Lower bounds for protrusion replacement by counting equivalence classes

Garnero et al. [SIAM J. Discrete Math. 2015, 29(4):1864--1894] 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 $\mathcal{R}_t$ of representatives. Any subgraph $H$ with a boundary of size $t$ can be replaced with a representative $H' \in \mathcal{R}_t$ 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 $\mathcal{R}_t$ 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 $\mathcal{R}_t$ for Independent Set or Dominating Set contains a graph with $\Omega(2^t / \sqrt{4t})$ 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 $(t+1)^{2^t}$

Uniform kernelization complexity of hitting forbidden minors

The F-MINOR-FREE DELETION problem asks, for a fixed set F and an input consisting of a graph G and integer k, whether κ vertices can be removed from G such that the resulting graph does not contain any member of F as a minor. At FOCS 2012, Fomin et al. showed that the special case when F contains at least one planar graph has a kernel of size f (F) · κg(F) for some functions f and g. They left open whether this PLANAR F-MINOR-FREE DELETION problem has kernels whose size is uniformly polynomial, of the form f (F) · κc for some universal constant c. We prove that some PLANAR F-MINOR-FREE DELETION problems do not have uniformly polynomial kernels (unless NP ⊆ coNP/poly), not even when parameterized by the vertex cover number. On the positive side, we consider the problem of determining whether κ vertices can be removed to obtain a graph of treedepth at most η. We prove that this problem admits uniformly polynomial kernels with O(κ6) vertices for every fixed η.</p

Feed-links for network extensions

Road network data is often incomplete, making it hard to perform network analysis. This paper discusses the problem of extending partial road networks with reasonable links, using the concept of dilation (also known as crow flight conversion coefficient). To this end, we study how to connect a point (relevant location) inside a polygon (face of the known part of the road network) to the boundary so that the dilation from that point to any point on the boundary is not too large. We provide algorithms and heuristics, and give a computational and experimental analysis

Vertex Cover Kernelization Revisited: Upper and Lower Bounds for a Refined Parameter

An important result in the study of polynomial-time preprocessing shows that there is an algorithm which given an instance (G,k) of Vertex Cover outputs an equivalent instance (G',k') in polynomial time with the guarantee that G' has at most 2k' vertices (and thus O((k')^2) edges) with k' <= k. Using the terminology of parameterized complexity we say that k-Vertex Cover has a kernel with 2k vertices. There is complexity-theoretic evidence that both 2k vertices and Theta(k^2) edges are optimal for the kernel size. In this paper we consider the Vertex Cover problem with a different parameter, the size fvs(G) of a minimum feedback vertex set for G. This refined parameter is structurally smaller than the parameter k associated to the vertex covering number vc(G) since fvs(G) <= vc(G) and the difference can be arbitrarily large. We give a kernel for Vertex Cover with a number of vertices that is cubic in fvs(G): an instance (G,X,k) of Vertex Cover, where X is a feedback vertex set for G, can be transformed in polynomial time into an equivalent instance (G',X',k') such that |V(G')| <= 2k and |V(G')| <= O(|X'|^3). A similar result holds when the feedback vertex set X is not given along with the input. In sharp contrast we show that the Weighted Vertex Cover problem does not have a polynomial kernel when parameterized by the cardinality of a given vertex cover of the graph unless NP is in coNP/poly and the polynomial hierarchy collapses to the third level.Comment: Published in "Theory of Computing Systems" as an Open Access publicatio

The Power of Data Reduction : Kernels for Fundamental Graph Problems

The purpose of this thesis is to give a mathematical analysis of the power of data reduction for dealing with fundamental NP-hard graph problems. It has often been observed that the use of heuristic reduction rules in a preprocessing phase gives significant performance gains when solving such problems. However, there is little scientific explanation for these empirically observed successes. We use the concept of kernelization, developed within the field of parameterized complexity theory, to give a mathematical analysis of the power of such data reduction techniques. A kernelization, or kernel, is a polynomial-time preprocessing algorithm that transforms an instance of a parameterized problem into an equivalent instance whose size depends only on the parameter. The concept of kernelization therefore formalizes efficient and provably effective preprocessing. In our analysis of fundamental graph problems we utilize various structural measures of graphs as the complexity parameter; these include the vertex cover number, the feedback vertex number, the treewidth, and the vertex-deletion distance to various well-studied graph classes. We parameterize four fundamental classes of graph problems by such graph-structural measures. We determine which of these parameterizations admit kernelizations for which the size of the output is bounded by a polynomial in the parameter. Towards this end, we also develop technical tools to prove that a parameterized problem does not admit a kernel of polynomial size, subject to certain complexity-theoretic assumptions. The four fundamental problems we study are Vertex Cover, Treewidth, Graph Coloring, and Longest Path. For the Vertex Cover problem we introduce novel reduction rules that provably reduce the size of an instance to at most O(k^3) vertices in polynomial time, where k is the size of a feedback vertex set of the input graph. We also prove that the existence of a kernel for the parameterization by the vertex-deletion distance to an outerplanar graph or a clique, leads to a collapse of the polynomial hierarchy and is therefore unlikely. In our analysis of the Treewidth problem, we prove that preprocessing rules that were initially developed for heuristic algorithms, lead to a polynomial kernel for Treewidth parameterized by the vertex cover number. By developing additional rules that eliminate almost-simplicial vertices and shrink clique-seeing paths, we obtain a polynomial kernel parameterized by the feedback vertex number. Finally, we prove that Treewidth and Pathwidth do not admit polynomial kernels parameterized by the vertex-deletion distance to a clique, unless the polynomial hierarchy collapses. We analyze the kernelization complexity of graph coloring problems with respect to parameterizations that measure the vertex-deletion to graph classes such as cographs and co-chordal graphs. We show that the existence of polynomial kernels is determined by the extremal properties of No-instances of the List Coloring problem on such graph classes. Finally, we investigate Longest Path and related problems, with structural parameterizations. We obtain polynomial kernels for parameterizations by the vertex cover number, the max leaf number, and the vertex-deletion distance to a cluster graph. These results are complemented by a lower bound for the parameterization by the deletion distance to an outerplanar grap

Turing kernelization for finding long paths and cycles in restricted graph classes

The k-PATH problem asks whether a given undirected graph has a (simple) path of length k. We prove that k-PATH has polynomial-size Turing kernels when restricted to planar graphs, graphs of bounded degree, claw-free graphs, or to K 3,t-minor-free graphs. This means that there is an algorithm that, given a k-PATH instance (G,k) belonging to one of these graph classes, computes its answer in polynomial time when given access to an oracle that solves k-PATH instances of size polynomial in k in a single step. Our techniques also apply to k-CYCLE, which asks for a cycle of length at least k

The Power of Data Reduction : Kernels for Fundamental Graph Problems

The purpose of this thesis is to give a mathematical analysis of the power of data reduction for dealing with fundamental NP-hard graph problems. It has often been observed that the use of heuristic reduction rules in a preprocessing phase gives significant performance gains when solving such problems. However, there is little scientific explanation for these empirically observed successes. We use the concept of kernelization, developed within the field of parameterized complexity theory, to give a mathematical analysis of the power of such data reduction techniques. A kernelization, or kernel, is a polynomial-time preprocessing algorithm that transforms an instance of a parameterized problem into an equivalent instance whose size depends only on the parameter. The concept of kernelization therefore formalizes efficient and provably effective preprocessing. In our analysis of fundamental graph problems we utilize various structural measures of graphs as the complexity parameter; these include the vertex cover number, the feedback vertex number, the treewidth, and the vertex-deletion distance to various well-studied graph classes. We parameterize four fundamental classes of graph problems by such graph-structural measures. We determine which of these parameterizations admit kernelizations for which the size of the output is bounded by a polynomial in the parameter. Towards this end, we also develop technical tools to prove that a parameterized problem does not admit a kernel of polynomial size, subject to certain complexity-theoretic assumptions. The four fundamental problems we study are Vertex Cover, Treewidth, Graph Coloring, and Longest Path. For the Vertex Cover problem we introduce novel reduction rules that provably reduce the size of an instance to at most O(k^3) vertices in polynomial time, where k is the size of a feedback vertex set of the input graph. We also prove that the existence of a kernel for the parameterization by the vertex-deletion distance to an outerplanar graph or a clique, leads to a collapse of the polynomial hierarchy and is therefore unlikely. In our analysis of the Treewidth problem, we prove that preprocessing rules that were initially developed for heuristic algorithms, lead to a polynomial kernel for Treewidth parameterized by the vertex cover number. By developing additional rules that eliminate almost-simplicial vertices and shrink clique-seeing paths, we obtain a polynomial kernel parameterized by the feedback vertex number. Finally, we prove that Treewidth and Pathwidth do not admit polynomial kernels parameterized by the vertex-deletion distance to a clique, unless the polynomial hierarchy collapses. We analyze the kernelization complexity of graph coloring problems with respect to parameterizations that measure the vertex-deletion to graph classes such as cographs and co-chordal graphs. We show that the existence of polynomial kernels is determined by the extremal properties of No-instances of the List Coloring problem on such graph classes. Finally, we investigate Longest Path and related problems, with structural parameterizations. We obtain polynomial kernels for parameterizations by the vertex cover number, the max leaf number, and the vertex-deletion distance to a cluster graph. These results are complemented by a lower bound for the parameterization by the deletion distance to an outerplanar grap

On structural parameterizations of hitting set : hitting paths in graphs using 2-SAT

Hitting Set is a classic problem in combinatorial optimization. Its input consists of a set system F over a finite universe U and an integer t; the question is whether there is a set of t elements that intersects every set in F. The Hitting Set problem parameterized by the size of the solution is a well-known W[2]-complete problem in parameterized complexity theory. In this paper we investigate the complexity of Hitting Set under various structural parameterizations of the input. Our starting point is the folklore result that Hitting Set is polynomial-time solvable if there is a tree T on vertex set U such that the sets in F induce connected subtrees of T. We consider the case that there is a treelike graph with vertex set U such that the sets in F induce connected subgraphs; the parameter of the problem is a measure of how treelike the graph is. Our main positive result is an algorithm that, given a graph G with cyclomatic number k, a collection P of simple paths in G, and an integer t, determines in time 2 5k(|G| + |P|) O(1)whether there is a vertex set of size t that hits all paths in P. It is based on a connection to the 2-SAT problem in multiple valued logic. For other parameterizations we derive W[1]-hardness and para-NP-completeness results

On sparsification for computing treewidth

We investigate whether an n -vertex instance (G,k) of Treewidth, asking whether the graph G has treewidth at most k , can efficiently be made sparse without changing its answer. By giving a special form of OR -cross-composition, we prove that this is unlikely: if there is an ¿>0 and a polynomial-time algorithm that reduces n -vertex Treewidth instances to equivalent instances, of an arbitrary problem, with O(n2-¿) bits, then NP ¿ coNP / poly and the polynomial hierarchy collapses to its third level. Our sparsification lower bound has implications for structural parameterizations of Treewidth: parameterizations by measures l that do not exceed the number of vertices cannot have kernels with O(l2-¿) bits for any ¿>0 , unless NP ¿ coNP / poly. Motivated by the question of determining the optimal kernel size for Treewidth parameterized by the size of a vertex cover X , we improve the O(|X|3) -vertex kernel from Bodlaender et al. (SIDMA 2013) to a kernel with O(|X|2) vertices. Our improved kernel is based on the novel notion of treewidth-invariant set. We use the q -expansion lemma of Fomin et al. (STACS 2011) to find such sets efficiently in graphs whose order is superquadratic in their vertex cover number. We believe that our new reduction rule will be useful in practice

Fine-grained parameterized complexity analysis of graph coloring problems

The q-Coloring problem asks whether the vertices of a graph can be properly colored with q colors. Lokshtanov et al. [SODA 2011] showed that q-Coloring on graphs with a feedback vertex set of size k cannot be solved in time O ∗ ((q−ε) k ) O∗((q−ε)k) , for any ε&gt;0 ε&gt;0 , unless the Strong Exponential-Time Hypothesis (SETH SETH ) fails. In this paper we perform a fine-grained analysis of the complexity of q-Coloring with respect to a hierarchy of parameters. We show that unless ETH ETH fails, there is no universal constant θ θ such that q-Coloring parameterized by vertex cover can be solved in time O ∗ (θ k ) O∗(θk) for all fixed q. We prove that there are O ∗ ((q−ε) k ) O∗((q−ε)k) time algorithms where k is the vertex deletion distance to several graph classes F F for which q-Coloring is known to be solvable in polynomial time, including all graph classes whose (q+1) (q+1) -colorable members have bounded treedepth. In contrast, we prove that if F F is the class of paths – some of the simplest graphs of unbounded treedepth – then no such algorithm can exist unless SETH SETH fails. This research was partially funded by the Networks programme via the Dutch Ministry of Education, Culture and Science through the Netherlands Organisation for Scientific Research. The research was done while the first author was at CWI, Amsterdam. The second author was supported by NWO Veni grant “Frontiers in Parameterized Preprocessing”