Efficiently listing bounded length st-paths

The problem of listing the $K$ shortest simple (loopless) $st$-paths in a graph has been studied since the early 1960s. For a non-negatively weighted graph with $n$ vertices and $m$ edges, the most efficient solution is an $O(K(mn + n^2 \log n))$ algorithm for directed graphs by Yen and Lawler [Management Science, 1971 and 1972], and an $O(K(m+n \log n))$ algorithm for the undirected version by Katoh et al. [Networks, 1982], both using $O(Kn + m)$ space. In this work, we consider a different parameterization for this problem: instead of bounding the number of $st$-paths output, we bound their length. For the bounded length parameterization, we propose new non-trivial algorithms matching the time complexity of the classic algorithms but using only $O(m+n)$ space. Moreover, we provide a unified framework such that the solutions to both parameterizations -- the classic $K$-shortest and the new length-bounded paths -- can be seen as two different traversals of a same tree, a Dijkstra-like and a DFS-like traversal, respectively.Comment: 12 pages, accepted to IWOCA 201

Efficient Enumeration of Induced Subtrees in a K-Degenerate Graph

In this paper, we address the problem of enumerating all induced subtrees in an input k-degenerate graph, where an induced subtree is an acyclic and connected induced subgraph. A graph G = (V, E) is a k-degenerate graph if for any its induced subgraph has a vertex whose degree is less than or equal to k, and many real-world graphs have small degeneracies, or very close to small degeneracies. Although, the studies are on subgraphs enumeration, such as trees, paths, and matchings, but the problem addresses the subgraph enumeration, such as enumeration of subgraphs that are trees. Their induced subgraph versions have not been studied well. One of few example is for chordless paths and cycles. Our motivation is to reduce the time complexity close to O(1) for each solution. This type of optimal algorithms are proposed many subgraph classes such as trees, and spanning trees. Induced subtrees are fundamental object thus it should be studied deeply and there possibly exist some efficient algorithms. Our algorithm utilizes nice properties of k-degeneracy to state an effective amortized analysis. As a result, the time complexity is reduced to O(k) time per induced subtree. The problem is solved in constant time for each in planar graphs, as a corollary

Linear Time Subgraph Counting, Graph Degeneracy, and the Chasm at Size Six

We consider the problem of counting all k-vertex subgraphs in an input graph, for any constant k. This problem (denoted SUB-CNT_k) has been studied extensively in both theory and practice. In a classic result, Chiba and Nishizeki (SICOMP 85) gave linear time algorithms for clique and 4-cycle counting for bounded degeneracy graphs. This is a rich class of sparse graphs that contains, for example, all minor-free families and preferential attachment graphs. The techniques from this result have inspired a number of recent practical algorithms for SUB-CNT_k. Towards a better understanding of the limits of these techniques, we ask: for what values of k can SUB_CNT_k be solved in linear time? We discover a chasm at k=6. Specifically, we prove that for k < 6, SUB_CNT_k can be solved in linear time. Assuming a standard conjecture in fine-grained complexity, we prove that for all k ? 6, SUB-CNT_k cannot be solved even in near-linear time

Badger: Complexity Analysis with Fuzzing and Symbolic Execution

Hybrid testing approaches that involve fuzz testing and symbolic execution have shown promising results in achieving high code coverage, uncovering subtle errors and vulnerabilities in a variety of software applications. In this paper we describe Badger - a new hybrid approach for complexity analysis, with the goal of discovering vulnerabilities which occur when the worst-case time or space complexity of an application is significantly higher than the average case. Badger uses fuzz testing to generate a diverse set of inputs that aim to increase not only coverage but also a resource-related cost associated with each path. Since fuzzing may fail to execute deep program paths due to its limited knowledge about the conditions that influence these paths, we complement the analysis with a symbolic execution, which is also customized to search for paths that increase the resource-related cost. Symbolic execution is particularly good at generating inputs that satisfy various program conditions but by itself suffers from path explosion. Therefore, Badger uses fuzzing and symbolic execution in tandem, to leverage their benefits and overcome their weaknesses. We implemented our approach for the analysis of Java programs, based on Kelinci and Symbolic PathFinder. We evaluated Badger on Java applications, showing that our approach is significantly faster in generating worst-case executions compared to fuzzing or symbolic execution on their own

The Connectivity of Boolean Satisfiability: Computational and Structural Dichotomies

Boolean satisfiability problems are an important benchmark for questions about complexity, algorithms, heuristics and threshold phenomena. Recent work on heuristics, and the satisfiability threshold has centered around the structure and connectivity of the solution space. Motivated by this work, we study structural and connectivity-related properties of the space of solutions of Boolean satisfiability problems and establish various dichotomies in Schaefer's framework. On the structural side, we obtain dichotomies for the kinds of subgraphs of the hypercube that can be induced by the solutions of Boolean formulas, as well as for the diameter of the connected components of the solution space. On the computational side, we establish dichotomy theorems for the complexity of the connectivity and st-connectivity questions for the graph of solutions of Boolean formulas. Our results assert that the intractable side of the computational dichotomies is PSPACE-complete, while the tractable side - which includes but is not limited to all problems with polynomial time algorithms for satisfiability - is in P for the st-connectivity question, and in coNP for the connectivity question. The diameter of components can be exponential for the PSPACE-complete cases, whereas in all other cases it is linear; thus, small diameter and tractability of the connectivity problems are remarkably aligned. The crux of our results is an expressibility theorem showing that in the tractable cases, the subgraphs induced by the solution space possess certain good structural properties, whereas in the intractable cases, the subgraphs can be arbitrary

Stronger 3-SUM Lower Bounds for Approximate Distance Oracles via Additive Combinatorics

The "short cycle removal" technique was recently introduced by Abboud, Bringmann, Khoury and Zamir (STOC '22) to prove fine-grained hardness of approximation. Its main technical result is that listing all triangles in an $n^{1/2}$-regular graph is $n^{2-o(1)}$-hard under the 3-SUM conjecture even when the number of short cycles is small; namely, when the number of $k$-cycles is $O(n^{k/2+\gamma})$ for $\gamma<1/2$. Abboud et al. achieve $\gamma\geq 1/4$ by applying structure vs. randomness arguments on graphs. In this paper, we take a step back and apply conceptually similar arguments on the numbers of the 3-SUM problem. Consequently, we achieve the best possible $\gamma=0$ and the following lower bounds under the 3-SUM conjecture: * Approximate distance oracles: The seminal Thorup-Zwick distance oracles achieve stretch $2k\pm O(1)$ after preprocessing a graph in $O(m n^{1/k})$ time. For the same stretch, and assuming the query time is $n^{o(1)}$ Abboud et al. proved an $\Omega(m^{1+\frac{1}{12.7552 \cdot k}})$ lower bound on the preprocessing time; we improve it to $\Omega(m^{1+\frac1{2k}})$ which is only a factor 2 away from the upper bound. We also obtain tight bounds for stretch $2+o(1)$ and $3-\epsilon$ and higher lower bounds for dynamic shortest paths. * Listing 4-cycles: Abboud et al. proved the first super-linear lower bound for listing all 4-cycles in a graph, ruling out $(m^{1.1927}+t)^{1+o(1)}$ time algorithms where $t$ is the number of 4-cycles. We settle the complexity of this basic problem by showing that the $\widetilde{O}(\min(m^{4/3},n^2) +t)$ upper bound is tight up to $n^{o(1)}$ factors. Our results exploit a rich tool set from additive combinatorics, most notably the Balog-Szemer\'edi-Gowers theorem and Rusza's covering lemma. A key ingredient that may be of independent interest is a subquadratic algorithm for 3-SUM if one of the sets has small doubling.Comment: Abstract shortened to fit arXiv requirement