Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 35 / 4

Represent a feasible allocation by a label for each item saying which bidder receives it. Every such label vector gives disjoint bundles covering all items, so the following heuristic optimization methods can search without violating allocation feasibility.

In local search, begin with any feasible partition and repeatedly make an improving move, such as transferring one item between bidders or exchanging two items. For a transfer $j:i\to r$, evaluate the changeStop when no allowed move improves the objective. Several starting partitions can avoid dependence on one local optimum, but no guarantee of a global optimum follows for general bids.

In simulated annealing, use the same feasible neighborhood but also accept a worsening move of loss $\Delta >0$ with probability $e^{-\Delta /T}$ at temperature $T$. Begin with a relatively high temperature, gradually cool, and retain the best allocation seen. Worsening moves can cross a barrier between local optima; a practical finite cooling schedule does not guarantee optimality.

In tabu search, retain a short memory of recent transfers or exchanges and temporarily prohibit their reversal. Select the best admissible move, possibly worsening the objective, to escape a local optimum. Permit an otherwise tabu move if it improves the best solution found, an aspiration rule. For the winner determination problem, bidder/item pairs or recent reverse transfers provide convenient tabu attributes. All three methods preserve the partition, but for arbitrary bids they are heuristics rather than algorithms with a proved constant approximation ratio.

The supplied hint interchanges its item and bidder indices in the first-value identity. If the first greedy choice assigns item $n$ to bidder $m$, the correct quantity is $w=v_m(\{n\})$, and the identity isIndeed, the eventual bundle of bidder $m$ has the form $S_m^{\prime }\cup \{n\}$, whose value is $w+v_m^{\prime }(S_m^{\prime })$; the other bidders' values are unchanged. Every later marginal contribution for bidder $m$ is exactly its marginal in the residual problem, and this is also true for every other bidder. Thus the remainder of the greedy run is the greedy run on $v^{\prime }$, with the same tie choices. The residual values are nonnegative and increasing; they remain submodular set functions as allowed in the question.

We prove the greedy half-approximation for submodular welfare by induction on the number of items. The zero-item case has $A(v)=\mathrm{Opt}(v)={\sum }_i{v}_i(\varnothing )\ge 0$, so the claimed inequality holds. This base case also handles the fact that the question does not explicitly normalize empty-bundle values to zero.

For the induction step, letbe the first greedy marginal. Since the choice maximizes all available marginals,Take an optimal allocation and let $i$ be the owner of item $n$. If $i\ne m$, remove that item from $i$'s bundle and give it to $m$. By submodularity, the loss to its old owner isMonotonicity makes the recipient's gain nonnegative. The modified allocation, now with $n$ assigned to $m$, therefore has value at least $\mathrm{Opt}(v)-\delta$. If $i=m$, no modification is necessary and this bound still holds. Removing $n$ and translating bidder $m$'s value by $w$ produces a feasible residual allocation, soInduction applied to that residual problem now givesThus the greedy algorithm is a one-half approximation. Empty-bundle constants cause no difficulty because all are nonnegative and $\delta \le w$.

At each of $n$ stages evaluate the marginal for at most $m$ bidders and $n$ remaining items. This requires $O(m{n}^2)$ value queries, and polynomial additional work with bundle bit-vectors and cached current values. Hence it is a polynomial time approximation algorithm in the value-oracle model, assuming polynomial time bid evaluations. If every bundle bid is explicitly listed, this query bound is polynomial in that larger explicit input as well. The exponentially many possible bundles do not require enumerating allocations.