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In the k-Disjoint Shortest Paths (k-DSP) problem, we are given a graph G (with positive edge weights) on n nodes and m edges with specified source vertices s_1, … , s_k, and target vertices t_1, … , t_k, and are tasked with determining if G contains vertex-disjoint (s_i,t_i)-shortest paths. For any constant k, it is known that k-DSP can be solved in polynomial time over undirected graphs and directed acyclic graphs (DAGs). However, the exact time complexity of k-DSP remains mysterious, with large gaps between the fastest known algorithms and best conditional lower bounds. In this paper, we obtain faster algorithms for important cases of k-DSP, and present better conditional lower bounds for k-DSP and its variants. Previous work solved 2-DSP over weighted undirected graphs in O(n⁷) time, and weighted DAGs in O(mn) time. For the main result of this paper, we present optimal linear time algorithms for solving 2-DSP on weighted undirected graphs and DAGs. Our linear time algorithms are algebraic however, and so only solve the detection rather than search version of 2-DSP (we show how to solve the search version in O(mn) time, which is faster than the previous best runtime in weighted undirected graphs, but only matches the previous best runtime for DAGs). We also obtain a faster algorithm for k-Edge Disjoint Shortest Paths (k-EDSP) in DAGs, the variant of k-DSP where one seeks edge-disjoint instead of vertex-disjoint paths between sources and their corresponding targets. Algorithms for k-EDSP on DAGs from previous work take Ω(m^k) time. We show that k-EDSP can be solved over DAGs in O(mn^{k-1}) time, matching the fastest known runtime for solving k-DSP over DAGs. Previous work established conditional lower bounds for solving k-DSP and its variants via reductions from detecting cliques in graphs. Prior work implied that k-Clique can be reduced to 2k-DSP in DAGs and undirected graphs with O((kn)²) nodes. We improve this reduction, by showing how to reduce from k-Clique to k-DSP in DAGs and undirected graphs with O((kn)²) nodes (halving the number of paths needed in the reduced instance). A variant of k-DSP is the k-Disjoint Paths (k-DP) problem, where the solution paths no longer need to be shortest paths. Previous work reduced from k-Clique to p-DP in DAGs with O(kn) nodes, for p = k + k(k-1)/2. We improve this by showing a reduction from k-Clique to p-DP, for p = k + ⌊k²/4⌋. Under the k-Clique Hypothesis from fine-grained complexity, our results establish better conditional lower bounds for k-DSP for all k ≥ 4, and better conditional lower bounds for p-DP for all p ≤ 4031. Notably, our work gives the first nontrivial conditional lower bounds 4-DP in DAGs and 4-DSP in undirected graphs and DAGs. Before our work, nontrivial conditional lower bounds were only known for k-DP and k-DSP on such graphs when k ≥ 6. 

Abstract In a Merlin–Arthur proof system, the proof verifier (Arthur) accepts valid proofs (from Merlin) with probability 1, and rejects invalid proofs with probability arbitrarily close to 1. The running time of such a system is defined to be the length of Merlin’s proof plus the running time of Arthur. We provide new Merlin–Arthur proof systems for some key problems in fine-grained complexity. In several cases our proof systems have optimal running time. Our main results include:Certifying that a list ofnintegers has no 3-SUM solution can be done in Merlin–Arthur time$$\tilde{O}(n)$$ $\tilde{O}\left(n\right)$. Previously, Carmosino et al. [ITCS 2016] showed that the problem has a nondeterministic algorithm running in$$\tilde{O}(n^{1.5})$$ $\tilde{O}\left(n^{1.5}\right)$time (that is, there is a proof system with proofs of length$$\tilde{O}(n^{1.5})$$ $\tilde{O}\left(n^{1.5}\right)$and a deterministic verifier running in$$\tilde{O}(n^{1.5})$$ $\tilde{O}\left(n^{1.5}\right)$time).Counting the number ofk-cliques with total edge weight equal to zero in ann-node graph can be done in Merlin–Arthur time$${\tilde{O}}(n^{\lceil k/2\rceil })$$ $\tilde{O}\left(n^{\lceil k/2\rceil }\right)$(where$$k\ge 3$$ $k\ge 3$). For oddk, this bound can be further improved for sparse graphs: for example, counting the number of zero-weight triangles in anm-edge graph can be done in Merlin–Arthur time$${\tilde{O}}(m)$$ $\tilde{O}\left(m\right)$. Previous Merlin–Arthur protocols by Williams [CCC’16] and Björklund and Kaski [PODC’16] could only countk-cliques in unweighted graphs, and had worse running times for smallk.Computing the All-Pairs Shortest Distances matrix for ann-node graph can be done in Merlin–Arthur time$$\tilde{O}(n^2)$$ $\tilde{O}\left(n^2\right)$. Note this is optimal, as the matrix can have$$\Omega (n^2)$$ $\Omega \left(n^2\right)$nonzero entries in general. Previously, Carmosino et al. [ITCS 2016] showed that this problem has an$$\tilde{O}(n^{2.94})$$ $\tilde{O}\left(n^{2.94}\right)$nondeterministic time algorithm.Certifying that ann-variablek-CNF is unsatisfiable can be done in Merlin–Arthur time$$2^{n/2 - n/O(k)}$$ $2^{n/2-n/O\left(k\right)}$. We also observe an algebrization barrier for the previous$$2^{n/2}\cdot \textrm{poly}(n)$$ $2^{n/2}·\text{poly}\left(n\right)$-time Merlin–Arthur protocol of R. Williams [CCC’16] for$$\#$$ $\#$SAT: in particular, his protocol algebrizes, and we observe there is no algebrizing protocol fork-UNSAT running in$$2^{n/2}/n^{\omega (1)}$$ $2^{n/2}/n^{\omega \left(1\right)}$time. Therefore we have to exploit non-algebrizing properties to obtain our new protocol.Certifying a Quantified Boolean Formula is true can be done in Merlin–Arthur time$$2^{4n/5}\cdot \textrm{poly}(n)$$ $2^{4n/5}·\text{poly}\left(n\right)$. Previously, the only nontrivial result known along these lines was an Arthur–Merlin–Arthur protocol (where Merlin’s proof depends on some of Arthur’s coins) running in$$2^{2n/3}\cdot \textrm{poly}(n)$$ $2^{2n/3}·\text{poly}\left(n\right)$time.Due to the centrality of these problems in fine-grained complexity, our results have consequences for many other problems of interest. For example, our work implies that certifying there is no Subset Sum solution tonintegers can be done in Merlin–Arthur time$$2^{n/3}\cdot \textrm{poly}(n)$$ $2^{n/3}·\text{poly}\left(n\right)$, improving on the previous best protocol by Nederlof [IPL 2017] which took$$2^{0.49991n}\cdot \textrm{poly}(n)$$ $2^{0.49991n}·\text{poly}\left(n\right)$time. 

The k-Detour problem is a basic path-finding problem: given a graph G on n vertices, with specified nodes s and t, and a positive integer k, the goal is to determine if G has an st-path of length exactly dist(s,t) + k, where dist(s,t) is the length of a shortest path from s to t. The k-Detour problem is NP-hard when k is part of the input, so researchers have sought efficient parameterized algorithms for this task, running in f(k)poly(n) time, for f(⋅) as slow-growing as possible. We present faster algorithms for k-Detour in undirected graphs, running in 1.853^k poly(n) randomized and 4.082^kpoly(n) deterministic time. The previous fastest algorithms for this problem took 2.746^k poly(n) randomized and 6.523^k poly(n) deterministic time [Bezáková-Curticapean-Dell-Fomin, ICALP 2017]. Our algorithms use the fact that detecting a path of a given length in an undirected graph is easier if we are promised that the path belongs to what we call a "bipartitioned" subgraph, where the nodes are split into two parts and the path must satisfy constraints on those parts. Previously, this idea was used to obtain the fastest known algorithm for finding paths of length k in undirected graphs [Björklund-Husfeldt-Kaski-Koivisto, JCSS 2017], intuitively by looking for paths of length k in randomly bipartitioned subgraphs. Our algorithms for k-Detour stem from a new application of this idea, which does not involve choosing the bipartitioned subgraphs randomly. Our work has direct implications for the k-Longest Detour problem, another related path-finding problem. In this problem, we are given the same input as in k-Detour, but are now tasked with determining if G has an st-path of length at least dist(s,t)+k. Our results for k-Detour imply that we can solve k-Longest Detour in 3.432^k poly(n) randomized and 16.661^k poly(n) deterministic time. The previous fastest algorithms for this problem took 7.539^k poly(n) randomized and 42.549^k poly(n) deterministic time [Fomin et al., STACS 2022]. 

Controllability for Simple Temporal Networks with Uncertainty (STNUs) has thus far been limited to three levels: strong, dynamic, and weak. Because of this, there is currently no systematic way for an agent to assess just how far from being controllable an uncontrollable STNU is. We use a new geometric interpretation of STNUs to introduce the degrees of strong and dynamic controllability – continuous metrics that measure how far a network is from being controllable. We utilize these metrics to approximate the probabilities that an STNU can be dispatched successfully offline and online respectively. We introduce new methods for predicting the degrees of strong and dynamic controllability for uncontrollable networks. In addition, we show empirically that both metrics are good predictors of the actual dispatch success rate.