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

Reconfiguration on sparse graphs

A vertex-subset graph problem Q defines which subsets of the vertices of an input graph are feasible solutions. A reconfiguration variant of a vertex-subset problem asks, given two feasible solutions S and T of size k, whether it is possible to transform S into T by a sequence of vertex additions and deletions such that each intermediate set is also a feasible solution of size bounded by k. We study reconfiguration variants of two classical vertex-subset problems, namely Independent Set and Dominating Set. We denote the former by ISR and the latter by DSR. Both ISR and DSR are PSPACE-complete on graphs of bounded bandwidth and W[1]-hard parameterized by k on general graphs. We show that ISR is fixed-parameter tractable parameterized by k when the input graph is of bounded degeneracy or nowhere-dense. As a corollary, we answer positively an open question concerning the parameterized complexity of the problem on graphs of bounded treewidth. Moreover, our techniques generalize recent results showing that ISR is fixed-parameter tractable on planar graphs and graphs of bounded degree. For DSR, we show the problem fixed-parameter tractable parameterized by k when the input graph does not contain large bicliques, a class of graphs which includes graphs of bounded degeneracy and nowhere-dense graphs

Parameterized Algorithmics for Computational Social Choice: Nine Research Challenges

Computational Social Choice is an interdisciplinary research area involving Economics, Political Science, and Social Science on the one side, and Mathematics and Computer Science (including Artificial Intelligence and Multiagent Systems) on the other side. Typical computational problems studied in this field include the vulnerability of voting procedures against attacks, or preference aggregation in multi-agent systems. Parameterized Algorithmics is a subfield of Theoretical Computer Science seeking to exploit meaningful problem-specific parameters in order to identify tractable special cases of in general computationally hard problems. In this paper, we propose nine of our favorite research challenges concerning the parameterized complexity of problems appearing in this context

Betting and Belief: Prediction Markets and Attribution of Climate Change

Despite much scientific evidence, a large fraction of the American public doubts that greenhouse gases are causing global warming. We present a simulation model as a computational test-bed for climate prediction markets. Traders adapt their beliefs about future temperatures based on the profits of other traders in their social network. We simulate two alternative climate futures, in which global temperatures are primarily driven either by carbon dioxide or by solar irradiance. These represent, respectively, the scientific consensus and a hypothesis advanced by prominent skeptics. We conduct sensitivity analyses to determine how a variety of factors describing both the market and the physical climate may affect traders' beliefs about the cause of global climate change. Market participation causes most traders to converge quickly toward believing the "true" climate model, suggesting that a climate market could be useful for building public consensus.Comment: All code and data for the model is available at http://johnjnay.com/predMarket/. Forthcoming in Proceedings of the 2016 Winter Simulation Conference. IEEE Pres

Separating sets of strings by finding matching patterns is almost always hard

© 2017 Elsevier B.V. We study the complexity of the problem of searching for a set of patterns that separate two given sets of strings. This problem has applications in a wide variety of areas, most notably in data mining, computational biology, and in understanding the complexity of genetic algorithms. We show that the basic problem of finding a small set of patterns that match one set of strings but do not match any string in a second set is difficult (NP-complete, W[2]-hard when parameterized by the size of the pattern set, and APX-hard). We then perform a detailed parameterized analysis of the problem, separating tractable and intractable variants. In particular we show that parameterizing by the size of pattern set and the number of strings, and the size of the alphabet and the number of strings give FPT results, amongst others

Polynomial Kernels for Generalized Domination Problems

In this paper, we study the parameterized complexity of a generalized domination problem called the [${\sigma}, {\rho}$] Dominating Set problem. This problem generalizes a large number of problems including the Minimum Dominating Set problem and its many variants. The parameterized complexity of the [${\sigma}, {\rho}$] Dominating Set problem parameterized by treewidth is well studied. Here the properties of the sets ${\sigma}$ and ${\rho}$ that make the problem tractable are identified [1]. We consider a larger parameter and investigate the existence of polynomial sized kernels. When ${\sigma}$ and ${\rho}$ are finite, we identify the exact condition when the [${\sigma}, {\rho}$] Dominating Set problem parameterized by vertex cover admits polynomial kernels. Our lower and upper bound results can also be extended to more general conditions and provably smaller parameters as well.Comment: 19 pages, 6 figure

Parameterized Complexity of Edge Interdiction Problems

We study the parameterized complexity of interdiction problems in graphs. For an optimization problem on graphs, one can formulate an interdiction problem as a game consisting of two players, namely, an interdictor and an evader, who compete on an objective with opposing interests. In edge interdiction problems, every edge of the input graph has an interdiction cost associated with it and the interdictor interdicts the graph by modifying the edges in the graph, and the number of such modifications is constrained by the interdictor's budget. The evader then solves the given optimization problem on the modified graph. The action of the interdictor must impede the evader as much as possible. We focus on edge interdiction problems related to minimum spanning tree, maximum matching and shortest paths. These problems arise in different real world scenarios. We derive several fixed-parameter tractability and W[1]-hardness results for these interdiction problems with respect to various parameters. Next, we show close relation between interdiction problems and partial cover problems on bipartite graphs where the goal is not to cover all elements but to minimize/maximize the number of covered elements with specific number of sets. Hereby, we investigate the parameterized complexity of several partial cover problems on bipartite graphs