Solving equations over small unary algebras

Przemyslaw Broniek

doi:10.46298/dmtcs.3474

Solving equations over small unary algebras

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.3474 ◽

2005 ◽

Vol DMTCS Proceedings vol. AF,... (Proceedings) ◽

Author(s):

Przemyslaw Broniek

Keyword(s):

Polynomial Time ◽

Partial Characterization ◽

Polynomial Equations ◽

Solving Equations ◽

International Audience ◽

Np Complete ◽

System Of Polynomial Equations

International audience We consider the problem of solving a system of polynomial equations over fixed algebra $A$ which we call MPolSat($A$). We restrict ourselves to unary algebras and give a partial characterization of complexity of MPolSat($A$). We isolate a preorder $P(A)$ to show that when $A$ has at most 3 elements then MPolSat($A$) is in $P$ when width of $P(A)$ is at most 2 and is NP-complete otherwise. We show also that if $P ≠ NP$ then the class of unary algebras solvable in polynomial time is not closed under homomorphic images.

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TAYLOR TERMS, CONSTRAINT SATISFACTION AND THE COMPLEXITY OF POLYNOMIAL EQUATIONS OVER FINITE ALGEBRAS

International Journal of Algebra and Computation ◽

10.1142/s0218196706003116 ◽

2006 ◽

Vol 16 (03) ◽

pp. 563-581 ◽

Cited By ~ 18

Author(s):

BENOIT LAROSE ◽

LÁSZLÓ ZÁDORI

Keyword(s):

Polynomial Time ◽

Constraint Satisfaction ◽

Finite Algebra ◽

Algorithmic Complexity ◽

Polynomial Equations ◽

Finite Algebras ◽

Complete Problem ◽

Np Complete ◽

System Of Polynomial Equations

We study the algorithmic complexity of determining whether a system of polynomial equations over a finite algebra admits a solution. We characterize, within various families of algebras, which of them give rise to an NP-complete problem and which yield a problem solvable in polynomial time. In particular, we prove a dichotomy result which encompasses the cases of lattices, rings, modules, quasigroups and also generalizes a result of Goldmann and Russell for groups [15].

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On paths, trails and closed trails in edge-colored graphs

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.586 ◽

2012 ◽

Vol Vol. 14 no. 2 (Graph Theory) ◽

Author(s):

Laurent Gourvès ◽

Adria Lyra ◽

Carlos A. Martinhon ◽

Jérôme Monnot

Keyword(s):

Polynomial Time ◽

Optimal Solution ◽

Colored Graph ◽

Edge Disjoint ◽

Colored Graphs ◽

International Audience ◽

Np Complete ◽

T Path ◽

Vertex Disjoint

Graph Theory International audience In this paper we deal from an algorithmic perspective with different questions regarding properly edge-colored (or PEC) paths, trails and closed trails. Given a c-edge-colored graph G(c), we show how to polynomially determine, if any, a PEC closed trail subgraph whose number of visits at each vertex is specified before hand. As a consequence, we solve a number of interesting related problems. For instance, given subset S of vertices in G(c), we show how to maximize in polynomial time the number of S-restricted vertex (resp., edge) disjoint PEC paths (resp., trails) in G(c) with endpoints in S. Further, if G(c) contains no PEC closed trails, we show that the problem of finding a PEC s-t trail visiting a given subset of vertices can be solved in polynomial time and prove that it becomes NP-complete if we are restricted to graphs with no PEC cycles. We also deal with graphs G(c) containing no (almost) PEC cycles or closed trails through s or t. We prove that finding 2 PEC s-t paths (resp., trails) with length at most L > 0 is NP-complete in the strong sense even for graphs with maximum degree equal to 3 and present an approximation algorithm for computing k vertex (resp., edge) disjoint PEC s-t paths (resp., trails) so that the maximum path (resp., trail) length is no more than k times the PEC path (resp., trail) length in an optimal solution. Further, we prove that finding 2 vertex disjoint s-t paths with exactly one PEC s-t path is NP-complete. This result is interesting since as proved in Abouelaoualim et. al.(2008), the determination of two or more vertex disjoint PEC s-t paths can be done in polynomial time. Finally, if G(c) is an arbitrary c-edge-colored graph with maximum vertex degree equal to four, we prove that finding two monochromatic vertex disjoint s-t paths with different colors is NP-complete. We also propose some related problems.

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The extended equivalence and equation solvability problems for groups

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.536 ◽

2011 ◽

Vol Vol. 13 no. 4 ◽

Author(s):

Gabor Horvath ◽

Csaba Szabo

Keyword(s):

Polynomial Time ◽

Equivalence Problem ◽

Nilpotent Groups ◽

60Th Birthday ◽

Special Issue ◽

Algorithms And Complexity ◽

International Audience ◽

Solvability Problem ◽

Np Complete ◽

Equation Solvability

special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexity International audience We prove that the extended equivalence problem is solvable in polynomial time for finite nilpotent groups, and coNP-complete, otherwise. We prove that the extended equation solvability problem is solvable in polynomial time for finite nilpotent groups, and NP-complete, otherwise.

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Acyclic colourings of graphs with bounded degree

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.483 ◽

2010 ◽

Vol Vol. 12 no. 1 (Graph and Algorithms) ◽

Author(s):

Mieczyslaw Borowiecki ◽

Anna Fiedorowicz ◽

Katarzyna Jesse-Jozefczyk ◽

Elzbieta Sidorowicz

Keyword(s):

Polynomial Time ◽

Maximum Degree ◽

Cubic Graph ◽

Bounded Degree ◽

Complete Problem ◽

International Audience ◽

Np Complete

Graphs and Algorithms International audience A k-colouring of a graph G is called acyclic if for every two distinct colours i and j, the subgraph induced in G by all the edges linking a vertex coloured with i and a vertex coloured with j is acyclic. In other words, there are no bichromatic alternating cycles. In 1999 Boiron et al. conjectured that a graph G with maximum degree at most 3 has an acyclic 2-colouring such that the set of vertices in each colour induces a subgraph with maximum degree at most 2. In this paper we prove this conjecture and show that such a colouring of a cubic graph can be determined in polynomial time. We also prove that it is an NP-complete problem to decide if a graph with maximum degree 4 has the above mentioned colouring.

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Uniquely monopolar-partitionable block graphs

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2073 ◽

2014 ◽

Vol Vol. 16 no. 2 (PRIMA 2013) ◽

Author(s):

Xuegang Chen ◽

Jing Huang

Keyword(s):

Polynomial Time Algorithm ◽

Time Algorithm ◽

Special Issue ◽

Block Graphs ◽

International Audience ◽

Split Graphs ◽

Np Complete ◽

Vertex Partitions ◽

General Graphs

Special issue PRIMA 2013 International audience As a common generalization of bipartite and split graphs, monopolar graphs are defined in terms of the existence of certain vertex partitions. It has been shown that to determine whether a graph has such a partition is NP-complete for general graphs and polynomial for several classes of graphs. In this paper, we investigate graphs that admit a unique such partition and call them uniquely monopolar-partitionable graphs. By employing a tree trimming technique, we obtain a characterization of uniquely monopolar-partitionable block graphs. Our characterization implies a polynomial time algorithm for recognizing them.

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P6- and triangle-free graphs revisited: structure and bounded clique-width

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.372 ◽

2006 ◽

Vol Vol. 8 ◽

Author(s):

Andreas Brandstädt ◽

Tilo Klembt ◽

Suhail Mahfud

Keyword(s):

Polynomial Time ◽

Dominating Set ◽

Vertex Cover ◽

Maximum Clique ◽

Stable Set ◽

Induced Matching ◽

International Audience ◽

Np Complete ◽

Maximum Induced Matching ◽

Second Order Logic

International audience The Maximum Weight Stable Set (MWS) Problem is one of the fundamental problems on graphs. It is well-known to be NP-complete for triangle-free graphs, and Mosca has shown that it is solvable in polynomial time when restricted to P6- and triangle-free graphs. We give a complete structure analysis of (nonbipartite) P6- and triangle-free graphs which are prime in the sense of modular decomposition. It turns out that the structure of these graphs is extremely simple implying bounded clique-width and thus, efficient algorithms exist for all problems expressible in terms of Monadic Second Order Logic with quantification only over vertex predicates. The problems Vertex Cover, MWS, Maximum Clique, Minimum Dominating Set, Steiner Tree, and Maximum Induced Matching are among them. Our results improve the previous one on the MWS problem by Mosca with respect to structure and time bound but also extends a previous result by Fouquet, Giakoumakis and Vanherpe which have shown that bipartite P6-free graphs have bounded clique-width. Moreover, it covers a result by Randerath, Schiermeyer and Tewes on polynomial time 3-colorability of P6- and triangle-free graphs.

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Low Degree Nullstellensatz Certificates for 3-Colorability

The Electronic Journal of Combinatorics ◽

10.37236/5103 ◽

2016 ◽

Vol 23 (1) ◽

Author(s):

Bo Li ◽

Benjamin Lowenstein ◽

Mohamed Omar

Keyword(s):

Linear Algebra ◽

Directed Graphs ◽

Polynomial Equations ◽

Combinatorial Characterization ◽

Seminal Paper ◽

Low Degree ◽

System Of Polynomial Equations

In a seminal paper, De Loera et. al introduce the algorithm NulLA (Nullstellensatz Linear Algebra) and use it to measure the difficulty of determining if a graph is not 3-colorable. The crux of this relies on a correspondence between 3-colorings of a graph and solutions to a certain system of polynomial equations over a field $\mathbb{k}$. In this article, we give a new direct combinatorial characterization of graphs that can be determined to be non-3-colorable in the first iteration of this algorithm when $\mathbb{k}=GF(2)$. This greatly simplifies the work of De Loera et. al, as we express the combinatorial characterization directly in terms of the graphs themselves without introducing superfluous directed graphs. Furthermore, for all graphs on at most $12$ vertices, we determine at which iteration NulLA detects a graph is not 3-colorable when $\mathbb{k}=GF(2)$.

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Complexity of conditional colouring with given template

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2092 ◽

2014 ◽

Vol Vol. 16 no. 3 (Graph Theory) ◽

Author(s):

Peter J. Dukes ◽

Steve Lowdon ◽

Gary Macgillivray

Keyword(s):

Graph Theory ◽

Polynomial Time ◽

Vertex Set ◽

International Audience ◽

Partitioning Problem ◽

Np Complete ◽

Graph Families ◽

General Collection

Graph Theory International audience We study partitions of the vertex set of a given graph into cells that each induce a subgraph in a given family, and for which edges can have ends in different cells only when those cells correspond to adjacent vertices of a fixed template graph H. For triangle-free templates, a general collection of graph families for which the partitioning problem can be solved in polynomial time is described. For templates with a triangle, the problem is in some cases shown to be NP-complete.

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