arXiv subjects

Andreas Grigorjew

Publications and source records attributed to Andreas Grigorjew.

7 recordsLinked to original sources

arXiv2026

Forcing a unique minimum spanning tree and a unique shortest path

A forcing set $S$ in a combinatorial problem is a set of elements such that there is a unique solution that contains all the elements in $S$. An anti-forcing set is the symmetric concept: a set $S$ of elements is called an anti-forcing set if there is a unique solution disjoint from $S$. There are extensive studies on the computational complexity of finding a minimum forcing set in various combinatorial problems, and the known results indicate that many problems are harder than their classical counterparts: the decision version of finding a minimum forcing set for perfect matchings is NP-complete [Adams et al., Discrete Mathematics, 2004], and that of finding a minimum forcing set for satisfying assignments for 3CNF formulas is $Σ_2^P$-complete [Hatami-Maserrat, Discrete Applied Mathematics, 2005]. In this paper, we investigate the complexity of finding minimum forcing and anti-forcing sets for the shortest $s$-$t$ path problem and the minimum-weight spanning tree problem. We show that, unlike the aforementioned results, these problems are tractable, with the exception of the decision version of finding a minimum anti-forcing set for shortest $s$-$t$ paths, which is NP-complete. To complement this intractability result, we design fixed-parameter tractable algorithms for finding a minimum anti-forcing set for shortest $s$-$t$ paths.

cs.DS↗

arXiv2026

Maximum Coverage $k$-Antichains and Chains: A Greedy Approach

Given an acyclic digraph $G = (V,E)$ and a positive integer $k$, the problem of Maximum Coverage $k$-Antichains (resp. Chains) denoted as MA-$k$ (resp. MC-$k$) asks to find $k$ sets of pairwise unreachable vertices, known as antichains (resp. $k$ subsequences of paths, known as chains), maximizing the number $α_k$ (resp. $β_k$) of vertices covered by these antichains (resp. chains). While MC-$k$ was solved in almost optimal $|E|^{1+o(1)}$ time~[Kogan and Parter, ICALP'22], the fastest algorithms for MA-$k$ are a $(k|E|)^{1+o(1)}$-time solution and a $|E|^{1+o(1)}$-time $1/2$ approximation~[Kogan and Parter, ESA'24]. We obtain the following for MA-$k$: - An algorithm running in $|E|^{1+o(1)}$ time, and an algorithm running in parameterized near-linear $\tilde{O}(α_k |E|)$ time. Our algorithms are simple solutions exploiting a paths-based proof of the Greene-Kleitman theorems leveraged by the greedy algorithm for set cover as well as recent advances in fast algorithms for flows and shortest paths. - An approximation algorithm running in parameterized linear time $O(α_1^2|V| + (α_1+k)|E|)$ with approximation ratio of $(1-1/e) > 0.63 > 1/2$, beating the state-of-the-art $1/2$ approximation. Our solution uses greedy for antichains and a simple strategy to amortize the cost of computing consecutive maximum antichains. We complement these results with two examples (one for chains and one for antichains) showing that, for every $k \ge 2$, greedy misses the tight $1/e$ portion of the optimal coverage for chains, and a $1/4$ portion for antichains. We also show that greedy is a $Ω(\log{|V|})$ factor away from minimality when required to cover all vertices: previously unknown for sets of chains or antichains.

cs.DS↗

arXiv2026

d-QBF with Few Existential Variables Revisited

Quantified Boolean Formula (QBF) is a notoriously hard generalization of \textsc{SAT}, especially from the point of view of parameterized complexity, where the problem remains intractable for most standard parameters. A recent work by Eriksson et al.~[IJCAI 24] addressed this by considering the case where the propositional part of the formula is in CNF and we parameterize by the number $k$ of existentially quantified variables. One of their main results was that this natural (but so far overlooked) parameter does lead to fixed-parameter tractability, if we also bound the maximum arity $d$ of the clauses of the given CNF. Unfortunately, their algorithm has a \emph{double-exponential} dependence on $k$ ($2^{2^k}$), even when $d$ is an absolute constant. Since the work of Eriksson et al.\ only complemented this with a SETH-based lower bound implying that a $2^{O(k)}$ dependence is impossible, this left a large gap as an open question. Our main result in this paper is to close this gap by showing that the double-exponential dependence is optimal, assuming the ETH: even for CNFs of arity $4$, QBF with $k$ existential variables cannot be solved in time $2^{2^{o(k)}}|ϕ|^{O(1)}$. Complementing this, we also consider the further restricted case of QBF with only two quantifier blocks ($\forall\exists$-QBF). We show that in this case the situation improves dramatically: for each $d\ge 3$ we show an algorithm with running time $k^{O_d(k ^{d-1})}|ϕ|^{O(1)}$ and a lower bound under the ETH showing our algorithm is almost optimal.

cs.CC↗

arXiv2026

EMERALD-UI: An interactive web application to unveil novel protein biology hidden in the suboptimal-alignment space

Life over the past four billion years has been shaped by proteins and their capacity to assemble into three dimensional conformations. Protein sequence alignments have been the enabling technology for exploring the evolution and functional adaptation of proteins across the tree of life. Recent advancements in scaling the prediction of three dimensional protein structures from primary sequence alone, revealed that different modes of conservation and function operate on the sequence and structure level. This difference in protein conservation patterns and their underlying functional change that could emerge in suboptimal alignment configurations is often ignored in optimal protein alignment approaches. We introduce EMERALD-UI, an open-source interactive web application which is designed to reveal unexplored biology by visualising stable structural conformations or protein regions hidden in the suboptimal alignment space. Availability: EMERALD-UI is available at https://algbio.github.io/emerald-ui/. Contact: hdrost001@dundee.ac.uk or alexandru.tomescu@helsinki.fi.

q-bio.QM↗

arXiv2025

Width Parameters for Minimum Flow Decomposition

Minimum flow decomposition (MFD) is the strongly NP-hard problem of finding a smallest set of integer weighted $s$-$t$ paths in an $s$-$t$ DAG $G$ whose weighted sum is equal to a given flow $f$ on $G$. Despite its many practical applications, we lack an understanding of graph structures that make MFD easy or hard. Recent progress is due to Cáceres et al. [ACM TALG 2024], who showed that the DAG width, the minimum number of paths to cover all edges, plays an essential role in the approximation of the problem. Our first set of results regard the computational complexity of MFD parameterised by the width. This question was previously open, because MFD on width-1 DAGs (paths) is trivially solvable, and the existing NP-hardness proofs use DAGs of unbounded width. We show that MFD on width-2 DAGs is already NP-hard and that MFD on width-3 DAGs is strongly NP-hard. Our main contribution complements these hardness bounds, as we show that weak NP-hardness is the best we can hope for on width-2 DAGs. In fact, we prove the more general statement that MFD with unary coded input can be solved in quasi-polynomial time on DAGs of constant parallel-width, which includes width-2 DAGs. The parallel-width of a DAG $G$ (par-width$(G)$) was defined by Deligkas and Meir [MFCS 2017] as the size of the largest minimal $s$-$t$ cut-set. We obtain these results by, a) interpreting flow decompositions as a sequence of certain digraph minor operations defined by Deligkas and Meir [MFCS 2017], and b) defining a new notion of width of a flow network, flow-width of $(G,f)$, defined as the minimum number of paths covering all edges of $G$, where every edge $e$ can be covered by at most $f(e)$ paths. Using (a) and (b), we show as an intermediate result, an improved upper bound $(\lfloor\log \Vert f\Vert\rfloor+1) \cdot \text{par-width}(G)$ for MFD, where $\Vert f\Vert$ is the largest flow weight of all edges.

cs.DS↗

arXiv2023

Accelerating ILP solvers for Minimum Flow Decompositions through search space and dimensionality reductions

Given a flow network, the Minimum Flow Decomposition (MFD) problem is finding the smallest possible set of weighted paths whose superposition equals the flow. It is a classical, strongly NP-hard problem that is proven to be useful in RNA transcript assembly and applications outside of Bioinformatics. We improve an existing ILP (Integer Linear Programming) model by Dias et al. [RECOMB 2022] for DAGs by decreasing the solver's search space using solution safety and several other optimizations. This results in a significant speedup compared to the original ILP, of up to 55-90x on average on the hardest instances. Moreover, we show that our optimizations apply also to MFD problem variants, resulting in similar speedups, going up to 123x on the hardest instances. We also developed an ILP model of reduced dimensionality for an MFD variant in which the solution path weights are restricted to a given set. This model can find an optimal MFD solution for most instances, and overall, its accuracy significantly outperforms that of previous greedy algorithms while being up to an order of magnitude faster than our optimized ILP.

q-bio.GN↗

arXiv2023

Width Helps and Hinders Splitting Flows

Minimum flow decomposition (MFD) is the NP-hard problem of finding a smallest decomposition of a network flow/circulation $X$ on a directed graph $G$ into weighted source-to-sink paths whose superposition equals $X$. We show that, for acyclic graphs, considering the \emph{width} of the graph (the minimum number of paths needed to cover all of its edges) yields advances in our understanding of its approximability. For the version of the problem that uses only non-negative weights, we identify and characterise a new class of \emph{width-stable} graphs, for which a popular heuristic is a \gwsimple-approximation ($|X|$ being the total flow of $X$), and strengthen its worst-case approximation ratio from $Ω(\sqrt{m})$ to $Ω(m / \log m)$ for sparse graphs, where $m$ is the number of edges in the graph. We also study a new problem on graphs with cycles, Minimum Cost Circulation Decomposition (MCCD), and show that it generalises MFD through a simple reduction. For the version allowing also negative weights, we give a $(\lceil \log \Vert X \Vert \rceil +1)$-approximation ($\Vert X \Vert$ being the maximum absolute value of $X$ on any edge) using a power-of-two approach, combined with parity fixing arguments and a decomposition of unitary circulations ($\Vert X \Vert \leq 1$), using a generalised notion of width for this problem. Finally, we disprove a conjecture about the linear independence of minimum (non-negative) flow decompositions posed by Kloster et al. [ALENEX 2018], but show that its useful implication (polynomial-time assignments of weights to a given set of paths to decompose a flow) holds for the negative version.

cs.DS↗