Tight Localizations of Feedback Sets

The classical NP-hard feedback arc set problem (FASP) and feedback vertex set problem (FVSP) ask for a minimum set of arcs $\varepsilon \subseteq E$ or vertices $\nu \subseteq V$ whose removal $G\setminus \varepsilon$, $G\setminus \nu$ makes a given multi-digraph $G=(V,E)$ acyclic, respectively. Though both problems are known to be APX-hard, approximation algorithms or proofs of inapproximability are unknown. We propose a new $\mathcal{O}(|V||E|^4)$-heuristic for the directed FASP. While a ratio of $r \approx 1.3606$ is known to be a lower bound for the APX-hardness, at least by empirical validation we achieve an approximation of $r \leq 2$. The most relevant applications, such as circuit testing, ask for solving the FASP on large sparse graphs, which can be done efficiently within tight error bounds due to our approach.Comment: manuscript submitted to AC

Hardness of Vertex Deletion and Project Scheduling

Assuming the Unique Games Conjecture, we show strong inapproximability results for two natural vertex deletion problems on directed graphs: for any integer $k\geq 2$ and arbitrary small $\epsilon > 0$, the Feedback Vertex Set problem and the DAG Vertex Deletion problem are inapproximable within a factor $k-\epsilon$ even on graphs where the vertices can be almost partitioned into $k$ solutions. This gives a more structured and therefore stronger UGC-based hardness result for the Feedback Vertex Set problem that is also simpler (albeit using the "It Ain't Over Till It's Over" theorem) than the previous hardness result. In comparison to the classical Feedback Vertex Set problem, the DAG Vertex Deletion problem has received little attention and, although we think it is a natural and interesting problem, the main motivation for our inapproximability result stems from its relationship with the classical Discrete Time-Cost Tradeoff Problem. More specifically, our results imply that the deadline version is NP-hard to approximate within any constant assuming the Unique Games Conjecture. This explains the difficulty in obtaining good approximation algorithms for that problem and further motivates previous alternative approaches such as bicriteria approximations.Comment: 18 pages, 1 figur

Exact Localisations of Feedback Sets

The feedback arc (vertex) set problem, shortened FASP (FVSP), is to transform a given multi digraph $G=(V,E)$ into an acyclic graph by deleting as few arcs (vertices) as possible. Due to the results of Richard M. Karp in 1972 it is one of the classic NP-complete problems. An important contribution of this paper is that the subgraphs $G_{\mathrm{el}}(e)$, $G_{\mathrm{si}}(e)$ of all elementary cycles or simple cycles running through some arc $e \in E$, can be computed in $\mathcal{O}\big(|E|^2\big)$ and $\mathcal{O}(|E|^4)$, respectively. We use this fact and introduce the notion of the essential minor and isolated cycles, which yield a priori problem size reductions and in the special case of so called resolvable graphs an exact solution in $\mathcal{O}(|V||E|^3)$. We show that weighted versions of the FASP and FVSP possess a Bellman decomposition, which yields exact solutions using a dynamic programming technique in times $\mathcal{O}\big(2^{m}|E|^4\log(|V|)\big)$ and $\mathcal{O}\big(2^{n}\Delta(G)^4|V|^4\log(|E|)\big)$, where $m \leq |E|-|V| +1$, $n \leq (\Delta(G)-1)|V|-|E| +1$, respectively. The parameters $m,n$ can be computed in $\mathcal{O}(|E|^3)$, $\mathcal{O}(\Delta(G)^3|V|^3)$, respectively and denote the maximal dimension of the cycle space of all appearing meta graphs, decoding the intersection behavior of the cycles. Consequently, $m,n$ equal zero if all meta graphs are trees. Moreover, we deliver several heuristics and discuss how to control their variation from the optimum. Summarizing, the presented results allow us to suggest a strategy for an implementation of a fast and accurate FASP/FVSP-SOLVER

Multi-Budgeted Directed Cuts

In this paper, we study multi-budgeted variants of the classic minimum cut problem and graph separation problems that turned out to be important in parameterized complexity: Skew Multicut and Directed Feedback Arc Set. In our generalization, we assign colors 1,2,...,l to some edges and give separate budgets k_1,k_2,...,k_l for colors 1,2,...,l. For every color i in {1,...,l}, let E_i be the set of edges of color i. The solution C for the multi-budgeted variant of a graph separation problem not only needs to satisfy the usual separation requirements (i.e., be a cut, a skew multicut, or a directed feedback arc set, respectively), but also needs to satisfy that |C cap E_i| <= k_i for every i in {1,...,l}. Contrary to the classic minimum cut problem, the multi-budgeted variant turns out to be NP-hard even for l = 2. We propose FPT algorithms parameterized by k=k_1 +...+ k_l for all three problems. To this end, we develop a branching procedure for the multi-budgeted minimum cut problem that measures the progress of the algorithm not by reducing k as usual, by but elevating the capacity of some edges and thus increasing the size of maximum source-to-sink flow. Using the fact that a similar strategy is used to enumerate all important separators of a given size, we merge this process with the flow-guided branching and show an FPT bound on the number of (appropriately defined) important multi-budgeted separators. This allows us to extend our algorithm to the Skew Multicut and Directed Feedback Arc Set problems. Furthermore, we show connections of the multi-budgeted variants with weighted variants of the directed cut problems and the Chain l-SAT problem, whose parameterized complexity remains an open problem. We show that these problems admit a bounded-in-parameter number of "maximally pushed" solutions (in a similar spirit as important separators are maximally pushed), giving somewhat weak evidence towards their tractability

A Tutorial on Clique Problems in Communications and Signal Processing

Since its first use by Euler on the problem of the seven bridges of K\"onigsberg, graph theory has shown excellent abilities in solving and unveiling the properties of multiple discrete optimization problems. The study of the structure of some integer programs reveals equivalence with graph theory problems making a large body of the literature readily available for solving and characterizing the complexity of these problems. This tutorial presents a framework for utilizing a particular graph theory problem, known as the clique problem, for solving communications and signal processing problems. In particular, the paper aims to illustrate the structural properties of integer programs that can be formulated as clique problems through multiple examples in communications and signal processing. To that end, the first part of the tutorial provides various optimal and heuristic solutions for the maximum clique, maximum weight clique, and $k$-clique problems. The tutorial, further, illustrates the use of the clique formulation through numerous contemporary examples in communications and signal processing, mainly in maximum access for non-orthogonal multiple access networks, throughput maximization using index and instantly decodable network coding, collision-free radio frequency identification networks, and resource allocation in cloud-radio access networks. Finally, the tutorial sheds light on the recent advances of such applications, and provides technical insights on ways of dealing with mixed discrete-continuous optimization problems