Dániel Marx

Lower bounds based on the Exponential Time Hypothesis

The Exponential Time Hypothesis (ETH) is a conjecture stating that, roughly speaking, n-variable 3-SAT cannot be solved in time 2o(n). In this chapter, we prove lower bounds based on ETH for the time needed to solve various problems. In many cases, these lower bounds match (up to small factors) the running time of the best known algorithms for the problem.

Parameterized Complexity and Approximation Algorithms

Approximation algorithms and parameterized complexity are usually considered to be two separate ways of dealing with hard algorithmic problems. In this paper, our aim is to investigate how these two fields can be combined to achieve better algorithms than what any of the two theories could offer. We discuss the different ways parameterized complexity can be extended to approximation algorithms, survey results of this type and propose directions for future research.

Parameterized graph separation problems

We consider parameterized problems where some separation property has to be achieved by deleting as few vertices as possible. The following five problems are studied: delete k vertices such that (a) each of the given l terminals is separated from the others, (b) each of the given l pairs of terminals is separated, (c) exactly l vertices are cut away from the graph, (d) exactly l connected vertices are cut away from the graph, (e) the graph is separated into at least l components. We show that if both k and l are parameters, then (a), (b) and (d) are fixed-parameter tractable, while (c) and (e) are W[1]-hard.

Constraint solving via fractional edge covers

Many important combinatorial problems can be modelled as constraint satisfaction problems, hence identifying polynomial-time solvable classes of constraint satisfaction problems received a lot of attention. In this paper, we are interested in structural properties that can make the problem tractable. So far, the largest structural class that is known to be polynomial-time solvable is the class of bounded hypertree width instances introduced by Gottlob et al. [20]. Here we identify a new class of polynomial-time solvable instances: those having bounded fractional edge cover number.Combining hypertree width and fractional edge cover number, we then introduce the notion of fractional hypertree width. We prove that constraint satisfaction problems with bounded fractional hypertree width can be solved in polynomial time (provided that a the tree decomposition is given in the input). We also prove that certain parameterized constraint satisfaction, homomorphism, and embedding problems are fixed-parameter tractable on instances having bounded fractional hypertree width.

Can You Beat Treewidth

It is well-known that constraint satisfaction problems (CSP) can be solved in time nO(k) if the treewidth of the primal graph of the instance is at most k and n is the size of the input. We show that no algorithm can be significantly better than this treewidth-based algorithm, even if we restrict the problem to some special class of primal graphs. Formally, let g be an arbitrary class of graphs and assume that there is an algorithm A solving binary CSP for instances whose primal graph is in g. We prove that if the running lime of A is f(G)nO(k/logk), where k is the treewidth of the primal graph G and f is an arbitrary function, then the Exponential Time Hypothesis fails. We prove the result also in the more general framework of the homomorphism problem for bounded-arity relational structures. For this problem, the treewidth of the core of the left-hand side structure plays the same role as the. treewidth of the primal graph above.

Fixed-parameter tractability of multicut parameterized by the size of the cutset

Given an undirected graph

Finding topological subgraphs is fixed-parameter tractable

G

On the Optimality of Planar and Geometric Approximation Schemes

, a collection {(s₁,t₁), ..., (s_l,t_l)} of pairs of vertices, and an integer p, the Edge Multicut problem ask if there is a set <i>S</i> of at most p edges such that the removal of S disconnects every s_i from the corresponding t_i. Vertex Multicut is the analogous problem where S is a set of at most p vertices. Our main result is that both problems can be solved in time 2^O(p³) ⋅ n^O(1), i.e., fixed-parameter tractable parameterized by the size p of the cutset in the solution. By contrast, it is unlikely that an algorithm with running time of the form f(p) ⋅ n^O(1) exists for the directed version of the problem, as we show it to be W[1]-hard parameterized by the size of the cutset.

Heuristic algorithms for joint configuration of the optical and electrical layer in multi-hop wavelength routing networks

We prove that for every fixed undirected graph <i>H</i>, there is an O(|V(G)|³) time algorithm that, given a graph <i>G</i>, tests if <i>G</i> contains <i>H</i> as a topological subgraph (that is, a subdivision of <i>H</i> is subgraph of <i>G</i>). This shows that topological subgraph testing is fixed-parameter tractable, resolving a longstanding open question of Downey and Fellows from 1992. As a corollary, for every <i>H</i> we obtain an O(|V(G)|³) time algorithm that tests if there is an immersion of <i>H</i> into a given graph <i>G</i>. This answers another open question raised by Downey and Fellows in 1992.

Closest Substring Problems with Small Distances

We show for several planar and geometric problems that the best known approximation schemes are essentially optimal with respect to the dependence on epsi. For example, we show that the 2^O(1/epsi)ldrn time approximation schemes for planar maximum independent set and for TSP on a metric defined bv a planar graph are essentially optimal: if there is a delta>0 such that any of these problems admits a 2^O((1/epsi) ^1-delta ⁾n^O(1) time PTAS, then the exponential tune hypothesis (ETH) fails. It is known that maximum independent set on unit disk graphs and the planar logic problems MPSAT. TMIN, TMAX admit n^O(1/epsi) time approximation schemes. We show that they are optimal in the sense that if there is a delta>0 such that any of these problems admits a 2^(1/epsi) ^O(1) n^O((1/epsi) ^1-delta ⁾ time PTAS, then ETH fails.