necessary cost of 1 has been efficiently subtracted with MaxResE) is a concise, easy to solve, satisfiable problem.

Lower bounds for modal logics

2007 ◽  
Vol 72 (3) ◽  
pp. 941-958 ◽  
Author(s):  
Pavel Hrubeš
Keyword(s):  
Lower Bound ◽  
Modal Logic ◽  
Lower Bounds ◽  
Modal Logics ◽  
Proof System ◽  
Speed Up ◽  
Exponential Lower Bound

AbstractWe give an exponential lower bound on number of proof-lines in the proof system K of modal logic, i.e., we give an example of K-tautologies ψ1, ψ2, … s.t. every K-proof of ψi must have a number of proof-lines exponential in terms of the size of ψi. The result extends, for the same sequence of K-tautologies, to the systems K4, Gödel–Löb's logic, S andS4. We also determine some speed-up relations between different systems of modal logic on formulas of modal-depth one.

Lower and upper bounds for the linear arrangement problem on interval graphs

2018 ◽  
Vol 52 (4-5) ◽  
pp. 1123-1145
Author(s):  
Alain Quilliot ◽  
Djamal Rebaine ◽  
Hélène Toussaint
Keyword(s):  
Lower Bound ◽  
Polynomial Time ◽  
Feasible Solution ◽  
Interval Graph ◽  
Upper Bounds ◽  
Unit Interval ◽  
Lower And Upper Bounds ◽  
Linear Arrangement ◽  
Arrangement Problem ◽  
Linear Arrangement Problem

We deal here with theLinear Arrangement Problem(LAP) onintervalgraphs, any interval graph being given here together with its representation as theintersectiongraph of some collection of intervals, and so with relatedprecedenceandinclusionrelations. We first propose a lower boundLB, which happens to be tight in the case ofunit intervalgraphs. Next, we introduce the restriction PCLAP of LAP which is obtained by requiring any feasible solution of LAP to be consistent with theprecedencerelation, and prove that PCLAP can be solved in polynomial time. Finally, we show both theoretically and experimentally that PCLAP solutions are a good approximation for LAP onintervalgraphs.

scholarly journals Sparse 0−1 Matrices and Forbidden Hypergraphs

1999 ◽  
Vol 8 (5) ◽  
pp. 417-427 ◽  
Author(s):  
CLAUDIA BERTRAM-KRETZBERG ◽  
THOMAS HOFMEISTER ◽  
HANNO LEFMANN
Keyword(s):  
Lower Bound ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Linearly Independent

We consider the problem of determining the maximum number N(m, k, r) of columns of a 0−1 matrix with m rows and exactly r ones in each column such that every k columns are linearly independent over ℤ2. For fixed integers k[ges ]4 and r[ges ]2, where k is even and gcd(k−1, r) = 1, we prove the lower bound N(m, k, r) = Ω(mkr/2(k−1) ·(ln m)1/k−1). This improves on earlier results from [14] by a factor Θ((ln m)1/k−1). Moreover, we describe a polynomial time algorithm achieving this new lower bound.

2-Exp Time lower bounds for propositional dynamic logics with intersection

2005 ◽  
Vol 70 (4) ◽  
pp. 1072-1086 ◽  
Author(s):  
Martin Lange ◽  
Carsten Lutz
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Open Problem ◽  
Dynamic Logic ◽  
Propositional Dynamic Logic ◽  
Tree Structures ◽  
Exponential Time ◽  
Double Exponential ◽  
Test Operator ◽  
Dynamic Logics

AbstractIn 1984. Danecki proved that satisfiability in IPDL, i.e., Propositional Dynamic Logic (PDL) extended with an intersection operator on programs, is decidabie in deterministic double exponential time. Since then, the exact complexity of IPDL has remained an open problem: the best known lower bound was the ExpTime one stemming from plain PDL until, in 2004. the first author established ExpSpace-hardness. In this paper, we finally close the gap and prove that IPDL is hard for 2-ExpTime. thus 2-ExpTime-complete. We then sharpen our lower bound, showing that it even applies to IPDL without the test operator interpreted on tree structures.

scholarly journals Counting matchings via capacity-preserving operators

2021 ◽  
pp. 1-26
Author(s):  
Leonid Gurvits ◽  
Jonathan Leake
Keyword(s):  
Differential Operator ◽  
Lower Bound ◽  
Recent Result ◽  
Polynomial Time ◽  
Bipartite Graphs ◽  
Perfect Matchings ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Polynomial Time Algorithms ◽  
Real Stability

Abstract The notion of the capacity of a polynomial was introduced by Gurvits around 2005, originally to give drastically simplified proofs of the van der Waerden lower bound for permanents of doubly stochastic matrices and Schrijver’s inequality for perfect matchings of regular bipartite graphs. Since this seminal work, the notion of capacity has been utilised to bound various combinatorial quantities and to give polynomial-time algorithms to approximate such quantities (e.g. the number of bases of a matroid). These types of results are often proven by giving bounds on how much a particular differential operator can change the capacity of a given polynomial. In this paper, we unify the theory surrounding such capacity-preserving operators by giving tight capacity preservation bounds for all nondegenerate real stability preservers. We then use this theory to give a new proof of a recent result of Csikvári, which settled Friedland’s lower matching conjecture.

scholarly journals Clique-transversal sets and weak 2-colorings in graphs of small maximum degree

2009 ◽  
Vol Vol. 11 no. 2 (Graph and Algorithms) ◽  
Author(s):  
Gábor Bacsó ◽  
Zsolt Tuza
Keyword(s):  
Lower Bound ◽  
Polynomial Time ◽  
Connected Graph ◽  
Maximum Degree ◽  
Free Graph ◽  
Polynomial Time Algorithms ◽  
Odd Cycle ◽  
Vertex Set ◽  
International Audience

Graphs and Algorithms International audience A clique-transversal set in a graph is a subset of the vertices that meets all maximal complete subgraphs on at least two vertices. We prove that every connected graph of order n and maximum degree three has a clique-transversal set of size left perpendicular19n/30 + 2/15right perpendicular. This bound is tight, since 19n/30 - 1/15 is a lower bound for infinitely many values of n. We also prove that the vertex set of any connected claw-free graph of maximum degree at most four, other than an odd cycle longer than three, can be partitioned into two clique-transversal sets. The proofs of both results yield polynomial-time algorithms that find corresponding solutions.

APPROXIMATING THE MAXIMUM ISOMORPHIC AGREEMENT SUBTREE IS HARD

2000 ◽  
Vol 11 (04) ◽  
pp. 579-590 ◽  
Author(s):  
Paola Bonizzoni ◽  
Gianluca Della Vedova ◽  
Giancarlo Mauri
Keyword(s):  
Lower Bound ◽  
Polynomial Time ◽  
Arbitrary Number ◽  
Evolutionary Trees ◽  
Factor Log ◽  
Approximation Complexity ◽  
Interval Weight ◽  
Maximum Interval

The Maximum Isomorphic Agreement Subtree [Formula: see text] problem is one of the simplest versions of the Maximum Interval Weight Agreement Subtree method [Formula: see text] which is used to compare phylogenies. More precisely [Formula: see text] allows to provide a subset of the species such that the exact distances between species in such subset are preserved among all evolutionary trees considered. In this paper, the approximation complexity of the MIT problem is investigated, showing that it cannot be approximated in polynomial time within factor log δn for any δ > 0 unless NP ⊆ DTIME(2polylog n) for instances containing three trees. Moreover, we show that such result can be strengthened whenever instances of the [Formula: see text] problem can contain an arbitrary number of trees, since [Formula: see text] shares the same approximation lower bound of [Formula: see text].

scholarly journals Matroid bases with cardinality constraints on the intersection

2021 ◽  
Author(s):  
Stefan Lendl ◽  
Britta Peis ◽  
Veerle Timmermans
Keyword(s):  
Lower Bound ◽  
Polynomial Time ◽  
Upper Bound ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Dual Algorithm ◽  
Matroid Intersection ◽  
Strongly Polynomial Time ◽  
Primal Dual ◽  
Strongly Polynomial

AbstractGiven two matroids $$\mathcal {M}_{1} = (E, \mathcal {B}_{1})$$ M 1 = ( E , B 1 ) and $$\mathcal {M}_{2} = (E, \mathcal {B}_{2})$$ M 2 = ( E , B 2 ) on a common ground set E with base sets $$\mathcal {B}_1$$ B 1 and $$\mathcal {B}_2$$ B 2 , some integer $$k \in \mathbb {N}$$ k ∈ N , and two cost functions $$c_{1}, c_{2} :E \rightarrow \mathbb {R}$$ c 1 , c 2 : E → R , we consider the optimization problem to find a basis $$X \in \mathcal {B}_{1}$$ X ∈ B 1 and a basis $$Y \in \mathcal {B}_{2}$$ Y ∈ B 2 minimizing the cost $$\sum _{e\in X} c_1(e)+\sum _{e\in Y} c_2(e)$$ ∑ e ∈ X c 1 ( e ) + ∑ e ∈ Y c 2 ( e ) subject to either a lower bound constraint $$|X \cap Y| \le k$$ | X ∩ Y | ≤ k , an upper bound constraint $$|X \cap Y| \ge k$$ | X ∩ Y | ≥ k , or an equality constraint $$|X \cap Y| = k$$ | X ∩ Y | = k on the size of the intersection of the two bases X and Y. The problem with lower bound constraint turns out to be a generalization of the Recoverable Robust Matroid problem under interval uncertainty representation for which the question for a strongly polynomial-time algorithm was left as an open question in Hradovich et al. (J Comb Optim 34(2):554–573, 2017). We show that the two problems with lower and upper bound constraints on the size of the intersection can be reduced to weighted matroid intersection, and thus be solved with a strongly polynomial-time primal-dual algorithm. We also present a strongly polynomial, primal-dual algorithm that computes a minimum cost solution for every feasible size of the intersection k in one run with asymptotic running time equal to one run of Frank’s matroid intersection algorithm. Additionally, we discuss generalizations of the problems from matroids to polymatroids, and from two to three or more matroids. We obtain a strongly polynomial time algorithm for the recoverable robust polymatroid base problem with interval uncertainties.

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