Data Reduction for Graph Coloring Problems

This paper studies the kernelization complexity of graph coloring problems with respect to certain structural parameterizations of the input instances. We are interested in how well polynomial-time data reduction can provably shrink instances of coloring problems, in terms of the chosen parameter. It is well known that deciding 3-colorability is already NP-complete, hence parameterizing by the requested number of colors is not fruitful. Instead, we pick up on a research thread initiated by Cai (DAM, 2003) who studied coloring problems parameterized by the modification distance of the input graph to a graph class on which coloring is polynomial-time solvable; for example parameterizing by the number k of vertex-deletions needed to make the graph chordal. We obtain various upper and lower bounds for kernels of such parameterizations of q-Coloring, complementing Cai's study of the time complexity with respect to these parameters. Our results show that the existence of polynomial kernels for q-Coloring parameterized by the vertex-deletion distance to a graph class F is strongly related to the existence of a function f(q) which bounds the number of vertices which are needed to preserve the NO-answer to an instance of q-List-Coloring on F.Comment: Author-accepted manuscript of the article that will appear in the FCT 2011 special issue of Information & Computatio

Polynomial kernels for hitting forbidden minors under structural parameterizations

We investigate polynomial-time preprocessing for the problem of hitting forbidden minors in a graph, using the framework of kernelization. For a fixed finite set of connected graphs F, the F-DELETION problem is the following: given a graph G and integer k, is it possible to delete k vertices from G to ensure the resulting graph does not contain any graph from F as a minor? Earlier work by Fomin, Lokshtanov, Misra, and Saurabh [FOCS'12] showed that when F contains a planar graph, an instance (G,k) can be reduced in polynomial time to an equivalent one of size kO(1). In this work we focus on structural measures of the complexity of an instance, with the aim of giving nontrivial preprocessing guarantees for instances whose solutions are large. Motivated by several impossibility results, we parameterize the F-DELETION problem by the size of a vertex modulator whose removal results in a graph of constant treedepth η. We prove that for each set F of connected graphs and constant η, the F-DELETION problem parameterized by the size of a treedepth-η modulator has a polynomial kernel. Our kernelization is fully explicit and does not depend on protrusion reduction or well-quasi-ordering, which are sources of algorithmic non-constructivity in earlier works on F-DELETION. Our main technical contribution is to analyze how models of a forbidden minor in a graph G with modulator X, interact with the various connected components of G−X. Using the language of labeled minors, we analyze the fragments of potential forbidden minor models that can remain after removing an optimal F-DELETION solution from a single connected component of G−X. By bounding the number of different types of behavior that can occur by a polynomial in |X|, we obtain a polynomial kernel using a recursive preprocessing strategy. Our results extend earlier work for specific instances of F-DELETION such as VERTEX COVER and FEEDBACK VERTEX SET. It also generalizes earlier preprocessing results for F-DELETION parameterized by a vertex cover, which is a treedepth-one modulator

A Turing kernelization dichotomy for structural parameterizations of F-Minor-Free Deletion

For a fixed finite family of graphs F, the F-MINOR-FREE DELETION problem takes as input a graph G and integer ℓ and asks whether a size-ℓ vertex set X exists such that G−X is F-minor-free. {K2}-MINOR-FREE DELETION and {K3}-MINOR-FREE DELETION encode VERTEX COVER and FEEDBACK VERTEX SET respectively. When parameterized by the feedback vertex number of G these two problems are known to admit a polynomial kernelization. We show {P3}-MINOR-FREE DELETION parameterized by the feedback vertex number is MK[2]-hard. This rules out the existence of a polynomial kernel assuming NP⊈coNP/poly. Our hardness result generalizes to any F containing only graphs with a connected component of at least 3 vertices, using as parameter the vertex-deletion distance to treewidth min⁡tw(F), where min⁡tw(F) denotes the minimum treewidth of the graphs in F. For all other families F we present a polynomial Turing kernelization. Our results extend to F-SUBGRAPH-FREE DELETION

Bodenschutz als Aufgabe der Landes- und Regionalplanung: Sitzung d. Sekt. I d. Akad. am 19./20. Februar 1986 in Osnabrueck

Includes 4 articlesSIGLEAvailable from Bibliothek des Instituts fuer Weltwirtschaft, ZBW, Duesternbrook Weg 120, D-24105 Kiel C 151309 / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Optimal sparsification for some binary CSPs using low-degree polynomials

This article analyzes to what extent it is possible to efficiently reduce the number of clauses in NP-hard satisfiability problems without changing the answer. Upper and lower bounds are established using the concept of kernelization. Existing results show that if NP coNP/poly, no efficient preprocessing algorithm can reduce n-variable instances of cnf-sat with d literals per clause to equivalent instances with O(nd-ϵ ) bits for any ϵ > 0. For the Not-All-Eqal sat problem, a compression to size O(nd-1) exists. We put these results in a common framework by analyzing the compressibility of CSPs with a binary domain. We characterize constraint types based on the minimum degree of multivariate polynomials whose roots correspond to the satisfying assignments, obtaining (nearly) matching upper and lower bounds in several settings. Our lower bounds show that not just the number of constraints, but also the encoding size of individual constraints plays an important role. For example, for Exact Satisfiability with unbounded clause length it is possible to efficiently reduce the number of constraints to n + 1, yet no polynomial-Time algorithm can reduce to an equivalent instance with O(n2-ϵ ) bits for any ϵ > 0, unless NP coNP/poly

Lower bounds for protrusion replacement by counting equivalence classes

Garnero et al. (2015) recently introduced a framework based on dynamic programming to make applications of the protrusion replacement technique constructive and to obtain explicit upper bounds on the involved constants. They show that for several graph problems, for every boundary size t one can find an explicit set R t of representatives. Any subgraph H with a boundary of size t can be replaced with a representative H ′ ∈R t such that the effect of this replacement on the optimum can be deduced from H and H ′ alone. Their upper bounds on the size of the graphs in R t grow triple-exponentially with t. In this paper we complement their results by lower bounds on the sizes of representatives, in terms of the boundary size t. For example, we show that each set of planar representatives R t for INDEPENDENT SET or DOMINATING SET contains a graph with Ω(2 t ∕4t) vertices. This lower bound even holds for sets that only represent the planar subgraphs of bounded pathwidth. To obtain our results we provide a lower bound on the number of equivalence classes of the canonical equivalence relation for INDEPENDENT SET on t-boundaried graphs. We also find an elegant characterization of the number of equivalence classes in general graphs, in terms of the number of monotone functions of a certain kind. Our results show that the number of equivalence classes is at most 2 2 t , improving on earlier bounds of the form (t+1) 2 t

Bridge-Depth Characterizes which Minor-Closed Structural Parameterizations of Vertex Cover Admit a Polynomial Kernel

We study the kernelization complexity of structural parameterizations of the Vertex Cover problem. Here, the goal is to find a polynomial-time preprocessing algorithm that can reduce any instance (G, k) of the Vertex Cover problem to an equivalent one, whose size is polynomial in the size of a predetermined complexity parameter of G. A long line of previous research deals with parameterizations based on the number of vertex deletions needed to reduce G to a member of a simple graph class F, such as forests, graphs of bounded tree-depth, and graphs of maximum degree two. We set out to find the most general graph classes F for which Vertex Cover parameterized by the vertex-deletion distance of the input graph to F admits a polynomial kernelization. We give a complete characterization of the minor-closed graph families F for which such a kernelization exists. We introduce a new graph parameter called bridge-depth, and prove that a polynomial kernelization exists if and only if F has bounded bridge-depth. The proof is based on an interesting connection between bridge-depth and the size of minimal blocking sets in graphs, which are vertex sets whose removal decreases the independence number

Preprocessing vertex-deletion problems: Characterizing graph properties by low-rank adjacencies

We consider the Π-FREE DELETION problem parameterized by the size of a vertex cover, for a range of graph properties Π. Given an input graph G, this problem asks whether there is a subset of at most k vertices whose removal ensures the resulting graph does not contain a graph from Π as an induced subgraph. We introduce the concept of characterizing a graph property Π by low-rank adjacencies, and use it as the cornerstone of a general kernelization theorem for Π-FREE DELETION parameterized by the size of a vertex cover. The resulting framework captures problems such as AT-FREE DELETION, WHEEL-FREE DELETION, and INTERVAL DELETION. Moreover, our new framework shows that the vertex-deletion problem to perfect graphs has a polynomial kernel when parameterized by vertex cover, thereby resolving an open question by Fomin et al. (2014) [18]

Preprocessing for Outerplanar Vertex Deletion: An Elementary Kernel of Quartic Size

In the F-Minor-Free Deletion problem one is given an undirected graph G, an integer k, and the task is to determine whether there exists a vertex set S of size at most k, so that G- S contains no graph from the finite family F as a minor. It is known that whenever F contains at least one planar graph, then F-Minor-Free Deletion admits a polynomial kernel, that is, there is a polynomial-time algorithm that outputs an equivalent instance of size kO(1) [Fomin, Lokshtanov, Misra, Saurabh; FOCS 2012]. However, this result relies on non-constructive arguments based on well-quasi-ordering and does not provide a concrete bound on the kernel size. We study the Outerplanar Deletion problem, in which we want to remove at most k vertices from a graph to make it outerplanar. This is a special case of F-Minor-Free Deletion for the family F= { K4, K2,3}. The class of outerplanar graphs is arguably the simplest class of graphs for which no explicit kernelization size bounds are known. By exploiting the combinatorial properties of outerplanar graphs we present elementary reduction rules decreasing the size of a graph. This yields a constructive kernel with O(k4) vertices and edges. As a corollary, we derive that any minor-minimal obstruction to having an outerplanar deletion set of size k has O(k4) vertices and edges