We prove that no randomized integral or fractional algorithm for online vertex cover under general vertex arrivals achieves a competitive ratio strictly below $1+\sqrt{e}/2\approx1.824360635$, even on bipartite graphs and against an oblivious adversary. This improves the previous lower bound of approximately $1.753$. Our proof extends the complete-bipartite alternating construction of Wang and Wong to an arbitrary number of alternations. The resulting adversary is described by a monotone integral recurrence. If the recurrence never violates the competitive budget, its iterates converge to an integrable fixed point; classifying all such fixed points forces the excess ratio to be at least $\sqrt{e}/2$. A truncated discrete recurrence and a Riemann-sum argument convert every strict continuous violation into a finite, algorithm-dependent but realization-oblivious input. We also exhibit a critical fixed point showing that $1+\sqrt{e}/2$ is the exact limit of this homogeneous complete-bipartite recurrence, rather than a numerical artifact.
We prove a tight impossibility result for online vertex cover under edge arrivals. No randomized integral or fractional algorithm achieves a competitive ratio strictly below $2$ against an oblivious adversary, even on bipartite graphs. Since the standard algorithm that takes both endpoints of every uncovered edge is $2...
For an $n\times n$ nonnegative matrix $A$, the Bethe permanent, which is computable in deterministic polynomial time, satisfies the tight universal comparison \[\operatorname{Bethe}(A) \leq \operatorname{per}(A) \leq 2^{n/2}\operatorname{Bethe}(A).\] The lower bound, due to Gurvits, is attained on forests. The upper bo...
We study the weighted feedback vertex set problem in tournaments. For every fixed integer $k\geq 2$, we give a deterministic $(2+1/k)$-approximation algorithm with running time $n^{2^{O(k)}}$, apart from polynomial dependence on the encoding length of the weights. Consequently, for every fixed $\varepsilon>0$, weighted...
This paper studies fundamental graph optimization problems under differential privacy (DP) and shows new, reconstruction-based lower bounds. We consider a graph $G = (V, E, \vec{w})$ where the vertex set $V$ and edges $E$ are public and the weights $\mathbf{w}:E\rightarrow \mathbb{R}$ must be kept differentially privat...
Jacob Imola, Rasmus Pagh, Lukas Retschmeier· 1 citation
We study prophet inequalities under intersections of $q$ partition matroids, where an online algorithm irrevocably selects elements with independent nonnegative values drawn from known distributions and revealed in an adversarial order. We prove an $\Omega(q/\log q)$ lower bound on the competitive ratio. Together with...
We study one-sided online bipartite matching with recourse. In this setting, one side of a bipartite graph is known in advance, while vertices on the other side arrive online together with their incident edges. After each arrival, the algorithm must maintain a maximum-cardinality matching while minimizing the total num...
Júlia Baligács, B. Bosek, Paweł Putra et al.· 0 citations
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