Abstract. We study the Generalized Min Sum Set Cover (GMSSC) problem, wherein given a collection of hyperedges [Formula: see text] with arbitrary covering requirements [Formula: see text], the goal is to find an ordering of the vertices to minimize the total cover time of the hyperedges; a hyperedge [Formula: see text] is considered covered by the first time when [Formula: see text] and many of its vertices appear in the ordering. We give a [Formula: see text] approximation algorithm for GMSSC, coming close to the best possible bound of 4, already for the classical special case (with all [Formula: see text]) of Min Sum Set Cover (MSSC) studied by Feige, Lovász, and Tetali, and improving upon the previous best known bound of [Formula: see text] due to Im, Sviridenko, and van der Zwaan. Our algorithm is based on transforming the LP solution by a suitable kernel and applying randomized rounding. As part of the analysis of our algorithm, we also derive an inequality on the lower tail of a sum of independent Bernoulli random variables, which might be of independent interest and broader utility. Min Sum Vertex Cover (MSVC) is a well-known special case of MSSC in which the input hypergraph is a graph (i.e., [Formula: see text]) and [Formula: see text] for every edge [Formula: see text]. We give a [Formula: see text] approximation for MSVC and show a matching integrality gap for the natural LP relaxation. This improves upon the previous best [Formula: see text] approximation of Barenholz, Feige, and Peleg. Finally, we revisit MSSC and consider the [Formula: see text] norm of cover-time of the hyperedges. Using a dual fitting argument, we show that the natural greedy algorithm achieves tight, up to NP-hardness, approximation guarantees of [Formula: see text] for all [Formula: see text], giving another proof of the result of Golovin, Gupta, Kumar, and Tangwongsan, and showing its tightness up to NP-hardness. For [Formula: see text], this gives yet another proof of the 4 approximation for MSSC.

We study the Quantum Approximate Optimization Algorithm ( QAOA ) in the context of the Max-Cut problem. Noisy quantum devices are only able to accurately execute QAOA at low circuit depths, while classically-challenging problem instances may call for a relatively high circuit-depth. This is due to the need to build correlations between reachable pairs of vertices in potentially large graphs [ 16 ]. To enhance the solving power of low-depth QAOA, we introduce a classical pre-processing step that initializes QAOA with a biased superposition of possible cuts in the graph, referred to as a warm-start . In particular, we initialize QAOA with a solution to a low-rank semidefinite programming relaxation of the Max-Cut problem. Our experimental results show that this variant of QAOA , called QAOA-warm , is able to outperform standard QAOA on lower circuit depths in solution quality and training time. While this improvement is partly due to the classical warm-start, we find strong evidence of further improvement using QAOA circuit at small depth. We provide experimental evidence of improved performance as well as theoretical properties of the proposed framework.

We study the generalized min sum set cover (GMSSC) problem, wherein given a collection of hyperedges E with arbitrary covering requirements {ke ∊ Z+ : e ∊ E}, the goal is to find an ordering of the vertices to minimize the total cover time of the hyperedges; a hyperedge e is considered covered by the first time when ke many of its vertices appear in the ordering. We give a 4.642 approximation algorithm for GMSSC, coming close to the best possible bound of 4, already for the classical special case (with all ke = 1) of min sum set cover (MSSC) studied by Feige, Lovász and Tetali [11], and improving upon the previous best known bound of 12.4 due to Im, Sviridenko and van der Zwaan [20]. Our algorithm is based on transforming the LP solution by a suitable kernel and applying randomized rounding. This also gives an LP-based 4 approximation for MSSC. As part of the analysis of our algorithm, we also derive an inequality on the lower tail of a sum of independent Bernoulli random variables, which might be of independent interest and broader utility. Another well-known special case is the min sum vertex cover (MSVC) problem, in which the input hypergraph is a graph (i.e., |e| = 2) and ke = 1, for every edge e ∊ E. We give a 16/9 ≃ 1.778 approximation for MSVC, and show a matching integrality gap for the natural LP relaxation. This improves upon the previous best 1.999946 approximation of Barenholz, Feige and Peleg [6]. (The claimed 1.79 approximation result of Iwata, Tetali and Tripathi [21] for the MSVC turned out have an unfortunate, seemingly unfixable, mistake in it.) Finally, we revisit MSSC and consider the ℓp norm of cover-time of the hyperedges. Using a dual fitting argument, we show that the natural greedy algorithm simultaneously achieves approximation guarantees of (p + 1)1+1/p, for all p ≥ 1, giving another proof of the result of Golovin, Gupta, Kumar and Tangwongsan [13], and showing its tightness up to NP-hardness. For p = 1, this gives yet another proof of the 4 approximation for MSSC.

This paper presents a low complexity while accurate Heart Rate (HR) estimation technique from signals captured by Photoplethysmographic (PPG) sensors worn on the wrist during intensive physical exercise. Wrist-type PPG signals experience severe Motion Artifacts (MA) that hinder efficient HR estimation especially during intensive physical exercises. To suppress the motion artifacts efficiently, simultaneous 3 dimensional acceleration signals are used as reference MAs. The proposed method achieves an Average Absolute Error (AAE) of 1.19 Beats Per Minute (BPM) on the 12 benchmark PPG recordings in which subjects run at speeds of up to 15 km/h. This method also achieves an AAE of 2.17 BPM on the whole benchmark database of 23 recordings that include both running and arm movement activities. This performance is comparable with state-of-the-art algorithms while at a significantly reduced computational cost which makes its standalone implementation on wearable devices feasible. The proposed algorithm achieves an average processing time of 32 milliseconds per input frames of length 8 seconds (2 channel PPG and 3D ACC signals) on a 3.2 GHz processor.

We study the problem of fair division of a heterogeneous resource among strategic players. Given a divisible heterogeneous cake, we wish to divide the cake among n players in a way that meets the following criteria: (I) every player(weakly) prefers his allocated cake to any other player’s share (such notion is known as envy-freeness), (II) the mechanism is strategy-proof (truthful), and (III) the number of cuts made on the cake is minimal. We provide methods, namely expansion process and expansion process with unlocking, for dividing the cake under different assumptions on the valuation functions of the players.